Open AccessArticle

Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function

Yusra Taj

Saira Zainab

Qin Xin

Ferdous M. O. Tawfiq

⁴,

Mohsan Raza

^5,*

and

Sarfraz Nawaz Malik

Department of Mathematics, COMSATS University Islamabad, Wah Campus, Wah Cantt 47040, Pakistan

School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan

Faculty of Science and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands, Denmark

⁴

Department of Mathematics, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia

⁵

Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan

Author to whom correspondence should be addressed.

Symmetry 2022, 14(10), 2200; https://doi.org/10.3390/sym14102200

Submission received: 11 September 2022 / Revised: 13 October 2022 / Accepted: 14 October 2022 / Published: 19 October 2022

(This article belongs to the Special Issue Applications of Symmetric Functions Theory to Certain Fields)

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Abstract

This study introduces a subclass

S_{q s}^{*}

of starlike functions associated with the q-analogue of the sine function defined in symmetric unit disk. This article comprises the investigation of sharp coefficient bounds, and the upper bound of the third-order Hankel determinant for this class. It also includes the findings of Zalcman and generalized Zalcman conjectures for functions of this class.

Keywords:

analytic functions; q-starlike functions; univalent function; Hankel determinant; symmetric matrix; Zalcman inequalities; q-derivative operator

1. Introduction and Preliminaries

In the study of analytic and univalent functions, coefficient problems play a vital role that helps in making many estimations about analytic functions. There is a wide range of coefficient problems that include coefficient bounds, necessary and sufficient conditions, covering results, Hankel determinants, Toeplitz determinants, and many coefficient inequalities and conjectures. The Hankel matrix was introduced by Hermann Hankel, and this matrix is a square symmetric matrix having the same entries in its skew diagonal. To relate the Hankel matrix with analytic functions, it is formed to have elements as the coefficients of certain power series of analytic functions. If an analytic function f, defined in the disk

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

assumes to satisfy the conditions

f (0) = 0

and

f^{'} (0) = 1,

then such functions are comprised in class

A

and they will have the following form of their Taylor series

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

(1)

The determinant of a matrix is an important number that helps in characterizing many properties of that matrix. Based on the coefficients of the series (1), the

j

th Hankel determinant for

f \in A

is given by

H_{j, n} (f) = |\begin{matrix} a_{n} & a_{n + 1} & \dots & a_{n + j - 1} \\ a_{n + 1} & a_{n + 2} & \dots & a_{n + j} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n + j - 1} & a_{n + j - 2} & \dots & a_{n + 2 j - 2} \end{matrix}|, j, n \in N .

The main goal in studying the Hankel determinant is to find the upper bound of the determinant. With certain variations in the values of j and

n,

the Hankel determinant

H_{j, n} (f)

takes the following forms

H_{2, 1} (f) = |\begin{matrix} a_{1} & a_{2} \\ a_{2} & a_{3} \end{matrix}| = a_{1} a_{3} - a_{2}^{2}, H_{2, 2} (f) = |\begin{matrix} a_{2} & a_{3} \\ a_{3} & a_{4} \end{matrix}| = a_{2} a_{4} - a_{3}^{2} .

The determinant

H_{2, 1}

is the renowned Fekete-Szegö functional, Ref. [1] and

H_{2, 2}

is the well-known and extensively studied second Hankel determinant. Also, the third-order Hankel determinant is written as

\begin{matrix} H_{3, 1} (f) & = & |\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{2} & a_{3} & a_{4} \\ a_{3} & a_{4} & a_{5} \end{matrix}| \\ = & a_{5} (a_{3} - a_{2}^{2}) - a_{4} (a_{4} - a_{2} a_{3}) + a_{3} (a_{2} a_{4} - a_{3}^{2}) . \end{matrix}

(2)

This implies that

|H_{3, 1} (f)| \leq |a_{5}| |a_{3} - a_{2}^{2}| + |a_{4}| |a_{4} - a_{2} a_{3}| + |a_{3}| |H_{2, 2} (f)| .

(3)

Also, the fourth-order determinant is given by

\begin{matrix} H_{4, 1} (f) & = & |\begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} \\ a_{2} & a_{3} & a_{4} & a_{5} \\ a_{3} & a_{4} & a_{5} & a_{6} \\ a_{4} & a_{5} & a_{6} & a_{7} \end{matrix}| \\ = & a_{4} \{a_{4} (a_{3} a_{5} - a_{4}^{2}) - a_{5} (a_{2} a_{5} - a_{3} a_{4}) + a_{6} (a_{4} - a_{2} a_{3})\} \\ - a_{5} \{a_{3} (a_{3} a_{5} - a_{4}^{2}) - a_{5} (a_{5} - a_{2} a_{4}) + a_{6} (a_{4} - a_{2} a_{3})\} \\ + a_{6} \{a_{3} (a_{2} a_{5} - a_{3} a_{4}) - a_{4} (a_{5} - a_{2} a_{4}) + a_{6} (a_{3} - a_{2}^{2})\} \\ - a_{7} \{a_{3} (a_{2} a_{4} - a_{3}^{2}) - a_{4} (a_{4} - a_{2} a_{3}) + a_{5} (a_{3} - a_{2}^{2})\} . \end{matrix}

In 1966, Pommerenke [2,3] studied the Hankel determinants for univalent functions, p-valent functions, and starlike functions. In 1976, Noonan and Thomas [4] analyzed the second Hankel determinants for p-valent functions. In

1983,

Noor [5] studied the Hankel determinants for close-to-convex univalent functions. Then, in 1987, Noor [6] studied the Hankel determinants for functions with bounded boundary rotations, and she studied the same for higher-order close-to-convex functions in [7]. Ehrenborg [8] studied the Hankel determinants for exponential polynomials in 2000. Following his work, in 2001, Layman [9] thoroughly studied the Hankel transform, along with its properties. In 2007, Janteng et al. [10] studied the Hankel determinants for starlike and convex functions with respect to symmetric points. Inspired by the research being performed in this area, many researchers have contributed in developing some interesting and useful results, which include some of the following recent developments.

In 2019, Mahmood et al. [11] found the third-order Hankel determinant for a family of q-starlike functions, defined by a q-Ruscheweyh derivative operator, and Srivastava et al. [12] performed the same work for close-to-convex functions associated with Lemniscate of Bernoulli. Arif et al. [13] found the bound of the third Hankel determinant for functions connected with the sine function, and Srivastava et al. [14] found the same bound of Hankel and Toeplitz determinants for q-starlike functions associated with the generalized conic domain. In 2020, Shafiq et al. [15] investigated the bound of the third Hankel determinant for q-starlike functions connected with k-Fibonacci numbers. Murugusundaramoorthy and Bulboacă [16] found the upper bound of Hankel determinants for certain analytic functions connected with the shell-shaped region. In 2021, Guney and Korfeci [17] studied the fourth-order Hankel determinant for analytic functions, which are defined by using the modified sigmoid function, Zhang and Tang [18] found the same bound for functions connected with the sine function, Srivastava et al. [19] investigated third Hankel for q-starlike functions associated with q-analogue of the exponential function, and Saliu and Noor [20] studied third Hankel for analytic functions which are defined by using the Sălăgean differential and Komatu integral operators. Recently, in 2022, Raza et al. [21] studied Hankel determinants for starlike functions connected with symmetric Booth Lemniscate, Khan et al. [22] found the bound of third-order Hankel determinants for logarithmic coefficients of starlike functions connected with Sine function, and Riaz et al. [23,24,25] studied the Hankel determinants for starlike and convex functions associated with the sigmoid function, lune, and cardioid domain.

Now, we intend to find the upper bound of the third-order Hankel determinant for a subclass of starlike univalent functions, denoted by

S_{q s}^{*}

, which is defined below. Also, certain coefficient inequalities named Zalcman and generalized Zalcman inequalities are also part of our investigations. Before introducing the class

S_{q s}^{*},

we need to know about some preliminary concepts, which are stated as follows.

A function f is called univalent in

E

if there exists a one-to-one correspondence between

E

and

f (E)

. That is, for

z_{1}, z_{2} \in E,

f (z_{1}) = f (z_{2})

leads to

z_{1} = z_{2} .

The class

S

consists of functions that are not only analytic, but univalent as well in

E

, and the conditions

f (0) = 0

f^{'} (0) = 1

normalize these functions. That means,

S = \{f \in A : f is univalent in E\} .

Starlikeness is a very important geometric property. To define a starlike domain, we join every point of the set with a fixed point through a straight line, and if all such straight lines lie entirely in that domain, then that domain is called starlike with respect to that fixed point to say,

w_{0} .

Geometrically speaking, if every point of the domain is visible from that fixed point

w_{0}

, then the domain will be starlike or star-shaped with respect to

w_{0} .

The function that maps

E

onto a domain that is starlike and whose fixed point is the origin is called the starlike function. Furthermore, all those functions of the class

S

that satisfy the condition

ℜ (\frac{z f^{'} (z)}{f (z)}) > 0, z \in E

form the class

S^{*}

of starlike univalent functions. The next important class is the class

P

whose elements are all those functions p, which is analytic, and the condition that normalizes these functions is

p (0) = 1

, such that

ℜ p (z) > 0, z \in E .

That is,

P = \{p : p (0) = 1 and ℜ p (z) > 0, z \in E\}

and having Taylor series expansion of the form

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

(4)

Let w be an analytic function in

E

, and it is called the Schwarz function, if

w (0) = 0

, such that

|w (z)| < 1

for

z \in E .

Let

f (z)

and

g (z)

be analytic functions in

E

, and if a Schwarz function w exists in

E

such that

f (z) = g (w (z)), z \in E,

(5)

then f is said to subordinate g and is denoted by

f ≺ g .

If the function g is univalent in

E

and

f (0) = g (0)

, then

f (E) \subset g (E) .

For more details, see [26,27,28]. By using the relation (5) one can write the formation of classes

P

and

S^{*}

as follows.

P = \{p : p (0) = 1 and p (z) ≺ \frac{1 + z}{1 - z}, z \in E\},

S^{*} = \{f \in S : \frac{z f^{'} (z)}{f (z)} ≺ \frac{1 + z}{1 - z}, z \in E\} .

Quantum calculus, also known as q-calculus, is just the same as classical calculus, but with a major difference, in that unlike the classical form, we do not use limits in q-calculus. We define derivatives as differences and antiderivatives as sums. By definition, the q-derivative of a function f, which is particularly complex-valued and is defined in the domain

D,

is given as follows

(D_{q} f) (z) = \{\begin{matrix} \frac{f (z) - f (q z)}{(1 - q) z}, & z \neq 0, \\ f^{'} (0), & z = 0, \end{matrix}

(6)

where

0 < q < 1 .

Also, we see that

lim_{q \to 1^{-}} (D_{q} f) (z) = lim_{q \to 1^{-}} \frac{f (z) - f (q z)}{(1 - q) z} = f^{'} (z),

provided that the function f is differentiable in domain

D .

The Maclaurin’s series representation of the function

D_{q} f

given in (1) is given by

D_{q} f (z) = \sum_{n = 0}^{\infty} {[n]}_{q} a_{n} z^{n - 1},

where

{[n]}_{q} = \{\begin{matrix} \frac{1 - q^{n}}{1 - q}, & if n \in C, \\ \sum_{n = 0}^{n - 1} q^{n}, & if n \in N . \end{matrix}

(7)

For more details, see [29,30]. To take a brief overview of the applications of q-calculus, we observe mainly that the q-derivative operator

D_{q}

is an important tool that is used to define and to thoroughly investigate the numerous subclasses of analytic functions. Similarly, using this derivative operator, a q-extension of the class of starlike functions was firstly given in [31]. However, the development that provided a strong foundation for the application of the q-calculus in the context of Geometric Function Theory was presented by Srivastava, and he achieved this by introducing the basic (or q-) hypergeometric functions; for details, see [32]. To access the recent work on q-derivative, we refer to the following recent developments.

Mahmood et al. [33] studied the coefficient problems of q-starlike functions associated with conic domains. Mahmood et al. [34] studied the geometric properties of certain analytic functions that are defined by using the q-integral operator. Raza et al. [35] studied the q-analogue of differential subordinations by considering the Janowski functions and Lemniscate of Bernoulli. Zainab et al. [36] studied q-starlike functions defined by the q-version of the Ruscheweyh differential operator. Riaz et al. [37] studied the q-starlike functions of negative order, and Saliu et al. [38] studied q-symmetric starlike functions of Janowski type. Moreover, to access the fractional version of certain derivatives like the Caputo fractional derivative, and the conformable fractional derivative, see [39,40] and the references therein. Motivated by this referred work, we now introduce the class

S_{q s}^{*}

of q-starlike functions associated with the q-analogue of the function

1 + sin (q z),

as follows.

Definition 1.

A function

f \in S

is said to be in the class

S_{q s}^{*},

if it satisfies the following condition

\frac{z D_{q} f (z)}{f (z)} ≺ 1 + sin (q z), z \in E .

(8)

That is,

S_{q s}^{*} = \{f \in S : \frac{z D_{q} f (z)}{f (z)} ≺ 1 + sin (q z), z \in E\} .

(9)

The class

S_{q s}^{*}

generalizes the class

S_{s}^{*}

of starlike functions associated with the function

1 + \sin (z)

and

lim_{q ⟶ 1^{-}} S_{q s}^{*} \equiv S_{s}^{*} .

The class

S_{s}^{*}

was introduced and studied by Cho et al. [41].

Now, we proceed to discuss certain coefficient problems for the class

S_{q s}^{*},

for which we need the following lemmas.

Lemma 1

([42,43]). If

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

then

2 c_{2} = c_{1}^{2} + x (4 - c_{1}^{2})

for some x

(|x| \leq 1)

and

4 c_{3} = c_{1}^{3} + 2 (4 - c_{1}^{2}) c_{1} x - (4 - c_{1}^{2}) c_{1} x^{2} + 2 (4 - c_{1}^{2}) (1 - {|x|}^{2}) z

for some z

(|z| \leq 1) .

Lemma 2.

Let the function

p \in P

be given by (4) Then,

|c_{3} - 2 B c_{1} c_{2} + D c_{1}^{3}| \leq 2

0 \leq B \leq 1

and

B (2 B - 1) \leq D \leq B .

Lemma 3

([43]). Let the function

p \in P

be given by (4) Then,

|c_{n}| \leq 2 (n \in N) .

(10)

and the inequality is sharp. Also,

|c_{n} - μ c_{k} c_{n - k}| \leq 2, n > k, μ \in [0, 1] .

(11)

Lemma 4

([44]). Let the function

p \in P

be given by (4),

0 < a < 1

0 < α < 1

and

8 a (1 - a) {{(α β - 2 λ)}^{2} + {(α (a + α) - β)}^{2}} + α (1 - α) {(β - 2 a α)}^{2} \leq 4 α^{2} a {(1 - α)}^{2} (1 - a) .

Then,

|λ c_{1}^{4} + a c_{2}^{2} + 2 α c_{1} c_{3} - \frac{3}{2} β c_{1}^{2} c_{2} - c_{4}| \leq 2 .

(12)

Lemma 5

([45]). Let

\bar{E} : = {z \in C : | z | \leq 1}

, and for real numbers A, B, C, let

Y (A, B, C) : = max \{| A + B x + C x^{2} {| + 1 - | x |}^{2} : x \in \bar{E}\} .

(13)

A C \geq 0,

then

Y (A, B, C) = \{\begin{matrix} | A | + | B | + | C |, & | B | \geq 2 (1 - | C |), \\ 1 + | A | + \frac{B^{2}}{4 (1 - | C |)}, & | B | < 2 (1 - | C |) . \end{matrix}

By making use of the above lemmas, we contribute to the study of coefficient problems by investigating the following for the class

S_{q s}^{*} .

First, four coefficient bounds $|a_{n}|, n = 2, 3, 4, 5 .$
The Zalcman inequality $|a_{n}^{2} - a_{2 n - 1}| \leq {(n - 1)}^{2}$ for $n = 2 .$
The generalized Zalcman inequality $|a_{n} a_{m} - a_{n + m - 1}| \leq (n - 1) (m - 1)$ for certain values of m and n.
The upper bounds of the second Hankel $|H_{2, 2} (f)|$ and the third Hankel determinant $|H_{3, 1} (f)| .$

2. Main Results

Theorem 1.

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

has the series form as given in (1), then

|a_{n}| \leq \frac{1}{\sum_{j = 0}^{n - 2} q^{j}}, 0 < q < 1, n = 2, 3, 4

(14)

and

|a_{5}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}}, 0 < q \leq 0.8651682397 .

The result is sharp.

Proof.

f \in S_{q s}^{*},

then from (5) and (8)

\frac{z D_{q} f (z)}{f (z)} = 1 + sin (q w (z)), z \in E,

(15)

where

w (z) = \frac{p (z) - 1}{1 + p (z)} .

If p follows the form of (4) then

w (z) = \frac{c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots}{2 + c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots} .

Using this, one can have

\begin{matrix} 1 + sin (q w (z)) & = & 1 + sin (q (\frac{c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots}{2 + c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \dots})) \\ = & 1 + q \frac{c_{1}}{2} z + q (\frac{c_{2}}{2} - \frac{c_{1}^{2}}{4}) z^{2} + (\frac{- q^{3} c_{1}^{3}}{48} + q (- \frac{c_{1} c_{2}}{4} + \frac{c_{3}}{2} - \frac{c_{1} c_{2}}{4} + \frac{c_{1}^{3}}{8})) z^{3} \\ + (q (- \frac{c_{2}^{2}}{4} + \frac{c_{1}^{2} c_{2}}{8} - \frac{c_{1} c_{3}}{2} + \frac{c_{4}}{2} + \frac{c_{1}^{2} c_{2}}{4} - \frac{c_{1}^{4}}{8}) - \frac{q^{3}}{8} (\frac{c_{1}^{2} c_{2}}{8} - \frac{c_{1}^{4}}{8})) z^{4} + \dots . \end{matrix}

(16)

Now, from (1) and (6), we consider

\begin{matrix} \frac{z D_{q} f (z)}{f (z)} & = & \frac{z \{(z + a_{2} z^{2} + a_{3} z^{3} + . . .) - (q z + q a_{2} z^{2} + q a_{3} z^{3} + \dots)\}}{z (1 - q) (z + a_{2} z^{2} + a_{3} z^{3} + . . .)} \\ = & 1 + a_{2} q z + \{q (q + 1) a_{3} - q a_{2}^{2}\} z^{2} + \{q (\sum_{j = 0}^{2} q^{j}) a_{4 -} q (2 + q) a_{2} a_{3} + q a_{2}^{3}\} z^{3} \\ + q \{a_{5} q^{3} + (- a_{2} a_{4} + a_{5}) q^{2} + (- a_{3}^{2} + a_{3} a_{2}^{2} - a_{2} a_{4} + a_{5}) q \\ + 3 a_{3} a_{2}^{2} - 2 a_{2} a_{4} - a_{2}^{4} - a_{3}^{2} + a_{5}\} z^{4} + \dots \end{matrix}

(17)

Thus, by using the above series, (15) takes the form

\begin{matrix} 1 + a_{2} q z + \{q (q + 1) a_{3} - q a_{2}^{2}\} z^{2} + \{q (\sum_{j = 0}^{2} q^{j}) a_{4 -} q (2 + q) a_{2} a_{3} + q a_{2}^{3}\} z^{3} + \\ + q \{a_{5} q^{3} + (- a_{2} a_{4} + a_{5}) q^{2} + (- a_{3}^{2} + a_{3} a_{2}^{2} - a_{2} a_{4} + a_{5}) q + 3 a_{3} a_{2}^{2} \\ - 2 a_{2} a_{4} - a_{2}^{4} - a_{3}^{2} + a_{5}\} z^{4} + \dots \\ = & 1 + q \frac{c_{1}}{2} z + q (\frac{c_{2}}{2} - \frac{c_{1}^{2}}{4}) z^{2} + (\frac{- q^{3} c_{1}^{3}}{48} + q (- \frac{c_{1} c_{2}}{4} + \frac{c_{3}}{2} - \frac{c_{1} c_{2}}{4} + \frac{c_{1}^{3}}{8})) z^{3} + \\ (q (- \frac{c_{2}^{2}}{4} + \frac{c_{1}^{2} c_{2}}{8} - \frac{c_{1} c_{3}}{4} + \frac{c_{4}}{2} + \frac{c_{1}^{2} c_{2}}{4} - \frac{c_{1} c_{3}}{4} - \frac{c_{1}^{4}}{8}) - \frac{q^{3}}{8} (\frac{c_{1}^{2} c_{2}}{8} - \frac{c_{1}^{4}}{8})) z^{4} \\ + \dots . \end{matrix}

(18)

The comparison of coefficients of

z,

z^{2},

z^{3}

, and

z^{4},

together with precise computation, yields the following:

\begin{matrix} a_{2} & = & \frac{c_{1}}{2}, \end{matrix}

(19)

\begin{matrix} a_{3} & = & \frac{c_{2}}{2 (q + 1)}, \end{matrix}

(20)

\begin{matrix} a_{4} & = & - \frac{q^{2} c_{1}^{3}}{48 (\sum_{j = 0}^{2} q^{j})} + \frac{c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} - \frac{q}{4 (q + 1) (\sum_{j = 0}^{2} q^{j})} c_{1} c_{2}, \end{matrix}

(21)

and

\begin{matrix} a_{5} & = & \frac{- 1}{2 (\sum_{j = 0}^{3} q^{j})} [- \frac{2 q^{4} + 2 q^{3} + q^{2}}{48 (\sum_{j = 0}^{2} q^{j})} c_{1}^{4} + \frac{q c_{2}^{2}}{2 (q + 1)} + \frac{q (q + 1) c_{1} c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} \\ - \frac{q^{2} (1 - q - q^{2}) c_{1}^{2} c_{2}}{8 (\sum_{j = 0}^{2} q^{j})} - c_{4}] . \end{matrix}

(22)

Applying (10) to (19) and (20), we obtain

|a_{2}| \leq 1

and

|a_{3}| \leq \frac{1}{q + 1} .

Now consider,

|a_{4}| = \frac{1}{2 (\sum_{j = 0}^{2} q^{j})} |c_{3} - \frac{q}{2 (q + 1)} c_{1} c_{2} - \frac{q^{2} c_{1}^{3}}{24}| .

Assuming the values

B = \frac{q}{4 (q + 1)}

and

D = - \frac{q^{2}}{24}

, which satisfy

B (2 B - 1) \leq D \leq B

for the application of Lemma 2, we get

|a_{4}| \leq \frac{1}{q^{2} + q + 1} = \frac{1}{\sum_{j = 0}^{2} q^{j}} .

Now, from (22) consider,

\begin{matrix} |a_{5}| & = & |\frac{1}{\sum_{j = 0}^{3} q^{j}} [\frac{2 q^{4} + 2 q^{3} + q^{2}}{96 (\sum_{j = 0}^{2} q^{j})} c_{1}^{4} - \frac{q c_{2}^{2}}{4 (q + 1)} - \frac{q (q + 1) c_{1} c_{3}}{4 (\sum_{j = 0}^{2} q^{j})} \\ - \frac{(- q^{5} - 2 q^{4} + q^{2}) c_{1}^{2} c_{2}}{8 q (q + 1) (\sum_{j = 0}^{2} q^{j})} + \frac{c_{4}}{2}]| \\ = & |- \frac{1}{2 (\sum_{j = 0}^{3} q^{j})}| |- \frac{2 q^{4} + 2 q^{3} + q^{2}}{48 (\sum_{j = 0}^{2} q^{j})} c_{1}^{4} + \frac{q c_{2}^{2}}{2 (q + 1)} \\ + \frac{q (q + 1) c_{1} c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} - \frac{(q^{5} + 2 q^{4} - q^{2}) c_{1}^{2} c_{2}}{4 q (q + 1) (\sum_{j = 0}^{2} q^{j})} - c_{4}| \\ = & \frac{1}{2 (\sum_{j = 0}^{3} q^{j})} |λ c_{1}^{4} + a c_{2}^{2} + 2 α c_{1} c_{3} - \frac{3}{2} β c_{1}^{2} c_{2} - c_{4}|, \end{matrix}

where

\begin{matrix} λ & : & = - \frac{2 q^{4} + 2 q^{3} + q^{2}}{48 (\sum_{j = 0}^{2} q^{j})}, a : = \frac{q}{2 (q + 1)}, \\ α & : & = \frac{q (q + 1)}{4 (\sum_{j = 0}^{2} q^{j})}, β : = \frac{q^{2} (1 - q - q^{2})}{12 (\sum_{j = 0}^{2} q^{j})} . \end{matrix}

We see that

0 < a < 1

0 < α < 1

. Now,

\begin{matrix} 8 a (1 - a) {{(α β - 2 λ)}^{2} + {(α (a + α) - β)}^{2}} + α (1 - α) {(β - 2 a α)}^{2} - 4 α^{2} a {(1 - α)}^{2} (1 - a) & = \\ \frac{\begin{matrix} (53 q^{11} + 318 q^{10} + 589 q^{9} + 2128 q^{8} + 3210 q^{7} + 3304 q^{6} \\ + 1867 q^{5} - 452 q^{4} - 2091 q^{3} - 2118 q^{2} - 1152 q - 288) q^{3} \end{matrix}}{2304 {(q + 1)}^{2} {(\sum_{j = 0}^{2} q^{j})}^{4}} & = & Ψ (q) \end{matrix}

A calculation shows that the equation

Ψ (q) \leq 0

when

0 < q \leq 0.8651682397 .

Hence,

Ψ (q) \leq 0 .

Now, by using Lemma 4, we can have

|a_{5}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}} .

For sharpness, consider the function

f_{n} : E \to C

, defined by

\frac{z D_{q} f_{n} (z)}{f_{n} (z)} = 1 + sin (q z^{n}), n = 2, 3, 4, 5 .

(23)

Then, it is easy to see that the function

f_{n} \in S_{q s}^{*} .

A simple calculations shows that

\begin{matrix} f_{2} (z) & = & z + \frac{q}{{[2]}_{q} - 1} z^{2} + \dots, \\ f_{3} (z) & = & z + \frac{q}{{[3]}_{q} - 1} z^{3} + \frac{q^{2}}{- {[3]}_{q} + {[5]}_{q} {[3]}_{q} + 1 - {[5]}_{q}} z^{5} + \dots, \\ f_{4} (z) & = & z + \frac{q}{{[4]}_{q} - 1} z^{4} + \dots, \\ f_{5} (z) & = & z + \frac{q}{{[5]}_{q} - 1} z^{5} + \dots . \end{matrix}

Hence, by using (7) the result is sharp. □

Considering

q ⟶ 1^{-}

in the above result, we obtain the following improved result than the one proven in [46].

Corollary 1.

f \in S_{s}^{*}

has the series form as given in (1), then,

|a_{n}| \leq \frac{1}{n - 1}, n = 2, 3, 4 .

3. Zalcman and Generalized Zalcman Conjecture

In 1960, Zalcman proposed a remarkable conjecture for univalent functions whose generalized version were given by Ma [47] in 1999. These conjectures are still open but have been proven for certain subclasses of univalent functions. Zalcman’s conjecture states that every f

\in S

having the form of (1) satisfies the following sharp inequality.

|a_{n}^{2} - a_{2 n - 1}| \leq {(n - 1)}^{2}, n \geq 2 .

(24)

Whereas, the generalized Zalcman conjecture states that the Taylor series coefficients from (1) of univalent functions f

\in S

satisfy the following inequality.

|a_{n} a_{m} - a_{n + m - 1}| \leq (n - 1) (m - 1), \forall m, n \in N, n \geq 2, m \geq 2 .

(25)

We intend to find these inequalities for the considered class

S_{q s}^{*}

for certain values of n and

m .

For

n = 2,

the inequality (24) takes the form

|a_{2}^{2} - a_{3}| \leq 1 .

Theorem 2.

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

has the series form as given in (1) Then,

|a_{3} - a_{2}^{2}| \leq \frac{1}{q + 1} .

(26)

The above inequality is sharp, which can be obtained with the function

f_{3},

given in (23).

Proof.

From (19) and (20), consider

|a_{3} - a_{2}^{2}| = \frac{1}{2 (q + 1)} |c_{2} - \frac{(q + 1) c_{1}^{2}}{2}| = \frac{1}{2 (q + 1)} |c_{2} - v c_{1}^{2}|,

where

v = (q + 1) / 2 .

Since

q \in (0, 1),

therefore,

0 < v < 1 .

Now, by using (11) for

n = 2

and

k = 1,

we obtain (26). □

Upon letting

q ⟶ 1^{-}

, the above result reduces to the following, proven in [46].

Corollary 2.

f \in S_{s}^{*}

has the series form as given in (1), then,

|a_{3} - a_{2}^{2}| \leq \frac{1}{2} .

For

n = 3, m = 2,

the inequality (25) reduces to

|a_{2} a_{3} - a_{4}| \leq 2 .

We discuss it as follows:

Theorem 3.

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

has the series form as given in (1) then,

|a_{4} - a_{3} a_{2}| \leq \frac{1}{q^{2} + q + 1}, q \in (0, \frac{\sqrt{3}}{2}) .

(27)

The result is sharp for the function

f_{4}

, given in (23).

Proof.

From (19), (20) and (21), consider

\begin{matrix} a_{4} - a_{3} a_{2} & = & - \frac{q^{2} c_{1}^{3}}{48 (\sum_{j = 0}^{2} q^{j})} + \frac{c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} - \frac{q}{4 (q + 1) (\sum_{j = 0}^{2} q^{j})} c_{1} c_{2} - \frac{c_{1} c_{2}}{4 (q + 1)}, \\ = & - \frac{q^{2} c_{1}^{3}}{48 (\sum_{j = 0}^{2} q^{j})} + \frac{c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} - \frac{(q^{2} + 2 q + 1)}{4 (q + 1) (\sum_{j = 0}^{2} q^{j})} c_{1} c_{2} . \end{matrix}

Taking the modulus, we get

\begin{matrix} |a_{4} - a_{3} a_{2}| & = & |- \frac{q^{2} c_{1}^{3}}{48 (\sum_{j = 0}^{2} q^{j})} + \frac{c_{3}}{2 (\sum_{j = 0}^{2} q^{j})} - \frac{(q^{2} + 2 q + 1)}{4 (q + 1) (\sum_{j = 0}^{2} q^{j})} c_{1} c_{2}| \\ = & |\frac{1}{2 (\sum_{j = 0}^{2} q^{j})}| |c_{3} - \frac{(q^{2} + 2 q + 1)}{2 (q + 1)} c_{1} c_{2} - \frac{q^{2} c_{1}^{3}}{24}| . \end{matrix}

Assuming the values

B = \frac{(q^{2} + 2 q + 1)}{2 (q + 1)}

and

D = - \frac{q^{2}}{24}

, we see that

B (2 B - 1) - D = \frac{1}{24} (4 q^{2} - 3) < 0, for q \in (0, \frac{\sqrt{3}}{2})

which shows that

B (2 B - 1) \leq D \leq B

. Thus, the application of Lemma 2 gives that

|a_{4} - a_{3} a_{2}| \leq \frac{1}{q^{2} + q + 1} .

□

Now, the following results investigate the inequality (24) for

m = n = 3 .

Theorem 4.

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

has the series form as given in (1) then,

|a_{3}^{2} - a_{5}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}}, q \in (0, 0.3898584501) .

The result is sharp for the function

f_{5}

, given in (23).

Proof.

From (20) and (22), consider

\begin{matrix} |a_{3}^{2} - a_{5}| & = & |{(\frac{c_{2}}{2 (q + 1)})}^{2} - (\frac{1}{\sum_{j = 0}^{3} q^{j}}) (\frac{2 q^{4} + 2 q^{3} + q^{2}}{96 (\sum_{j = 0}^{2} q^{j})} c_{1}^{4} - \\ - \frac{q c_{2}^{2}}{4 (q + 1)} - \frac{q (q + 1) c_{1} c_{3}}{4 (\sum_{j = 0}^{2} q^{j})} - \frac{(q^{5} + 2 q^{4} + q^{2}) c_{1}^{2} c_{2}}{8 q (q + 1) (\sum_{j = 0}^{2} q^{j})} + \frac{c_{4}}{2})| \\ = & \frac{1}{2 (\sum_{j = 0}^{3} q^{j})} |(- \frac{q^{2} (1 + 2 q + 2 q^{2})}{48 (\sum_{j = 0}^{2} q^{j})} c_{1}^{4} + \frac{q^{2} + q + 1}{2 (q + 1)} c_{2}^{2} \\ + \frac{q (q + 1)}{2 (\sum_{j = 0}^{2} q^{j})} c_{1} c_{3} + \frac{q^{2} (q^{2} + q - 1)}{8 (\sum_{j = 0}^{2} q^{j})} c_{1}^{2} c_{2} - c_{4})| \\ = & \frac{1}{2 (\sum_{j = 0}^{3} q^{j})} |λ c_{1}^{4} + a c_{2}^{2} + 2 α c_{1} c_{3} - \frac{3}{2} β c_{1}^{2} c_{2} - c_{4}|, \end{matrix}

where

\begin{matrix} λ & : & = - \frac{q^{2} (1 + 2 q + 2 q^{2})}{48 (\sum_{j = 0}^{2} q^{j})}, a : = \frac{q^{2} + q + 1}{2 (q + 1)}, \\ α & : & = \frac{q (q + 1)}{4 (\sum_{j = 0}^{2} q^{j})}, β : = - \frac{q^{2} (q^{2} + q - 1)}{8 (\sum_{j = 0}^{2} q^{j})} . \end{matrix}

We see that

0 < a < 1

0 < α < 1

. Now,

\begin{matrix} 8 a (1 - a) {{(α β - 2 λ)}^{2} + {(α (a + α) - β)}^{2}} + α (1 - α) {(β - 2 a α)}^{2} - 4 α^{2} a {(1 - α)}^{2} (1 - a) & = \\ \frac{\begin{matrix} q^{2} (- 770 q^{3} - 4011 q^{4} - 8430 q^{5} - 11127 q^{6} - 10282 q^{7} - 6180 q^{8} \\ - 1810 q^{9} + 845 q^{10} + 72 + 1340 q^{11} + 835 q^{12} + 296 q^{13} + 50 q^{14} + 348 q + 416 q^{2}) \end{matrix}}{2304 {(q + 1)}^{2} {(\sum_{j = 0}^{2} q^{j})}^{4}} & = & Ψ_{1} (q) \end{matrix}

A calculation shows that the equation

Ψ_{1} (q) \leq 0

when

0 < q \leq 0.3898584501 .

Hence,

Ψ_{1} (q) \leq 0 .

Now by using Lemma 4, we can have

|a_{3}^{2} - a_{5}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}}

and

\frac{1}{\sum_{j = 0}^{3} q^{j}},

for

q \in (0, 0.3898584501) .

This shows that the inequality (24) is satisfied for

n = 3 .

□

The following result investigates the inequality (25) for

n = 4, m = 2 .

Theorem 5.

If the function

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

has the form (1) then,

|a_{4} a_{2} - a_{5}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}}, q \in (0, 0.1889972572] .

Proof.

From (19), (21), and (22), consider

\begin{matrix} |a_{5} - a_{2} a_{4}| & = & \frac{1}{2 (\sum_{j = 0}^{3} q^{j})} |\frac{- q^{2} (q + 2)}{48} c_{1}^{4} + \frac{q c_{2}^{2}}{2 (q + 1)} + \frac{(q + 1) c_{1} c_{3}}{2} + \frac{q (q - 2) c_{1}^{2} c_{2}}{8} - c_{4}| \\ = & \frac{1}{2 (\sum_{j = 0}^{3} q^{j})} |λ c_{1}^{4} + a c_{2}^{2} + 2 α c_{1} c_{3} - \frac{3}{2} β c_{1}^{2} c_{2} - c_{4}|, \end{matrix}

where

\begin{matrix} λ & : & = \frac{- q^{2} (q + 2)}{48}, a : = \frac{q}{2 (q + 1)}, \\ α & : & = \frac{(q + 1)}{4}, β : = \frac{q (2 - q)}{12} . \end{matrix}

We see that

0 < a < 1

0 < α < 1

. Now,

\begin{matrix} 8 a (1 - a) {{(α β - 2 λ)}^{2} + {(α (a + α) - β)}^{2}} + α (1 - α) {(β - 2 a α)}^{2} - 4 α^{2} a {(1 - α)}^{2} (1 - a) & = \\ \frac{q (40 q^{5} - 146 q^{4} + 82 q^{6} + 751 q^{3} + 17 q^{7} + 1930 q^{2} + 276 q - 126)}{2304 {(q + 1)}^{2}} & = & Ψ_{2} (q) \end{matrix}

A calculation shows that the equation

Ψ_{2} (q) \leq 0

when

0 < q \leq 0.1889972572 .

For the application of (12) we have

|a_{5} - a_{2} a_{4}| \leq \frac{1}{\sum_{j = 0}^{3} q^{j}},

which completes the proof. □

4. Hankel Determinants

Theorem 6.

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

has the series form as given in (1) then,

|H_{2, 2} (f)| \leq \frac{1}{{(q + 1)}^{2}} .

(28)

The result is sharp for the function

f_{3}

given in (23).

Proof.

From (19), (20), and (21), consider

H_{2, 2} (f) = a_{2} a_{4} - a_{3}^{2} = - \frac{q^{2} c_{1}^{4}}{96 (\sum_{j = 0}^{2} q^{j})} - \frac{q}{8 (q + 1) (\sum_{j = 0}^{2} q^{j})} c_{1}^{2} c_{2} + \frac{c_{1} c_{3}}{4 (\sum_{j = 0}^{2} q^{j})} - \frac{c_{2}^{2}}{4 {(q + 1)}^{2}} .

Since the class

S_{q s}^{*}

and the functional

H_{2, 2}

are rotationally invariant, we can consider that

c : = c_{1} \in [0, 2] .

Then, using Lemma 1 and after simplification, we obtain

H_{2, 2} (f) = \frac{1}{8 (\sum_{j = 0}^{2} q^{j})} (\begin{matrix} - \frac{q^{2} (q^{2} + 2 q + 7) c^{4}}{12 {(q + 1)}^{2}} + \frac{q (4 - c^{2}) (1 - q) x c^{2}}{2 {(q + 1)}^{2}} \\ - \frac{(q c^{2} + 4 (\sum_{j = 0}^{2} q^{j})) (4 - c^{2}) x^{2}}{2 {(q + 1)}^{2}} + c (4 - c^{2}) (1 - {|x|}^{2}) z \end{matrix})

where x and z are such that

| x | \leq 1

| z | \leq 1

(1) Suppose first that

c = 2

. Then,

| H_{2, 2} (f) | = \frac{q^{2} (q^{2} + 2 q + 7)}{6 {(q + 1)}^{2} (\sum_{j = 0}^{2} q^{j})} < \frac{1}{{(q + 1)}^{2}} .

(2) When

c = 0

| H_{2, 2} (f) | \leq \frac{| x |}{{(q + 1)}^{2}} \leq \frac{1}{{(q + 1)}^{2}} .

(3) Next, assume that

c \in (0, 2)

and the application of the triangle inequality gives

| H_{2, 2} (f) | \leq \frac{c (4 - c^{2})}{8 (\sum_{j = 0}^{2} q^{j})} Ψ (A, B, C),

where

Ψ (A, B, C) : = | A + B x + C x^{2} {| + 1 - | x |}^{2}, | x | \leq 1,

with

A : = - \frac{q^{2} (q^{2} + 2 q + 7) c^{3}}{12 {(q + 1)}^{2} (4 - c^{2})}, B : = \frac{q (1 - q) c}{2 {(q + 1)}^{2}}, C : = - \frac{(q c^{2} + 4 (\sum_{j = 0}^{2} q^{j}))}{2 c {(q + 1)}^{2}} .

Clearly,

A C = \frac{q^{2} (q c^{2} + 4 (\sum_{j = 0}^{2} q^{j})) (q^{2} + 2 q + 7) c^{2}}{24 {(q + 1)}^{4} (4 - c^{2})} > 0, c \in (0, 2), q \in (0, 1) .

We now show that

| B | \geq 2 (1 - | C |) .

For this, we define the function

φ (c) = 2 {(q + 1)}^{2} c (| B | - 2 (1 - | C |)) = q (3 - q) c^{2} - 4 (q^{2} + 2 q + 1) c + 8 (\sum_{j = 0}^{2} q^{j}) .

Now,

\begin{matrix} φ^{'} (c) & = & 2 q (3 - q) c - 4 (q^{2} + 2 q + 1), \\ φ^{″} (c) & = & 2 q (3 - q) > 0, q \in (0, 1) . \end{matrix}

This shows that the function

φ^{'}

is increasing and

max φ^{'} (c) = φ^{'} (2) = - 8 q^{2} + 4 q - 4 < 0, q \in (0, 1) .

This implies that

φ^{'} (c) < 0, c \in (0, 2), q \in (0, 1) .

Hence,

φ

is a decreasing function and

min φ (c) = φ (2) = 4 q (1 - q) > 0, q \in (0, 1) .

We conclude that

(| B | - 2 (1 - | C |)) > 0 .

Then, by Lemma 5,

| H_{2, 2} (f) | \leq \frac{c (4 - c^{2})}{8 (\sum_{j = 0}^{2} q^{j})} (| A | + | B | + | C |) = g (c),

(29)

where

g (c) : = - \frac{1}{96 {(q + 1)}^{2} (\sum_{j = 0}^{2} q^{j})} (\begin{matrix} (- 7 q^{2} - 2 q^{3} + 6 q (1 - q) + 6 q - q^{4}) c^{4} \\ + (- 24 q (1 - q) + 24 q^{2} + 24) c^{2} - 96 (1 + q + q^{2}) \end{matrix}) .

Now,

g^{'} (c) = \frac{1}{96 {(q + 1)}^{2} (\sum_{j = 0}^{2} q^{j})} (L c^{3} + M c),

where

\begin{matrix} L & = & - (- 28 q^{2} - 8 q^{3} + 24 q (1 - q) + 24 q - 4 q^{4}) < 0, q \in (0, 1), \\ M & = & - (48 - 48 q (1 - q) + 48 q^{2}) < 0, q \in (0, 1) . \end{matrix}

We conclude that g is a decreasing function and

g (c) \leq g (0) = \frac{1}{{(q + 1)}^{2}}, q \in (0, 1) .

Hence,

| H_{2, 2} (f) | \leq \frac{1}{{(q + 1)}^{2}} .

The result is sharp, which can be obtained with the function

f_{3},

given in (23) □

Upon letting

q ⟶ 1^{-}

, the above result reduces to the following, proven in [46].

Corollary 3.

f \in S_{s}^{*}

has the series form as given in (1), then,

|a_{2} a_{4} - a_{3}^{2}| \leq \frac{1}{4} .

Theorem 7.

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

has the series form as given in (1) then,

|H_{3, 1} (f)| \leq \frac{1}{(\sum_{j = 0}^{3} q^{j}) (q + 1)} + \frac{1}{{(\sum_{j = 0}^{2} q^{j})}^{2}} + \frac{1}{{(q + 1)}^{3}}, q \in (0, 0.86) .

(30)

The proof follows easily by using the inequalities of Theorem 1, Theorem 6, Theorem 3, and Theorem 2 in (3).

5. Conclusions

This work has introduced a new class

S_{q s}^{*}

of q-starlike functions linked with the q-analogue of the sine function through subordination relation. This class generalizes the class

S_{s}^{*}

of starlike functions. The investigations of certain coefficient inequalities like sharp coefficient bounds, the upper bound of the third-order Hankel determinant, Zalcman inequalities, and generalized Zalcman inequalities for the class

S_{q s}^{*}

have been included in this work. It also improves the coefficient bounds and the upper bound of the third-order Hankel determinant for the class

S_{s}^{*} .

Moreover, for future work, the defined class

S_{q s}^{*}

can be further investigated for finding the upper bounds of higher-order Hankel and Toeplitz determinants. Meanwhile, the obtained coefficient bounds may be used for studying many other unaddressed coefficient problems for this class.

Author Contributions

Conceptualization, Y.T., S.N.M. and Q.X.; methodology, Y.T., M.R. and Q.X.; software, S.Z.; validation, S.Z. and F.M.O.T.; formal analysis, M.R. and S.N.M.; investigation, Y.T., M.R. and Q.X.; resources, F.M.O.T.; data curation, F.M.O.T.; writing—original draft preparation, Y.T. and S.N.M.; writing—review and editing, S.N.M. and M.R.; visualization, F.M.O.T.; supervision, S.N.M.; project administration, S.Z.; funding acquisition, F.M.O.T. and S.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Acknowledgments

This work is supported by Researchers Supporting Project number (RSP2022R440), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare that there are no conflicts of interests regarding the publication of this article.

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Taj, Y.; Zainab, S.; Xin, Q.; Tawfiq, F.M.O.; Raza, M.; Malik, S.N. Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function. Symmetry 2022, 14, 2200. https://doi.org/10.3390/sym14102200

AMA Style

Taj Y, Zainab S, Xin Q, Tawfiq FMO, Raza M, Malik SN. Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function. Symmetry. 2022; 14(10):2200. https://doi.org/10.3390/sym14102200

Chicago/Turabian Style

Taj, Yusra, Saira Zainab, Qin Xin, Ferdous M. O. Tawfiq, Mohsan Raza, and Sarfraz Nawaz Malik. 2022. "Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function" Symmetry 14, no. 10: 2200. https://doi.org/10.3390/sym14102200

APA Style

Taj, Y., Zainab, S., Xin, Q., Tawfiq, F. M. O., Raza, M., & Malik, S. N. (2022). Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function. Symmetry, 14(10), 2200. https://doi.org/10.3390/sym14102200

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Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function

Abstract

1. Introduction and Preliminaries

2. Main Results

3. Zalcman and Generalized Zalcman Conjecture

4. Hankel Determinants

5. Conclusions

Author Contributions

Funding

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