Open AccessArticle

Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions

Shahid Khan

Nazar Khan

Aftab Hussain

Serkan Araci

^4,*

Bilal Khan

⁵

and

Hamed H. Al-Sulami

Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan

Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

⁴

Department of Economics, Faculty of Economics Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep TR-27410, Turkey

⁵

School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

Author to whom correspondence should be addressed.

Symmetry 2022, 14(4), 803; https://doi.org/10.3390/sym14040803

Submission received: 22 March 2022 / Revised: 7 April 2022 / Accepted: 11 April 2022 / Published: 12 April 2022

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

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Abstract

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in the open unit disk E is given. Certain properties of this subclass, such as its structural formula, necessary and sufficient conditions, coefficient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results, are investigated. Some new and known consequences of our main results as corollaries are also highlighted.

Keywords:

quantum (or q-) calculus; symmetric quantum (or q-) calculus; symmetric q-derivative; q-starlike functions; symmetric conic domains

MSC:

Primary 05A30, 30C45; Secondary 11B65, 47B38

1. Introduction and Definitions

Let

A

denote the class of all analytic functions f which are analytic in the open unit disk

E = {z : z \in C and |z| < 1}

and series expansion is

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n} .

(1)

The usage of normalize functions in geometric function theory is significant. They are used to map everything inside a unit circle, where everything is convergent. Using this concept, many different subclasses of analytic functions have been defined and studied.

Furthermore, let

S \subset A

represents the set of all univalent in E (see [1,2]). The classes of uniformly convex

(UCV)

and uniformly starlike

(UST)

functions were introduced by Goodman [3] and are defined analytically as:

f \in UCV ⟺ f \in A and |\frac{z f^{″} (z)}{f^{'} (z)}| < ℜ \{1 + \frac{z f^{″} (z)}{f^{'} (z)}\} z \in E

and

f \in UST ⟺ f \in A and |\frac{z f^{'} (z)}{f (z)} - 1| < ℜ \{\frac{z f^{'} (z)}{f (z)}\}, z \in E .

The classes of k-uniformly convex

(k - UCV)

and k-uniformly starlike

(k - UST)

functions were introduced by Kanas and Wisniowska in [4] and are defined analytically as:

f (z) \in k - UST ⟺ f (z) \in A and 1 > k |\frac{z f^{'} (z)}{f (z)} - 1| - ℜ \{\frac{z f^{'} (z)}{f (z)}\}, z \in E,

and

f (z) \in k - UCV ⟺ f (z) \in A and 1 > k |\frac{z f^{″} (z)}{f^{'} (z)}| - ℜ \{1 + \frac{z f^{^{″}} (z)}{f^{^{'}} (z)}\}, z \in E .

Note that

f (z) \in k - UCV ⟺ z f^{'} (z) \in k - ST .

The above-defined function classes were studied and investigated by a number of well-known mathematicians; see for details [1,2,3,4,5,6].

Two analytic functions

f

and g are subordinate to each other, (written as

f (z) ≺ g (z)

), if there exists a Schwarz function

s (z)

, which is analytic in E with

s (0) = 0 and | s (z) | < 1

such that

f (z) = g (s (z)) .

Additionally, if

g (z) \in S

in E, then we have (see [1,2])

f (z) ≺ g (z) ⟺ f (0) = g (0) and f (E) \subset g (E) .

The convolution of two analytic functions

f (z) = \sum_{n = 0}^{\infty} a_{n} z^{n} and g (z) = \sum_{n = 0}^{\infty} b_{n} z^{n}, (z \in E)

is defined as:

(f * g) (z) = \sum_{n = 0}^{\infty} a_{n} b_{n} z^{n} .

Let

P

represent the set of all of Carathéodory functions and every analytic function

p \in P

, having series of the form

p (z) = 1 + \sum_{n = 1}^{\infty} c_{n} z^{n}

and satisfying the condition

ℜ (p (z)) > 0, z \in E .

In terms of these analytic functions, many different subclasses of starlike and convex functions have been defined already.

We have discussed above that Kanas and Wisniowska [4] introduced the class of k-uniformly convex functions

(k - UCV)

and the corresponding class of k-starlike functions

(k - ST)

and then defined these classes subject to the conic domain

Ω_{k},

(

k \geq 0

) as follows:

Ω_{k} = \{u + i v : u > k \sqrt{{(u - 1)}^{2} + v^{2}}\},

(2)

Ω_{k} = \{w : ℜ w > k |w - 1|\} .

For

k = 0,

then (2) becomes the right half plane; for

0 < k < 1

, it becomes a hyperbola, for

k = 1

it becomes a parabola and for

k > 1,

it becomes an ellipse. The generalization of theses discussed by Shams et al. in [7] is subject to the conic domain

Ω_{k, γ}

(k \geq 0),

γ \in C \ {0}

, which is

Ω_{k, γ} = \{u + i v : u > k \sqrt{{(u - 1)}^{2} + v^{2}} + γ\},

Ω_{k, γ} = \{w : ℜ w > k |w - 1| + γ\} .

For this conic domain, the function

\tilde{p_{k, γ}}

(z)

plays the role of the extremal function for the conic domain

Ω_{k, γ}

\tilde{p_{k, γ}} (z) = \{\begin{matrix} \frac{1 + z}{1 - z} & for k = 0, \\ 1 + \frac{2 γ}{π^{2}} {(log \frac{1 + \sqrt{z}}{1 - \sqrt{z}})}^{2} & for k = 1, \\ 1 + \frac{2 γ}{1 - k^{2}} H_{1} (k, z) & for 0 < k < 1, \\ 1 + \frac{γ}{k^{2} - 1} H_{1} (k, z) + \frac{γ}{1 - k^{2}} & for k > 1, \end{matrix}

(3)

where

H_{1} (k, z) = {sinh}^{2} \{(\frac{2}{π} arccos k) arctan h \sqrt{z}\}

H_{2} (k, z) = sin (\frac{π}{2 K (i)} \int_{0}^{\frac{u (z)}{\sqrt{t}}} \frac{1}{\sqrt{1 - x^{2}} \sqrt{1 - {(i x)}^{2}}} d x)

and

i \in (0, 1)

k = cosh (\frac{π K^{^{'}} (i)}{4 K (i)}),

K (i)

is the first kind of Legendre’s complete elliptic integral. For details (see [4]). Indeed, from (3), we have

\tilde{p_{k, α}} (z) = 1 + Q_{1} z + Q_{2} z^{2} + . . .,

(4)

where

\begin{matrix} Q_{1} & = & \{\begin{matrix} \frac{2 γ {(\frac{2}{π} arccos k)}^{2}}{1 - k^{2}} & for 0 \leq k < 1, \\ \frac{8 γ}{π^{2}} & for k = 1, \\ H_{3} (k, z) & for k > 1, \end{matrix} \end{matrix}

(5)

\begin{matrix} Q_{2} & = & \{\begin{matrix} \frac{{(\frac{2}{π} arccos k)}^{2} + 2}{3} Q_{1} & for 0 \leq k < 1, \\ \frac{2}{3} Q_{1} & for k = 1, \\ H_{4} (k, z) Q_{1} & for k > 1, \end{matrix} \end{matrix}

(6)

and

\begin{matrix} H_{3} (k, z) & = & \frac{π^{2} γ}{4 (1 + t) \sqrt{t} K^{2} (t) (k^{2} - 1)}, \\ H_{4} (k, z) & = & \frac{4 K^{2} (t) (t^{2} + 6 t + 1) - π^{2}}{24 K^{2} (t) (1 + t) \sqrt{t}} . \end{matrix}

In the twentieth century the researchers have developed a great interest in the study of theory of q-calculus and its numerous applications in the fields of mathematics and physics. In defining the q-analogous of the derivative and integral operator and providing few of their applications, Jackson [8] was one of the pioneer researchers. It has several applications in number theory, combinatorics, orthogonal polynomials, fundamental hyper-geometric functions, and other fields of mathematics, including quantum mechanics, mechanics and relativity theory. Furthermore, quantum calculus has been proven to be a branch of the more comprehensive mathematical area of time scale calculus. For both discrete and continuous domains, time scales provide a coherent framework for studying dynamic equations. Many researchers provided useful applications for q-analysis in domains of mathematics; see [9,10,11,12,13,14,15,16,17,18]. Currently, operators of basic (or q-) calculus and fractional q-calculus and their applications have been elaborated by Srivastava in [19]. Considering the importance of q-operator calculus theory, researchers have intensively explored their applications in various fields; see [20,21,22,23,24,25].

In many areas of mathematics, the symmetric properties of functions play an important role in problem solving. Symmetry’s importance has been demonstrated in a variety of fields, including biology, chemistry, and psychology. Recently, in [26], Zhang et al. studied the application of a q-symmetric difference operator in harmonic univalent functions. Additionally, Khan et al. [27] defined the symmetric conic domain by using symmetric q-calculus operator theory and investigated q-starlike functions in this domain. Very recently, Khan et al. [28] defined and explored a new subclass of analytic and bi-univalent function involving certain q-Chebyshev polynomials by means of a symmetric q-operator. The symmetric q-calculus finds its applications in different fields, especially in quantum mechanics; see for example [29,30].

Here we present a few basic definitions, as well as a concept of q-calculus and symmetric q-calculus which will help us in further studies.

Definition 1

([31]). The q-number

{[t]}_{q}

for

q \in (0, 1)

is defined as:

{[t]}_{q} = \{\frac{1 - q^{t}}{1 - q}, (t \in C) .

For

(t = n \in N)

, then

{[t]}_{q} = \sum_{k = 0}^{n - 1} q^{k}

and the q-gamma function is defined by:

\frac{Γ_{q} (t + 1)}{Γ_{q} (t)} = {[t]}_{q}

and

Γ_{q} (1) = 1 .

Definition 2

([27]). For

n \in N,

the symmetric q-number is defined as:

{\tilde{[n]}}_{q} = \frac{q^{- n} - q^{n}}{q^{- 1} - q}, {\tilde{[0]}}_{q} = 0 .

Remark 1.

The symmetric q-number can not reduce to a q-number.

Definition 3

([27]). The q-factorial

{[n]}_{q}!

for

q \in (0, 1)

is defined as:

{[n]}_{q}! = \prod_{k = 1}^{n} {[k]}_{q}, (n \in N)

(7)

and

{[0]}_{q}! = 1 .

Definition 4

([27]). For any

n \in Z^{+} \cup \{0\},

the symmetric q-number shift factorial is defined as:

{\tilde{[n]}}_{q}! = \{\begin{matrix} {\tilde{[n]}}_{q} {\tilde{[n - 1]}}_{q} {\tilde{[n - 2]}}_{q} . . . {\tilde{[2]}}_{q} {\tilde{[1]}}_{q}, n \geq 1 \\ 1 n = 0 . \end{matrix}

Note that

lim_{q \to 1 -} {\tilde{[n]}}_{q}! = n! .

Definition 5

([32]). Let

f \in A

; then, the symmetric q-derivative operator is defined by:

{\tilde{(D)}}_{q} f (z) = \frac{1}{z} (\frac{f (q z) - f (q^{- 1} z)}{q - q^{- 1}}), z \in E .

(8)

We can write (8) as:

{\tilde{(D)}}_{q} f (z) = 1 + \sum_{n = 1}^{\infty} {\tilde{[n]}}_{q} a_{n} z^{n - 1} .

Note that

{\tilde{(D)}}_{q} z^{n} = {\tilde{[n]}}_{q} z^{n - 1}, {\tilde{(D)}}_{q} \{\sum_{n = 1}^{\infty} a_{n} z^{n}\} = \sum_{n = 1}^{\infty} {\tilde{[n]}}_{q} a_{n} z^{n - 1},

and

lim_{q \to 1 -} {\tilde{(D)}}_{q} f (z) = f^{'} (z) .

Definition 6

([27]). Let

f \in A

and

0 < q < 1;

then

f \in \tilde{S_{q}^{*}}

f (0) = f^{'} (0) = 1

and

|\frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} - \frac{1}{1 - \frac{q}{q^{- 1}}}| \leq \frac{1}{1 - \frac{q}{q^{- 1}}} .

(9)

Using the definition of subordination, we can write the condition in (9) as:

\frac{w {\tilde{(D)}}_{q} g (z)}{g (z)} ≺ \frac{1 + z}{1 - \frac{q}{q^{- 1}} z} .

Recently, several classes of analytic and univalent functions in different types of domains have investigated in [21,22,27,33,34,35,36,37,38,39,40,41,42,43]. For example, let

p (z)

be an analytic in E and

p (0) = 1 .

Then:

(i): The image domain of E under $p (z)$ lies in right half plane if $p (z) ≺ \frac{1 + z}{1 - z}$ (see [3]);
(ii): For $- 1 \leq B < A \leq 1,$ the image of E under $p (z)$ lies inside a circle centered on real axis if $p (z) ≺ \frac{1 + A z}{1 + B z}$ (see [44]);
(iii): For $0 \leq α < 1,$ in [4,34], Kanas showed that the image of E under $p (z)$ lies inside the conic domain $Ω_{k}$ and $Ω_{k, α}$ if

$p (z) ≺ p_{k, α} (z);$
(iv): For $γ \in C \ {0},$ in [7], Shams et al. showed that the image of U under $p (z)$ lies inside the conic domain $Ω_{k, γ}$ if

$p (z) ≺ p_{k, γ} (z) .$

By taking motivation from the above-cited works, we define the following domain:

Definition 7.

Let

k \in [0, \infty),

q \in (0, 1)

and

γ \in C \ {0} .

A function

p (z)

is said to be in the class k-

{\tilde{P}}_{q, q^{- 1}, γ} (z)

if and only if

p (z) ≺ \tilde{p_{k, q, q^{- 1}, γ}} (z)

(10)

where

\tilde{p_{k, q, q^{- 1}, γ}} = \frac{2 q^{- 1} \tilde{p_{k, γ}} (z)}{(q^{- 1} + q) + (q^{- 1} - q) \tilde{p_{k, γ}} (z)}

(11)

and

\tilde{p_{k, γ}} (z)

is given by (3). Geometrically, the function

p (z) \in k

{\tilde{P}}_{q, q^{- 1}, γ} (z)

takes on all values from the domain

\tilde{Ω_{k, q, q^{- 1}, γ}}

which is defined as:

\tilde{Ω_{k, q, q^{- 1}, γ}} = γ \tilde{Ω_{k, q, q^{- 1}}} + (1 - γ),

(12)

where

\tilde{Ω_{k, q, q^{- 1}}} = \{w : ℜ (\frac{(q^{- 1} + q) w (z)}{(q - q^{- 1}) w (z) + 2 q^{- 1}}) > k |\frac{(q^{- 1} + q) w (z)}{(q - q^{- 1}) w (z) + 2 q^{- 1}} - 1|\} .

Remark 2.

(i) First of all, we see that

lim_{q \to 1 -} \tilde{Ω_{k, q, q^{- 1}, γ}} = Ω_{k, γ} (γ \in C \ {0})

where

Ω_{k, γ}

is the conic domain considered by Shams et al. [7]. Secondly, we have

lim_{q \to 1 -} \tilde{Ω_{k, q, q^{- 1}, 1}} = Ω_{k},

where

Ω_{k}

is the conic domain considered by Kanas and Wisniowska [34]. Thirdly, we have

lim_{q \to 1 -} k - P_{q, q^{- 1}, 1} = P (p_{k}),

where

P (p_{k})

is the well-known class introduced by Kanas and Wisniowska [34]. Lastly, we have

lim_{q \to 1 -} 0 - P_{q, q^{- 1}, 1} = P,

where

P

is the well-known class of analytic functions with a positive real part.

Using the above-defined domain, we now define the following subclass of certain analytic functions:

Definition 8.

An analytic function

f \in

UST (q, q^{- 1}, γ)

if it satisfies the condition

ℜ \{1 + \frac{1}{γ} (J (q, q^{- 1}, f (z)) - 1)\} > k |\frac{1}{γ} (J (q, q^{- 1}, f (z)) - 1)|,

(13)

or equivalently

J (q, q^{- 1}, f (z)) \in k - {\tilde{P}}_{q, q^{- 1}, γ},

(14)

where

J (q, q^{- 1}, f (z)) = \frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} .

(15)

Remark 3.

First of all, we have (see [34])

lim_{q \to 1 -} k - UST (q, q^{- 1}, γ) = S_{k, γ}^{*} .

Secondly, it could be seen that (see [4])

lim_{q \to 1 -} k - UST (q, q^{- 1}, 1) = k - UCV .

Geometrically, the function

f \in A

is in the class k-

UST (q, q^{- 1}, γ)

, if and only if the function

J (q, q^{- 1}, f (z))

takes all values in the conic domain

\tilde{Ω_{k, q, q^{- 1}, γ}}

given by (12). Taking this geometrical interpretation into consideration, one can rephrase the above definition as:

Definition 9.

An analytic function

f \in

UST (q, q^{- 1}, γ)

if and only if

J (q, q^{- 1}, f (z)) ≺ \tilde{p_{k, q, q^{- 1}, γ}} (z),

(16)

where

\tilde{p_{k, q, q^{- 1}, γ}} (z)

is defined by (11).

We also fixed k-

{UST}^{-} (q, q^{- 1}, γ) = k

UST (q, q^{- 1}, γ) \cap T,

where

T

is the subclass of k-

UST (q, q^{- 1}, γ)

consisting of functions of the form

f (z) = z - \sum_{n = 2}^{\infty} a_{n} z^{n}, a_{n} \geq 0, \forall n \geq 2 .

(17)

Our further investigation is organized as follows. In Section 2, we give some supporting results in form of Lemmas, which will help in order to obtain our new results in Section 3. In Section 3, we obtain our main results and also give some of their special cases in the form of Corollaries and Remarks. In Section 4, we conclude our present investigation and also give some future direction to the interested readers toward the prospect that this kind of result will be obtained for other new subclasses of analytic functions.

2. A Set of Lemmas

We need the following lemmas in order to prove our main results.

Lemma 1

(See [45]). Let

p (z) = 1 + \sum_{n = 1}^{\infty} p_{n} z^{n} ≺ F (z) = 1 + \sum_{n = 1}^{\infty} d_{n} z^{n} .

F (z)

is convex univalent in

E,

then

|p_{n}| \leq |d_{1}|, n \geq 1 .

Lemma 2.

Let

k \in [0, \infty)

be fixed and

\tilde{p_{k, q, q^{- 1}, γ}} (z) = \frac{2 q^{- 1} \tilde{p_{k, γ}} (z)}{(q^{- 1} + q) + (q^{- 1} - q) \tilde{p_{k, γ}} (z)} .

Then

\tilde{p_{k, q, q^{- 1}, γ}} (z) = 1 + \frac{2 q^{- 1}}{q^{- 1} + q} Q_{1} z + \{\frac{2 q^{- 1}}{q^{- 1} + q} Q_{2} - \frac{2 (q^{- 1} - q)}{q^{- 1} + q} Q_{1}^{2}\} z^{2} + \dots,

where

Q_{1},

and

Q_{2}

is given by (5) and (6)

Proof.

From (11), we have

\begin{matrix} \tilde{p_{k, q, q^{- 1}, γ}} (z) & = \frac{2 q^{- 1} \tilde{p_{k, γ}} (z)}{(q^{- 1} + q) + (q^{- 1} - q) \tilde{p_{k, γ}} (z)} \\ = \frac{2 q^{- 1}}{(q^{- 1} + q)} \{\tilde{p_{k, γ}} (z)\} - \frac{2 q^{- 1} (q^{- 1} - q)}{{(q^{- 1} + q)}^{2}} {(\tilde{p_{k, γ}} (z))}^{2} \\ + \frac{2 q^{- 1} {(q^{- 1} - q)}^{2}}{{(q^{- 1} + q)}^{3}} {(\tilde{p_{k, γ}} (z))}^{3} \\ - \frac{2 q^{- 1} {(q^{- 1} - q)}^{3}}{{(q^{- 1} + q)}^{4}} {(\tilde{p_{k, γ}} (z))}^{4} + \dots . \end{matrix}

(18)

By using (4) in (18), we have

\begin{matrix} \tilde{p_{k, q, q^{- 1}, γ}} (z) \\ = \sum_{n = 1}^{\infty} \frac{2 q^{- 1} {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n - 1}}{{(q^{- 1} + q)}^{n}} \\ + \sum_{n = 1}^{\infty} \frac{2 q^{- 1} n {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n - 1}}{{(q^{- 1} + q)}^{n}} Q_{1} z \\ + \{\sum_{n = 1}^{\infty} \frac{2 q^{- 1} n {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n - 1}}{{(q^{- 1} + q)}^{n}} Q_{2} \\ - \sum_{n = 1}^{\infty} \frac{2 q^{- 1} (2 n - 1) {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n}}{{(q^{- 1} + q)}^{n + 1}} Q_{1}^{2}\} z^{2} + \dots . \end{matrix}

(19)

The series

\sum_{n = 1}^{\infty} \frac{2 q^{- 1} {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n - 1}}{{(q^{- 1} + q)}^{n}},

\sum_{n = 1}^{\infty} \frac{2 q^{- 1} n {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n - 1}}{{(q^{- 1} + q)}^{n}},

and

\sum_{n = 1}^{\infty} \frac{2 q^{- 1} (2 n - 1) {(- 1)}^{n - 1} {(q^{- 1} - q)}^{n}}{{(q^{- 1} + q)}^{n + 1}}

are convergent and convergent to

1,

\frac{2 q^{- 1}}{q^{- 1} + q},

and

\frac{2 q^{- 1} (q^{- 1} - q)}{(q^{- 1} + q)}

respectively.

Therefore, (19) becomes

\tilde{p_{k, q, q^{- 1}, γ}} (z) = 1 + \frac{2 q^{- 1}}{q^{- 1} + q} Q_{1} z + \{\frac{2 q^{- 1}}{q^{- 1} + q} Q_{2} - \frac{2 q^{- 1} (q^{- 1} - q)}{(q^{- 1} + q)} Q_{1}^{2}\} z^{2} + \dots .

(20)

This complete the proof of Lemma 2. □

Remark 4.

For

q \to 1 -,

Lemma 2 reduces to the lemma introduced by Sim et al. [46].

Lemma 3.

Let

p (z) = 1 + \sum_{n = 1}^{\infty} p_{n} z^{n} \in k

{\tilde{P}}_{q, q^{- 1}, γ};

then

|p_{n}| \leq \frac{2 q^{- 1}}{q^{- 1} + q} |Q_{1}|, n \geq 1 .

Proof.

By definition (7), a function

p (z) \in k

{\tilde{P}}_{q, q^{- 1}, γ} (z)

if and only if

p (z) ≺ \tilde{p_{k, q, q^{- 1}, α}} (z),

(21)

where

k \in [0, \infty),

and

\tilde{p_{k, q, q^{- 1}, γ}} (z)

is given by (11).

By using (20) in (21), we have

p (z) ≺ 1 + \frac{2 q^{- 1}}{q^{- 1} + q} Q_{1} z + \{\frac{2 q^{- 1}}{q^{- 1} + q} Q_{2} - \frac{2 q^{- 1} (q^{- 1} - q)}{(q^{- 1} + q)} Q_{1}^{2}\} z^{2} + \dots .

(22)

Now by using Lemma 1 on (22), we have

|p_{n}| \leq \frac{2 q^{- 1}}{q^{- 1} + q} |Q_{1}| .

Hence, the proof of Lemma 3 is complete. □

Remark 5.

For

q \to 1 -,

then Lemma 3 reduces to the lemma introduced by Noor et al. [47].

Lemma 4

([48]). Let

h (z) = 1 + \sum_{n = 1}^{\infty} c_{n} z^{n}

and

h (z)

be analytic in E and satisfy

ℜ {h (z)} > 0

for z in E. Then,

|c_{2} - v c_{1}^{2}| \leq 2 max \{1, |2 v - 1|\}, \forall v \in C .

3. Main Results

We now state and prove our main results. The first theorem of this section gives necessary conditions for an analytic function f of the form (1) to belong to the newly defined class k-

UST (q, q^{- 1}, γ) .

Theorem 1.

If an analytic function f is of the form (1), and it satisfies

\begin{matrix} \sum_{n = 2}^{\infty} \{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\} |a_{n}| \\ \leq (q + q^{- 1}) |γ|, \end{matrix}

(23)

then

f \in k

UST (q, q^{- 1}, γ) .

Proof.

Suppose that 23 holds; then, it is sufficient to show that

|\frac{k}{γ} (J (q, q^{- 1}, f (z)) - 1)| - ℜ \{\frac{1}{γ} (J (q, q^{- 1}, f (z))) - 1)\} \leq 1 .

Using (15), we have

\begin{matrix} |\frac{k}{γ} (\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} - 1)| \\ - ℜ \{\frac{1}{γ} (\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} - 1)\} \\ \leq \frac{k}{|γ|} |\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} - 1| + \frac{1}{|γ|} |\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} - 1|, \\ \leq \frac{(k + 1)}{|γ|} |\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} - 1| \\ = \frac{2 q^{- 1} (k + 1)}{|γ|} |\frac{\sum_{n = 2}^{\infty} ({\tilde{[n]}}_{q} - 1) a_{n} z^{n}}{(q + q^{- 1}) + \sum_{n = 2}^{\infty} \{(q - q^{- 1}) {\tilde{[n]}}_{q} + 2 q^{- 1}\} a_{n} z^{n}}|, \\ \leq \frac{2 q^{- 1} (k + 1)}{|γ|} \{\frac{\sum_{n = 2}^{\infty} |{\tilde{[n]}}_{q} - 1| |a_{n}|}{(q + q^{- 1}) - \sum_{n = 2}^{\infty} |(q - q^{- 1}) {\tilde{[n]}}_{q} + 2 q^{- 1}| |a_{n}|}\} . \end{matrix}

(24)

The expression (41) is bounded above by 1.

\frac{2 q^{- 1} (k + 1)}{|γ|} \{\frac{\sum_{n = 2}^{\infty} |{\tilde{[n]}}_{q} - 1| |a_{n}|}{(q + q^{- 1}) - \sum_{n = 2}^{\infty} |(q - q^{- 1}) {\tilde{[n]}}_{q} + 2 q^{- 1}| |a_{n}|}\} < 1 .

After some simple calculations, we have

\begin{matrix} \sum_{n = 2}^{\infty} \{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\} |a_{n}| \\ \leq (q + q^{- 1}) |γ| . \end{matrix}

Hence, we complete the proof of Theorem 1. □

When

q \to 1 -

and

γ = 1 - α

with

0 \leq α < 1

, we have the following known result (see [7]).

Corollary 1.

An analytic function f of the form (1) belongs to the class k-

UST (α)

if it satisfies the condition

\sum_{n = 2}^{\infty} \{n (k + 1) - (k + α)\} |a_{n}| \leq 1 - α,

where

0 \leq α < 1

and

k \geq 0

Inequality (23) gives the following corollary:

Corollary 2.

Let

0 \leq k < \infty,

q \in (0, 1)

and

γ \in C \ {0} .

If the inequality

|a_{n}| \leq \frac{(q + q^{- 1}) |γ|}{\{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\}}, n \geq 2,

holds for

f (z) = z + a_{n} z^{n},

then

k

UST (q, q^{- 1}, γ) .

In particular,

\begin{matrix} f (z) & = z + \frac{(q + q^{- 1}) |γ|}{\{2 q^{- 1} (k + 1) |{\tilde{[2]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[2]}}_{q}| + 2 q^{- 1}\}\}} z^{2} \\ \in k - UST (q, q^{- 1}, γ) \end{matrix}

and

|a_{2}| = \frac{(q + q^{- 1}) |γ|}{\{2 q^{- 1} (k + 1) |{\tilde{[2]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[2]}}_{q}| + 2 q^{- 1}\}\}} .

Coefficient estimates for the class k-

UST (q, q^{- 1}, γ)

are given in the next Theorem.

Theorem 2.

f \in k

UST (q, q^{- 1}, γ)

and is of the form 1, then

|a_{2}| \leq \frac{2 q^{- 1} |Q_{1}|}{(q^{- 1} + q)}

(25)

and

|a_{n}| \leq \prod_{j = 0}^{n - 2} (\frac{|Q_{1} - {\tilde{[j]}}_{q}|}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j} f o r n \geq 3,

(26)

where

Q_{1}

and

φ_{j}

are defined by (5) and (29).

Proof.

Let

\frac{(q^{- 1} + q) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z {\tilde{(D)}}_{q} f (z)}{f (z)} + 2 q^{- 1}} = p (z) .

(27)

After some simple simplification of Equation (27), we have

\begin{matrix} (q^{- 1} + q) z + \sum_{n = 2}^{\infty} (q^{- 1} + q) {\tilde{[n]}}_{q} a_{n} z^{n} \\ = \{(q^{- 1} + q) z + \sum_{n = 2}^{\infty} ((q - q^{- 1}) {\tilde{[n]}}_{q} + 2 q^{- 1}) a_{n} z^{n}\} (1 + \sum_{n = 1}^{\infty} c_{n} z^{n}) . \end{matrix}

Equating coefficients of

z^{n}

on both sides, we have

2 q^{- 1} ({\tilde{[n]}}_{q} - 1) a_{n} = \sum_{j = 1}^{n - 1} ({\tilde{[j - 1]}}_{q} (q - q^{- 1}) + 2 q^{- 1}) a_{n - j} c_{j}, a_{1} = 1 .

This implies that

|a_{n}| \leq \frac{1}{2 q^{- 1} ({\tilde{[n]}}_{q} - 1)} \sum_{j = 1}^{n - 1} \{{\tilde{[j - 1]}}_{q} (q - q^{- 1}) + 2 q^{- 1}\} |a_{n - j}| |c_{j}| .

By using Lemma 3, we have

|a_{n}| \leq \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[n]}}_{q} - 1)} \sum_{j = 1}^{n - 1} \{{\tilde{[j - 1]}}_{q} (q - q^{- 1}) + 2 q^{- 1}\} |a_{j}|,

|a_{n}| \leq \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[n]}}_{q} - 1)} \sum_{j = 1}^{n - 1} φ_{j - 1} |a_{j}|,

(28)

where

φ_{j - 1} = {\tilde{[j - 1]}}_{q} (q - q^{- 1}) + 2 q^{- 1} .

(29)

Now we prove that

\frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[n]}}_{q} - 1)} \sum_{j = 1}^{n - 1} φ_{j - 1} |a_{j}| \leq \prod_{j = 0}^{n - 2} (\frac{|Q_{1} - {\tilde{[j]}}_{q}|}{(q^{- 1} + q) ({\tilde{[j + 1]}}_{q})}) φ_{j} .

(30)

For this, we use the induction method. For

n = 2

from (28), we have

|a_{2}| \leq \frac{2 q^{- 1} |Q_{1}|}{(q^{- 1} + q)}, φ_{0} = 2 q^{- 1}

From (26), we have

|a_{2}| \leq \frac{2 q^{- 1} |Q_{1}|}{(q^{- 1} + q)}, φ_{0} = 2 q^{- 1} .

For

n = 3

, from (28), we have

\begin{matrix} |a_{3}| & \leq \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} (φ_{0} + φ_{1} |a_{2}|), \\ \leq \frac{2 q^{- 1} |Q_{1}|}{(q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} (1 + |Q_{1}|), φ_{1} = (q^{- 1} + q) . \end{matrix}

From (26), we have

\begin{matrix} |a_{3}| & \leq \frac{|Q_{1}| φ_{0}}{(q^{- 1} + q)} \{(\frac{|Q_{1} - {\tilde{[1]}}_{q}|}{(q^{- 1} + q) {\tilde{[2]}}_{q}}) φ_{1}\}, \\ \leq \frac{|Q_{1}| φ_{0}}{(q^{- 1} + q)} \{(\frac{|Q_{1}| + {\tilde{[1]}}_{q}}{(q^{- 1} + q) {\tilde{[2]}}_{q}}) φ_{1}\}, \\ = \frac{|Q_{1}| φ_{1}}{(q^{- 1} + q) {\tilde{[2]}}_{q}} (\frac{|Q_{1}| φ_{0}}{(q^{- 1} + q)} + \frac{φ_{0}}{(q^{- 1} + q)}), \\ = \frac{|Q_{1}| φ_{1}}{(q^{- 1} + q) {\tilde{[2]}}_{q}} (\frac{|Q_{1}| φ_{0}}{(q^{- 1} + q)} + \frac{φ_{0}}{(q^{- 1} + q)}), \\ = \frac{2 q^{- 1} |Q_{1}|}{(q^{- 1} + q) {\tilde{[2]}}_{q}} (|Q_{1}| + 1) . \end{matrix}

Let the hypothesis be true for

n = m

. From (28), we have

|a_{m}| \leq \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \sum_{j = 1}^{m - 1} φ_{j - 1} |a_{j}| .

From (26), we have

\begin{matrix} |a_{m}| & \leq \prod_{j = 0}^{m - 2} (\frac{|Q_{1} - {\tilde{[j]}}_{q}|}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j}, n \geq 2, \\ \leq \prod_{j = 0}^{m - 2} (\frac{|Q_{1}| + {\tilde{[j]}}_{q}}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j}, n \geq 2 . \end{matrix}

By the induction hypothesis, we have

\frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \sum_{j = 1}^{m - 1} φ_{j - 1} |a_{j}| \leq \prod_{j = 0}^{m - 2} (\frac{|Q_{1}| + {\tilde{[j]}}_{q}}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j} .

(31)

Multiplying

\frac{|Q_{1}| + ({\tilde{[m]}}_{q} - 1)}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)}

on both sides of (31), we have

\begin{matrix} \prod_{j = 0}^{m - 2} (\frac{|Q_{1}| + {\tilde{[j]}}_{q}}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j}, \\ \geq \frac{|Q_{1}| + ({\tilde{[m]}}_{q} - 1)}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \{\frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \sum_{j = 1}^{m - 1} φ_{j - 1} |a_{j}|\}, \\ = \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \{\frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} + 1\} \sum_{j = 1}^{m - 1} φ_{j - 1} |a_{j}|, \\ \geq \frac{|Q_{1}|}{(q^{- 1} + q) {({\tilde{[m]}}_{q} - 1)}_{q}} \{|a_{m}| + \sum_{j = 1}^{m - 1} φ_{j - 1} |a_{j}|\}, \\ = \frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \sum_{j = 1}^{m} φ_{j - 1} |a_{j}| . \end{matrix}

That is

\frac{|Q_{1}|}{(q^{- 1} + q) ({\tilde{[m]}}_{q} - 1)} \sum_{j = 1}^{m} φ_{j - 1} |a_{j}| \leq \prod_{j = 0}^{m - 2} (\frac{|Q_{1}| + {\tilde{[j]}}_{q}}{(q^{- 1} + q) {\tilde{[j + 1]}}_{q}}) φ_{j},

which shows that inequality (31) is true for

n = m + 1

. Hence, the proof of Theorem 2 is now completed. □

Corollary 3.

f (z) \in k

UST (γ)

and is of the form 1, then

|a_{n}| \leq \prod_{j = 0}^{n - 2} (\frac{|Q_{1} - j|}{(j + 1)}) f o r n \geq 3 .

For

q \to 1 -

, Kanas and Wisniowska proved Corollary (3) in [4].

In the next Theorem, we state and prove the Fekete–Szegö-type result for our defined function class k-

UST (q, q^{- 1}, γ)

Theorem 3.

Let

0 \leq k < \infty,

q \in (0, 1)

be fixed, and let

f (z) \in k

UST (q, q^{- 1}, γ)

and be of the form (1). Then for a complex number μ,

|a_{3} - μ a_{2}^{2}| \leq \frac{|Q_{1}|}{2 (q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} max \{1, |2 v - 1|\},

(32)

where v is given by (37).

Proof.

f (z) \in k

UST (q, q^{- 1}, γ),

then we have

J (q, q^{- 1}, f (z)) ≺ \tilde{p_{k, q, q^{- 1}, γ}} (z)

and if there exists a Schwarz function

w (z)

, we have

\frac{(q^{- 1} + q) \frac{z D_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z D_{q} f (z)}{f (z)} + 2 q^{- 1}} = \tilde{p_{k, q, q^{- 1}, γ}} (w (z)) .

(33)

Let the function

h (z) \in P

, defined by:

h (z) = \frac{1 + w (z)}{1 - w (z)},

this gives

w (z) = \frac{c_{1}}{2} z + \frac{1}{2} (c_{2} - \frac{c_{1}^{2}}{2}) z^{2} + \dots

and

\tilde{p_{k, q, q^{- 1}, γ}} (w (z)) = 1 + \frac{q^{- 1} Q_{1} c_{1}}{(q^{- 1} + q)} z + \frac{q^{- 1}}{(q^{- 1} + q)} ϝ_{1} (q, q^{- 1}) z^{2} + \dots,

(34)

where

ϝ_{1} (q, q^{- 1}) = \{\frac{Q_{2} c_{1}^{2}}{2} + (c_{2} - \frac{c_{1}^{2}}{2}) Q_{1} - \frac{(q^{- 1} - q) Q_{1}^{2} c_{1}^{2}}{2}\} .

Now, from L.H.S of (33), we have

\frac{(q^{- 1} + q) \frac{z D_{q} f (z)}{f (z)}}{(q - q^{- 1}) \frac{z D_{q} f (z)}{f (z)} + 2 q^{- 1}} = 1 + \frac{2 q^{- 1} ({\tilde{[2]}}_{q} - 1)}{(q^{- 1} + q)} a_{2} z + ϝ_{2} (q, q^{- 1}) z^{2} + . . .,

(35)

where

ϝ_{2} (q, q^{- 1}) = \{2 q^{- 1} ({\tilde{[3]}}_{q} - 1) a_{3} - \frac{2 q^{- 1}}{(q^{- 1} + q)} ({\tilde{[2]}}_{q} - 1) \{(q - q^{- 1}) {\tilde{[2]}}_{q} + 2 q^{- 1}\} a_{2}^{2}\} .

By using (34) and (35) in (33), we obtain

a_{2} = \frac{Q_{1} c_{1}}{2 ({\tilde{[2]}}_{q} - 1)}

and

a_{3} = \frac{1}{2 (q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} \{ϝ_{1} (q, q^{- 1}) + \frac{\{(q - q^{- 1}) {\tilde{[2]}}_{q} + 2 q^{- 1}\} Q_{1}^{2} c_{1}^{2}}{2 ({\tilde{[2]}}_{q} - 1)}\} .

For any complex number

μ

we have

|a_{3} - μ a_{2}^{2}| = \frac{|Q_{1}|}{2 (q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} \{c_{2} - v c_{1}^{2}\},

(36)

where

v = \frac{1}{2} \{1 - \frac{Q_{2}}{Q_{1}} + (q^{- 1} - q) Q_{1} - ϝ_{3} (q, q^{- 1})\}

(37)

and

ϝ_{3} (q, q^{- 1}) = \frac{\{(q - q^{- 1}) {[2]}_{q} + 2 q^{- 1}\} Q_{1}}{({\tilde{[2]}}_{q} - 1)} + μ \frac{(q^{- 1} + q) ({\tilde{[3]}}_{q} - 1) Q_{1}}{{({\tilde{[2]}}_{q} - 1)}^{2}} .

Now by using Lemma 4 on 36

,

we have

|a_{3} - μ a_{2}^{2}| \leq \frac{|Q_{1}|}{2 (q^{- 1} + q) ({\tilde{[3]}}_{q} - 1)} max \{1, |2 v - 1|\} .

Hence, we complete the proof of Theorem 3. □

Next we investigate the necessary and sufficient conditions for

f (z)

of the form (17) to be in the class k-

{UST}^{-} (q, q^{- 1}, γ)

Theorem 4.

Let

k \in [0, \infty),

q \in (0, 1)

and

γ \in C \ {0} .

A function

f (z)

of the form (17) will belong to the class k-

{UST}^{-} (q, q^{- 1}, γ)

which can be expressed as:

\begin{matrix} \sum_{n = 2}^{\infty} \{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\} |a_{n}| \\ \leq (q + q^{- 1}) |γ| . \end{matrix}

(38)

The result is sharp for the function

f (z) = z - \frac{(q + q^{- 1}) |γ|}{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}} z^{n} .

Proof.

In view of Theorem 1, it remains to prove the necessity.

f (z) \in k

{UST}^{-} (q, q^{- 1}, γ),

then infect that

|ℜ (z)| \leq |z|;

for any z, we have

\begin{matrix} |1 + \frac{1}{γ (q + q^{- 1})} (\frac{\sum_{n = 2}^{\infty} 2 q^{- 1} ({\tilde{[n]}}_{q} - 1) a_{n} z^{n - 1}}{1 - \sum_{n = 2}^{\infty} \frac{1}{q + q^{- 1}} \{(q - q^{- 1}) {[n]}_{q} + 2 q^{- 1}\} a_{n} z^{n - 1}})| \\ \geq |\frac{k}{γ (q + q^{- 1})} \{\frac{\sum_{n = 2}^{\infty} 2 q^{- 1} ({\tilde{[n]}}_{q} - 1) a_{n} z^{n - 1}}{1 - \sum_{n = 2}^{\infty} \frac{1}{(q + q^{- 1})} \{(q - q^{- 1}) {[n]}_{q} + 2 q^{- 1}\} a_{n} z^{n - 1}}\}| . \end{matrix}

(39)

Letting

z \to 1 -

along the real axis, we obtain the desired inequality (38). Hence we complete the proof of Theorem 4. □

Corollary 4.

Let the function

f (z)

of the form (17) be in the class k-

{UST}^{-} (q, q^{- 1}, γ)

. Then

a_{n} \leq \frac{(q + q^{- 1}) |γ|}{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}}, n \geq 2 .

(40)

Corollary 5.

Let the function

f (z)

of the form (17) be in the class k-

{UST}^{-} (q, q^{- 1}, γ)

. Then

a_{2} = \frac{(q + q^{- 1}) |γ|}{2 q^{- 1} (k + 1) |{\tilde{[2]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[2]}}_{q}| + 2 q^{- 1}\}} .

(41)

Theorem 5.

Let

k \in [0, \infty),

q \in (0, 1)

and

γ \in C \ {0}

and let

f_{1} (z) = z,

and

f_{n} (z) = z - R (n, q) z^{n}, n \geq 3 .

(42)

Then

f \in k

{UST}^{-} (q, q^{- 1}, γ),

if and only if f can be expressed in the form of

f (z) = \sum_{n = 1}^{\infty} λ_{n} f_{n} (z), λ_{n} > 0, and \sum_{n = 1}^{\infty} λ_{n} = 1,

(43)

where

R (n, q)

is given by (44).

Proof.

Suppose that

\begin{matrix} f (z) & = & \sum_{n = 1}^{\infty} λ_{n} f_{n} (z) = λ_{1} f_{1} (z) + \sum_{n = 2}^{\infty} λ_{n} f_{n} (z), \\ = & λ_{1} f_{1} (z) + \sum_{n = 2}^{\infty} λ_{n} \{z - R (n, q) z^{n}\}, \\ = & λ_{1} z + \sum_{n = 2}^{\infty} λ_{n} z - \sum_{n = 2}^{\infty} λ_{n} R (n, q) z^{n}, \\ = & (\sum_{n = 1}^{\infty} λ_{n}) z - \sum_{n = 2}^{\infty} λ_{n} R (n, q) z^{n}, \\ = & z - \sum_{n = 2}^{\infty} λ_{n} R (n, q) z^{n}, \end{matrix}

where

R (n, q) = \frac{(q + q^{- 1}) |γ|}{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}} .

(44)

Then

\begin{matrix} \sum_{n = 2}^{\infty} λ_{n} R (n, q) \times \frac{1}{R (n, q)} \\ = \sum_{n = 2}^{\infty} λ_{n} = \sum_{n = 1}^{\infty} λ_{n} - λ_{1} = 1 - λ_{1} \leq 1, \end{matrix}

and we find k-

{UST}^{-} (q, q^{- 1}, γ) .

Conversely, suppose that k-

{UST}^{-} (q, q^{- 1}, γ) .

Since

|a_{n}| \leq R (n, q),

we can set

λ_{n} = \frac{1}{R (n, q)} |a_{n}|,

and

λ_{1} = 1 - \sum_{n = 2}^{\infty} λ_{n} .

Then

\begin{matrix} f (z) & = z + \sum_{n = 2}^{\infty} a_{n} z^{n} \\ = z + \sum_{n = 2}^{\infty} λ_{n} R (n, q) z^{n}, \\ = z + \sum_{n = 2}^{\infty} λ_{n} (z + f_{n} (z)) = z + \sum_{n = 2}^{\infty} λ_{n} z + \sum_{n = 2}^{\infty} λ_{n} f_{n} (z), \\ = (1 - \sum_{n = 2}^{\infty} λ_{n}) z + \sum_{n = 2}^{\infty} λ_{n} f_{n} (z) = λ_{1} z + \sum_{n = 2}^{\infty} λ_{n} f_{n} (z) = \sum_{n = 1}^{\infty} λ_{n} f_{n} (z) . \end{matrix}

The proof of Theorem 5 is complete. □

Theorem 6.

Let

k \in [0, \infty),

q \in (0, 1),

|z| = r < 1

and

γ \in C \ {0} .

Let

f

of the form (17) be in the class

k - {UST}^{-} (q, q^{- 1}, γ) .

Then

r - R (2, q) r^{2} \leq |f (z)| \leq r + R (2, q) r^{2} .

(45)

Equality in (45) is attained for the function f given by the formula

f (z) = z + R (2, q) z^{2},

(46)

where

R (2, q) = \frac{(q + q^{- 1}) |γ|}{\{2 q^{- 1} (k + 1) |{\tilde{[2]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[2]}}_{q}| + 2 q^{- 1}\}\}} .

Proof.

Since

f \in k

{UST}^{-} (q, q^{- 1}, γ),

in view of Theorem 4, we find

\begin{matrix} \{2 q^{- 1} (k + 1) |{\tilde{[2]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[2]}}_{q}| + 2 q^{- 1}\}\} \sum_{n = 2}^{\infty} a_{n} \\ \leq \sum_{n = 2}^{\infty} \{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\} |a_{n}| \\ \leq (q + q^{- 1}) |γ| . \end{matrix}

This gives

\sum_{n = 2}^{\infty} a_{n} \leq R (2, q) .

Therefore

|f (z)| \leq |z| + \sum_{n = 2}^{\infty} a_{n} {|z|}^{n} \leq r + R (2, q) r^{2},

and

|f (z)| \geq |z| - \sum_{n = 2}^{\infty} a_{n} {|z|}^{n} \geq r - R (2, q) r^{2} .

By letting

r \to 1 -,

we get the required result. Hence, the proof of Theorem 6 is complete. □

Theorem 7.

Let

k \in [0, \infty),

q \in (0, 1),

|z| = r < 1

and

γ \in C \ {0} .

Let

f

be of the form (17) in the class k-

{UST}^{-} (q, q^{- 1}, γ) .

Then

1 - 2 R (2, q) r \leq |f^{'} (z)| \leq 1 + 2 R (2, q) r .

(47)

Proof.

Differentiating f and using triangle inequality for the modulus, we obtain

|f^{'} (z)| \leq 1 + \sum_{n = 2}^{\infty} n a_{n} {|z|}^{n - 1} \leq 1 + r \sum_{n = 2}^{\infty} n a_{n},

(48)

and

|f^{'} (z)| \geq 1 - \sum_{n = 2}^{\infty} n a_{n} {|z|}^{n - 1} \geq 1 - r \sum_{n = 2}^{\infty} n a_{n} .

(49)

Assertion (47) follows from (48) and (49); in view of the rather simple consequence, we get the following inequality

\sum_{n = 2}^{\infty} n a_{n} \leq 2 R (2, q) .

Hence, proof of Theorem 7 is complete. □

Theorem 8.

Under convex linear combination, the class k-

{UST}^{-} (q, q^{- 1}, γ)

is closed.

Proof.

Let

f (z)

and

g (z)

be in class k-

{UST}^{-} (q, q^{- 1}, γ) .

Suppose

f (z)

is given by (17) and

g (z) = z - \sum_{n = 2}^{\infty} d_{n} z^{n},

(50)

where

a_{n},

d_{n} \geq 0 .

It is sufficient to show that for

0 \leq λ \leq 1

, the function

H (z) = λ f (z) + (1 - λ) g (z) \in k - {UST}^{-} (q, q^{- 1}, γ) .

(51)

From (17), (50) and (51), we have

H (z) = z - \sum_{n = 2}^{\infty} \{λ a_{n} + (1 - λ) d_{n}\} z^{n} .

(52)

As we know that f and g are in class k-

{UST}^{-} (q, q^{- 1}, γ)

(0 \leq λ \leq 1)

, by using Theorem 4, we obtain

\begin{matrix} \sum_{n = 2}^{\infty} \{2 q^{- 1} (k + 1) |{\tilde{[n]}}_{q} - 1| + |γ| \{|(q - q^{- 1}) {\tilde{[n]}}_{q}| + 2 q^{- 1}\}\} \{λ a_{n} + (1 - λ) d_{n}\} \\ \leq (1 + q) |γ| . \end{matrix}

(53)

Again, by Theorem 4 and inequality (53), we have

H (z) \in k

{UST}^{-} (q, q^{- 1}, γ) .

Hence, the proof of Theorem 8 is complete. □

4. Conclusions

In this paper, motivated significantly by a number of recent works, we used the concept of symmetric quantum calculus and conic regions to define a new domain

\tilde{Ω_{k, q, q^{- 1}, γ}}

, which generalizes the symmetric conic domains. By using a certain generalized symmetric conic domain

\tilde{Ω_{k, q, q^{- 1}, γ}}

, we defined and investigated a new subclass of normalized analytic q-starlike functions in the open unit disk E, and we have successfully derived several properties and characteristics of a newly defined subclass of analytic functions. For the verification and validity of our main results, we have also pointed out relevant connections of our main results with those in several earlier related works on this subject.

To conclude our present investigation, we would like to remark that one may attempt the results presented in this paper for different subclasses of analytic functions in different domains. In particular, one may define a new subclass of symmetric q-starlike functions associated with this newly defined domain and can obtain the same results.

Author Contributions

The authors have equally contributed to accomplish this research work. All authors have read and agreed to the published version of the manuscript.

Funding

This research receive no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors agree with the contents of the manuscript, and there are no conflict of interest among the authors.

References

Goodman, A.W. Univalent Functions; Polygonal Publishing House: Washington, NJ, USA, 1983; Volume I. [Google Scholar]
Goodman, A.W. Univalent Functions; Polygonal Publishing House: Washington, NJ, USA, 1983; Volume II. [Google Scholar]
Goodman, A.W. On uniformly convex functions. Ann. Polon. Math. 1991, 56, 87–92. [Google Scholar] [CrossRef] [Green Version]
Kanas, S.; Wisniowska, A. Conic domains and k-starlike functions. Rev. Roum. Math. Pure Appl. 2000, 45, 647–657. [Google Scholar]
Rønning, F. Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118, 189–196. [Google Scholar] [CrossRef] [Green Version]
Ma, W.; Minda, D. Uniformly convex functions. Ann. Polon. Math. 1992, 57, 165–175. [Google Scholar] [CrossRef] [Green Version]
Shams, S.; Kulkarni, S.R.; Jahangiri, J.M. Classes of uniformly starlike and convex functions. Int. J. Math. Math. Sci. 2004, 55, 2959–2961. [Google Scholar] [CrossRef]
Jackson, F.H. On q-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Edinb. 1909, 46, 253–281. [Google Scholar] [CrossRef]
Islam, S.; Khan, M.G.; Ahmad, B.; Arif, M.; Chinram, R. q-Extension of Starlike Functions Subordinated with a Trigonometric Sine Function. Mathematics 2020, 8, 1676. [Google Scholar] [CrossRef]
Zainab, S.; Raza, M.; Xin, Q.; Jabeen, M.; Malik, S.N.; Riaz, S. On q-Starlike Functions Defined by q-Ruscheweyh Differential Operator in Symmetric Conic Domain. Symmetry 2021, 13, 1947. [Google Scholar] [CrossRef]
Jackson, F.H. On q-definite integrals. Q. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
Khan, S.; Hussain, S.; Zaighum, M.A.; Darus, M. A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator. Math. Slovaca 2019, 69, 825–832. [Google Scholar] [CrossRef]
Mahmood, S.; Khan, I.; Srivastava, H.M.; Malik, S.N. Inclusion relations for certain families of integral operators associated with conic regions. J. Inequalities Appl. 2019, 2019, 59. [Google Scholar] [CrossRef]
Mahmood, S.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M. Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative. J. Funct. Spaces 2018, 2018, 8492072. [Google Scholar] [CrossRef] [Green Version]
Ahmad, Q.Z.; Khan, N.; Raza, M.; Tahir, M.; Khan, B. Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions. Filomat 2019, 33, 3385–3397. [Google Scholar] [CrossRef]
Rehman, M.S.; Ahmad, Q.Z.; Khan, B.; Tahir, M.; Khan, N. Generalisation of certain subclasses of analytic and univalent functions, Maejo Internat. J. Sci. Technol. 2019, 13, 1–9. [Google Scholar]
Shi, L.; Ahmad, B.; Khan, N.; Khan, M.G.; Araci, S.; Mashwani, W.K.; Khan, B. Coefficient Estimates for a Subclass of Meromorphic Multivalent q-Close-to-Convex Functions. Symmetry 2021, 13, 1840. [Google Scholar] [CrossRef]
Uçar, H.E.O. Coefficient inequality for q-starlike Functions. Appl. Math. Comput. 2016, 76, 122–126. [Google Scholar]
Srivastava, H.M. Operators of basic (or q -) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. Sci. 2020, 44, 327–344. [Google Scholar] [CrossRef]
Srivastava, H.M. Univalent functions, fractional calculus, and associated generalized hypergeometric functions. In Univalent functions, fractional Calculus, and Their Applications; Srivastava, H.M., Owa, S., Eds.; John Wiley & Sons: New York, NY, USA, 1989. [Google Scholar]
Khan, B.; Liu, Z.-G.; Srivastava, H.M.; Khan, N.; Tahir, M. Applications of higher-order derivatives to subclasses of multivalent q-starlike functions. Maejo Int. J. Sci. Technol. 2021, 15, 61–72. [Google Scholar]
Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B. Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 2019, 7, 181. [Google Scholar] [CrossRef] [Green Version]
Ahmad, B.; Khan, M.G.; Frasin, B.A.; Aouf, M.K.; Abdeljawad, T.; Mashwani, W.K.; Arif, M. On q-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain. AIMS Math. 2021, 6, 3037–3052. [Google Scholar] [CrossRef]
Ahmad, B.; Khan, M.G.; Darus, M.; Khan Mashwani, W.; Arif, M. Applications of Some Generalized Janowski Meromorphic Multivalent Functions. J. Math. 2021, 2021, 6622748. [Google Scholar] [CrossRef]
Khan, N.; Srivastava, H.M.; Rafiq, A.; Arif, M.; Arjika, S. Some applications of q-difference operator involving a family of meromorphic harmonic functions. Adv. Differ. Equ. 2021, 2021, 471. [Google Scholar] [CrossRef]
Zhan, C.; Khan, S.; Hussain, A.; Khan, N.; Hussain, S.; Khan, N. Applications of q-difference symmetric operator in harmonic univalent functions. AIMS Math. 2021, 7, 667–680. [Google Scholar] [CrossRef]
Khan, S.; Hussain, S.; Naeem, M.; Darus, M.; Rasheed, A. A Subclass of q-starlike functions defined by using a symmetric q-derivative operator and related with generalized symmetric conic domains. Mathematics 2021, 9, 917. [Google Scholar] [CrossRef]
Khan, B.; Liu, Z.-G.; Shaba, T.G.; Araci, S.; Khan, N.; Khan, M.G. Applications of q-Derivative Operator to the Subclass of Bi-Univalent Functions Involving q-Chebyshev Polynomials. J. Math. 2022, 2022, 8162182. [Google Scholar] [CrossRef]
Cruz, A.M.D.; Martins, N. The q-symmetric variational calculus. Comput. Math. Appl. 2012, 64, 2241–2250. [Google Scholar] [CrossRef] [Green Version]
Lavagno, A. Basic-deformed quantum mechanics. Rep. Math. Phys 2009, 64, 79–88. [Google Scholar] [CrossRef] [Green Version]
Gasper, G.; Rahman, M. Volume 35 of Encyclopedia of Mathematics and its applications. In Basic Hpergeometric Series; Ellis Horwood: Chichester, UK, 1990. [Google Scholar]
Kamel, B.; Yosr, S. On some symmetric q-special functions. Le Matematiche 2013, 68, 107–122. [Google Scholar]
Kanas, S.; Raducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
Kanas, S.; Wisniowska, A. Conic regions and k-uniform convexity. J. Comput. Appl. Math. 1999, 105, 327–336. [Google Scholar] [CrossRef] [Green Version]
Noor, K.I. Applications of certain operators to the classes related with generalized Janowski functions. Integral Transform. Spec. Funct. 2010, 21, 557–567. [Google Scholar] [CrossRef]
Noor, K.I.; Malik, S.N. On coefficient inequalities of functions associated with conic domains. Comput. Math. Appl. 2011, 62, 2209–2217. [Google Scholar] [CrossRef] [Green Version]
Noor, K.I.; Malik, S.N.; Arif, M.; Raza, M. On bounded boundary and bounded radius rotation related with Janowski function. World Appl. Sci. J. 2011, 12, 895–902. [Google Scholar]
Sokół, J. Classes of multivalent functions associated with a convolution operator. Comput. Math. Appl. 2010, 60, 1343–1350. [Google Scholar] [CrossRef] [Green Version]
Zhang, X.; Khan, S.; Hussain, S.; Tang, H.; Shareef, Z. New subclass of q-starlike functions associated with generalized conic domain. AIMS Math. 2020, 5, 4830–4848. [Google Scholar] [CrossRef]
Andrei, L.; Caus, V.-A. A Generalized Class of Functions Defined by the q-Difference Operator. Symmetry 2021, 13, 2361. [Google Scholar] [CrossRef]
Cătaş, A.A. On the Fekete–Szegö Problem for Meromorphic Functions Associated with p,q-Wright Type Hypergeometric Function. Symmetry 2021, 13, 2143. [Google Scholar] [CrossRef]
Swamy, S.R.; Alb Lupaş, A. Bi-univalent Function Subfamilies Defined by q-Analogue of Bessel Functions Subordinate to (p,q)-Lucas Polynomials. WSEAS Trans. Math. 2022, 21, 98–106. [Google Scholar] [CrossRef]
Akgül, A.; Cotîrlă, L.-I. Coefficient Estimates for a Family of Starlike Functions Endowed with Quasi Subordination on Conic Domain. Symmetry 2022, 14, 582. [Google Scholar] [CrossRef]
Janowski, W. Some extremal problems for certain families of analytic functions. Ann. Polon. Math. 1973, 28, 297–326. [Google Scholar] [CrossRef] [Green Version]
Rogosinski, W. On the coefficients of subordinate functions. Proc. Lond. Math. Soc 1943, 48, 48–82. [Google Scholar] [CrossRef]
Sim, Y.J.; Kwon, O.S.; Cho, N.E.; Srivastava, H.M. Some classes of analytic functions associated with conic regions. Taiwan J. Math. 2012, 16, 387–408. [Google Scholar] [CrossRef]
Noor, K.I.; Arif, M.; Ul-Haq, W. On k-uniformly close-to-convex functions of complex order. Appl. Math. Comput. 2009, 215, 629–635. [Google Scholar] [CrossRef]
Ma, W.; Minda, D. A unified treatment of some special classes of univalent functions. In Proceedings of the Conferene on Complex Analysis; Li, Z., Ren, F., Yang, L., Zhang, S., Eds.; International Press Inc.: New York, NY, USA, 1992; pp. 157–169. [Google Scholar]

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Khan, S.; Khan, N.; Hussain, A.; Araci, S.; Khan, B.; Al-Sulami, H.H. Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions. Symmetry 2022, 14, 803. https://doi.org/10.3390/sym14040803

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Khan S, Khan N, Hussain A, Araci S, Khan B, Al-Sulami HH. Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions. Symmetry. 2022; 14(4):803. https://doi.org/10.3390/sym14040803

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Khan, Shahid, Nazar Khan, Aftab Hussain, Serkan Araci, Bilal Khan, and Hamed H. Al-Sulami. 2022. "Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions" Symmetry 14, no. 4: 803. https://doi.org/10.3390/sym14040803

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Khan, S., Khan, N., Hussain, A., Araci, S., Khan, B., & Al-Sulami, H. H. (2022). Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions. Symmetry, 14(4), 803. https://doi.org/10.3390/sym14040803

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Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions

Abstract

1. Introduction and Definitions

2. A Set of Lemmas

3. Main Results

4. Conclusions

Author Contributions

Funding

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Data Availability Statement

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References

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