Open AccessArticle

Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli

Hari M. Srivastava

^1,2

Qazi Zahoor Ahmad

Maslina Darus

⁴

Nazar Khan

^3,*,

Bilal Khan

Naveed Zaman

³ and

Hasrat Hussain Shah

⁵

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

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

⁴

School of Mathematical Sciences, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

⁵

Department of Mathematical Sciecnes, Balochistan University of Information Technology, Engineering and Management Sciences, Quetta 87300, Pakistan

Author to whom correspondence should be addressed.

Mathematics 2019, 7(9), 848; https://doi.org/10.3390/math7090848

Submission received: 26 June 2019 / Revised: 2 September 2019 / Accepted: 5 September 2019 / Published: 14 September 2019

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)

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Abstract

In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk

U

that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the third Hankel determinant. Various other related results are also considered.

Keywords:

analytic functions; close-to-convex functions; subordination; lemniscate of Bernoulli Hankel determinant

MSC:

primary 05A30, 30C45; secondary 11B65, 47B38

1. Introduction

H (U)

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

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

where

C

is the set of complex numbers. We also let

A

be the class of analytic functions having the following form:

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

(1)

and which are normalized by the following conditions:

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

We denote by

S

the class of functions in

A

, which are univalent in

U

A function

f \in A

is called starlike in

U

if it satisfies the following inequality:

ℜ (\frac{z f^{'} (z)}{f (z)}) > 0 (\forall z \in U) .

The class of all such functions is denoted by

S^{*} .

For

f \in S^{*},

one can find that (see [1]):

|a_{n}| ≦ n for n = 2, 3, \dots

(2)

Next, by

K

, we denote the class of close-to-convex functions in

U

that satisfy the following inequality:

ℜ (\frac{z f^{'} (z)}{g (z)}) > 0 (\forall z \in U),

for some

g \in

S^{*} .

An example of a function, which is close-to-convex in

U

, is given by:

F (z) = \frac{z - e^{2 i α} cos α z^{2}}{{(1 - e^{i α} z)}^{2}} (0 < α < π)

which maps

U

onto the complex z-plane excluding a vertical slit (see [2] where some interesting properties of this function are obtained).

Moreover, by

{SL}^{*}

, we denote the class of functions

f \in A

that satisfy the following inequality:

|{(\frac{z f^{'} (z)}{f (z)})}^{2} - 1| < 1 (\forall z \in U) .

Thus a function

f \in {SL}^{*}

is such that

\frac{z f^{'} (z)}{f (z)}

lies in the region bounded by the right half of the lemniscate of Bernoulli given by the following relation:

|w^{2} - 1| < 1,

where

w = \frac{z f^{'} (z)}{f (z)} .

The above defined class was introduced by Sokół et al. (see [3]) and studied by the many authors (see, for example, [4,5,6]).

Next, if two functions f and g are analytic in

U

, we say that the function f is subordinate to the function g and write:

f ≺ g or f (z) ≺ g (z),

if there exists a Schwarz function

w (z)

that is analytic in

U

with:

w (0) = 0 and |w (z)| < 1,

such that:

f (z) = g (w (z)) .

Furthermore, if the function g is univalent in

U

, then we have the following equivalence (see, for example, [7]; see also [8]):

f (z) ≺ g (z) (z \in U) ⇄ f (0) = g (0) and f (U) \subset g (U) .

We next denote by

P

the class of analytic functions p which are normalized by

p (0) = 1

and have the following form:

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

(3)

such that:

ℜ (p (z)) > 0 (\forall z \in U) .

In recent years, several interesting subclasses of analytic and multivalent functions have been introduced and investigated (see, for example, [9,10,11,12,13,14,15,16]). Motivated and inspired by recent and ongoing research, we introduce and investigate here a new subclass of close-to-convex functions in

U

which are associated with the lemniscate of Bernoulli by using some techniques similar to those that were used earlier by Sokół and Stankiewicz (see [3]).

Definition 1.

A function f of the form of Equation (1) is said to be in the class

{KL}^{*}

if and only if:

|{(\frac{z f^{'} (z)}{g (z)})}^{2} - 1| < 1

(4)

for some

g \in S^{*}

. Equivalently, we have:

\frac{z f^{'} (z)}{g (z)} ≺ \sqrt{1 + z} (\forall z \in U)

for some

g \in S^{*} .

Thus, clearly, a function

f \in {KL}^{*}

is such that

\frac{z f^{'} (z)}{g (z)}

lies in the region bounded by the right half of the lemniscate of Bernoulli given by the following relation:

|w^{2} - 1| < 1 .

A closer look at the above series development of f suggests that many properties of the function f may be affected (or implied) by the size of its coefficients. The coefficient problem has been reformulated in the more special manner of estimating

|a_{n}|

, that is, the modulus of the nth coefficient. In 1916, Bieberbach conjectured that the nth coefficient of a univalent function is less or equal to that of the Koebe function.

Closely related to the Bieberbach conjecture is the problem of finding sharp estimates for the coefficients of odd univalent functions, which has the most general form of the square-root transformation of a function

f \in S :

l (z) = \sqrt{f (z^{2})} = z + c_{3} z^{3} + c_{5} z^{5} \dots

For odd univalent functions, Littlewood and Parley in 1932 proved that, for each postive integer n, the modulus

|c_{2 n + 1}|

is less than an absolute constant M. For

M = 1

, the bound becomes the Littlewood–Parley conjecture.

Let

n ≧ 0

and

q ≧ 1

. Then the qth Hankel determinant is defined as follows:

H_{q} (n) = |\begin{array}{l} a_{n} & a_{n + 1} & . & . & . & a_{n + q - 1} \\ a_{n + 1} & . & . \\ . & . & . \\ . & . & . \\ . & . & . \\ a_{n + q - 1} & . & . & . & . & a_{n + 2 (q - 1)} \end{array}|

The Hankel determinant plays a vital role in the theory of singularities [17] and is useful in the study of power series with integer coefficients (see [18,19,20]). Noteworthy, several authors obtained the sharp upper bounds on

H_{2} (2)

(see, for example, [5,21,22,23,24,25,26,27,28,29]) for various classes of functions. It is a well-known fact for the Fekete-Szegö functional that:

|a_{3} - a_{2}^{2}| = H_{2} (1) .

This functional is further generalized as follows:

|a_{3} - μ a_{2}^{2}|

for some real or complex number

μ

. Fekete and Szegö gave sharp estimates of

|a_{3} - μ a_{2}^{2}|

for

μ

real and

f \in S

, the class of normalized univalent functions in

N

. It is also known that the functional

|a_{2} a_{4} - a_{3}^{2}|

is equivalent to

H_{2} (2)

. Babalola [30] studied the Hankel determinant

H_{3} (1)

for some subclasses of analytic functions. In the present investigation, our focus is on the Hankel determinant

H_{3} (1)

for the above-defined function class

{KL}^{*} .

2. A Set of Lemmas

Lemma 1.

(see [31]) Let:

p (z) = 1 + p_{1} z + p_{2} z^{2} + \dots

be in the class

P

of functions with positive real part in

U

. Then, for any number υ:

|p_{2} - υ p_{1}^{2}| \{\begin{matrix} - 4 υ + 2 & (υ 0) \\ 2 & (0 υ 1) \\ 4 υ - 2 & (υ 1) . \end{matrix}

(5)

When

υ < 0

υ > 1

, the equality holds true in Equation (5) if and only if:

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

or one of its rotations. If

0 < υ < 1,

then the equality holds true in Equation (5) if and only if:

p (z) = \frac{1 + z^{2}}{1 - z^{2}}

or one of its rotations. If

υ = 0,

the equality holds true in Equation (5) if and only if:

p (z) = (\frac{1 + ρ}{2}) \frac{1 + z}{1 - z} + (\frac{1 - ρ}{2}) \frac{1 - z}{1 + z} (0 ρ 1)

or one of its rotations. If

υ = 1,

then the equality in Equation (5) holds true if

p (z)

is a reciprocal of one of the functions such that the equality holds true in the case when

υ = 0 .

Lemma 2.

[32,33] Let:

p (z) = 1 + p_{1} z + p_{2} z^{2} + \dots

be in the class

P

of functions with positive real part in

U

. Then:

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

for some x

(|x| ≦ 1)

and:

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

for some z

(|z| ≦ 1) .

Lemma 3.

[1] Let:

p (z) = 1 + p_{1} z + p_{2} z^{2} + \dots

be in the class

P

of functions with positive real part in

U

. Then:

|p_{k}| ≦ 2 (k \in N) .

The inequality is sharp.

3. Main Results and Their Demonstrations

In this section, we will prove our main results.

Theorem 1.

Let

f \in {KL}^{*}

and be of the form of Equation (1). Then:

|a_{3} - μ a_{2}^{2}| ≦ \{\begin{matrix} \frac{1}{48} (62 - 75 μ) & (μ < \frac{38}{75}) \\ \frac{1}{2} & (\frac{38}{75} ≦ μ ≦ \frac{86}{75}) \\ \frac{1}{48} (75 μ - 62) & (μ > \frac{86}{75}) . \end{matrix}

It is asserted also that:

|a_{3} - μ a_{2}^{2}| + \frac{1}{3} (3 μ - \frac{38}{25}) {|a_{2}|}^{2} ≦ \frac{1}{2} (\frac{38}{75} < μ ≦ \frac{62}{75})

and:

|a_{3} - μ a_{2}^{2}| + \frac{1}{3} (\frac{86}{25} - 3 μ) {|a_{2}|}^{2} ≦ \frac{1}{2} (\frac{62}{75} < μ ≦ \frac{86}{75}) .

Proof.

f \in {KL}^{*}

, then it follows from definition that:

\frac{z f^{'} (z)}{g (z)} ≺ ϕ (z) (for some g \in S^{*}),

(6)

where:

ϕ (z) = {(1 + z)}^{\frac{1}{2}} .

Define a function

p (z)

by:

p (z) = \frac{1 + w (z)}{1 - w (z)} = 1 + p_{1} z + p_{2} z^{2} + \dots .

It is clear that

p (z) \in P

. This implies that:

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

In addition, from Equation (6), we have:

\frac{z f^{'} (z)}{g (z)} ≺ ϕ (z)

with:

ϕ (w (z)) = {(\frac{2 p (z)}{p (z) + 1})}^{\frac{1}{2}} .

We now have:

\begin{matrix} {(\frac{2 p (z)}{p (z) + 1})}^{\frac{1}{2}} & = 1 + \frac{1}{4} p_{1} z + [\frac{1}{4} p_{2} - \frac{5}{32} p_{1}^{2}] z^{2} + [\frac{1}{4} p_{3} - \frac{5}{16} p_{1} p_{2} + \frac{13}{128} p_{1}^{3}] z^{3} \\ + [\frac{1}{4} p_{4} - \frac{5}{16} p_{1} p_{3} + \frac{39}{128} p_{2} p_{1}^{2} - \frac{5}{32} p_{2}^{2} - \frac{141}{2048} p_{1}^{4}] z^{4} + \dots . \end{matrix}

Similarly, we get:

\begin{matrix} \frac{z f^{'} (z)}{g (z)} & = 1 + [2 a_{2} - b_{2}] z + [3 a_{3} - 2 a_{2} b_{2} - b_{3} + b_{2}^{2}] z^{2} \\ + [4 a_{4} - 2 a_{2} b_{3} - 3 a_{3} b_{2} + 2 b_{2} b_{3} + 2 a_{2} b_{2}^{2} - b_{4} - b_{2}^{3}] z^{3} + \dots . \end{matrix}

Therefore, upon comparing the corresponding coefficients and by using Equation (2), we find that:

\begin{matrix} a_{2} = \frac{5}{8} p_{1,} \end{matrix}

(7)

\begin{matrix} a_{3} = \frac{1}{4} p_{2} + \frac{19}{96} p_{1}^{2} \end{matrix}

(8)

\begin{matrix} a_{4} = \frac{7}{48} p_{3} + \frac{9}{64} p_{1} p_{2} + \frac{91}{1536} p_{1}^{3} . \end{matrix}

(9)

We thus obtain:

|a_{3} - μ a_{2}^{2}| = \frac{1}{4} |p_{2} - \frac{1}{48} (75 μ - 38) p_{1}^{2}| .

(10)

Finally, by applying Lemma 1 in conjunction with Equation (10), we obtain the result asserted by Theorem 1. □

Theorem 2.

Let

f \in {KL}^{*}

and be of the form of Equation (1). Then:

|a_{2} a_{4} - a_{3}^{2}| ≦ \frac{9105}{36416} .

(11)

Proof.

Making use of Equations (7)–(9), we have:

\begin{matrix} a_{2} a_{4} - a_{3}^{2} & = (\frac{35}{384} p_{1} p_{3} + \frac{45}{512} p_{1}^{2} p_{2} + \frac{455}{12288} p_{1}^{4}) - {(\frac{1}{4} p_{2} + \frac{19}{96} p_{1}^{2})}^{2} \\ = \frac{35}{384} p_{1} p_{3} - \frac{1}{16} p_{2}^{2} - \frac{17}{1536} p_{1}^{2} p_{2} - \frac{79}{36864} p_{1}^{4} \\ = \frac{1}{36864} (3360 p_{1} p_{3} - 2304 p_{2}^{2} - 408 p_{1}^{2} p_{2} - 79 p_{1}^{4}) . \end{matrix}

With the value of

p_{2}

and

p_{3}

from Lemma 2, using triangular inequality and replacing

|x| < 1

ρ

and

p_{1}

p,

we have:

\begin{matrix} |a_{2} a_{4} - a_{3}^{2}| & = \frac{1}{36864} [19 p^{4} + 1680 p (4 - p^{2}) + 324 (4 - p^{2}) p^{2} ρ \\ + ρ^{2} (4 - p^{2}) (264 p^{2} - 1680 p + 2304)] \\ = F (p, ρ) . \end{matrix}

(12)

Differentiating Equation (12) with respect to

ρ

, we have:

\frac{\partial F}{\partial ρ} = \frac{1}{36864} [324 (4 - p^{2}) p^{2} + 2 ρ (4 - p^{2}) (264 p^{2} - 1680 p + 2304)] .

It is clear that:

\frac{\partial F (p, ρ)}{\partial ρ} > 0,

which shows that

F (p, ρ)

is an increasing function on the closed interval

[0, 1] .

This implies that the maximum value occurs at

ρ = 1,

that is:

max {F (p, ρ)} = F (p, 1) = G (p) .

We now have:

G (p) = \frac{1}{36864} [- 569 p^{4} + 48 p^{2} + 9216] .

(13)

Differentiating Equation (13) with respect to p, we have:

G^{'} (p) = \frac{1}{36864} [- 2276 p^{3} + 96 p]

Differentiating the above equation again with respect to p, we have:

G^{″} (p) = \frac{1}{36864} [- 6828 p^{2} + 96] < 0 .

For

p = 0

, this shows that the maximum value of

G (p)

occurs at

p = 0 .

Hence we obtain:

|a_{2} a_{4} - a_{3}^{2}| ≦ \frac{9105}{36416},

which completes the proof of Theorem 2. □

Theorem 3.

Let

f \in {KL}^{*}

and of the form of Equation (1). Then:

|a_{2} a_{3} - a_{4}| ≦ \frac{7}{24} .

Proof.

We make use of Equations (7)–(9), along with Lemma 2. Since

p_{1} ≦ 2

, by Lemma 3, let

p_{1} = p

and assume without restriction that

p \in [0, 2] .

Then, taking the absolute value and applying the triangle inequality with

ρ = |x|

, we obtain:

\begin{matrix} |a_{2} a_{3} - a_{4}| & ≦ \frac{1}{1536} \{55 p^{3} + 100 p ρ (4 - p^{2}) + 112 (4 - p^{2}) \\ + 56 ρ^{2} (p - 2) (4 - p^{2})\} \\ = : F (ρ) . \end{matrix}

Differentiating

F (ρ)

with respect to

ρ

, we have:

F^{'} (ρ) = \frac{1}{1536} \{100 p (4 - p^{2}) + 112 ρ (p - 2) (4 - p^{2})\} .

For

0 < ρ < 1

and fixed

p \in (0, 2),

it can easily be seen that:

\frac{\partial F}{\partial ρ} < 0 .

This shows that

F_{1} (p, ρ)

is a decreasing function of

ρ

, which contradicts our assumption. Therefore, we have:

max F (p, ρ) = F (p, 0) = G (p) .

This implies that:

G^{'} (p) = \frac{1}{1536} \{165 p^{2} - 224 p\}

and:

G^{″} (p) = \frac{1}{1536} \{330 p - 224\} < 0

for

p = 0

. Thus, clearly,

p = 0

is the point of maximum. Hence we get the required result asserted by Theorem 3. □

To prove Theorem 4, we need Lemma 4.

Lemma 4.

If a function f of the form of Equation (1) is in the class

{KL}^{*},

then:

\begin{matrix} |a_{2}| ≦ \frac{5}{4}, |a_{3}| ≦ \frac{31}{24}, |a_{4}| ≦ \frac{85}{64} and |a_{5}| ≦ \frac{859}{640} . \end{matrix}

These estimates are sharp.

Proof.

The proof of Lemma 4 is similar to that of a known result which was proved by Sokół (see [6]). Therefore, we here choose to omit the details involved in the proof of Lemma 4. □

Theorem 4.

Let

f \in {KL}^{*}

and be of the form of Equation (1). Then:

|H_{3} (1)| ≦ \frac{1509169}{1092480} .

Proof.

Since:

|H_{3} (1)| ≦ |a_{3}| |(a_{2} a_{4} - a_{3}^{2})| + |a_{4}| |(a_{2} a_{3} - a_{4})| + |a_{5}| |(a_{1} a_{3} - a_{2}^{2})| .

(14)

By Theorem 2, we have:

|a_{2} a_{4} - a_{3}^{2}| ≦ \frac{9105}{36416} .

(15)

In addition, by Theorem 3, we get:

|a_{2} a_{3} - a_{4}| ≦ \frac{7}{24} .

(16)

Now, using the fact that

a_{1} = 1,

as well as Theorem 1 with

μ = 1

, Lemma 4, Equations (15) and (16) in conjunction with Equation (14), we have the required result asserted by Theorem 4. □

4. Conclusions

Using the concept of the principle of subordination, we have introduced a new subclass of close-to-convex functions in

U

, associated with the limniscate of Bernoulli. We have then derived the upper bound on

H_{3} (1)

for this subclass of close-to-convex functions in

U

, which is associated with the limniscate of Bernoulli. Our main results are stated and proved as Theorems 1–4. These general results are motivated essentially by the earlier works which are pointed out in this presentation.

Author Contributions

Conceptualization, Q.Z.A. and N.K.; methodology, N.K.; software, B.K.; validation, H.M.S.; formal analysis, H.M.S.; Writing—Original draft preparation, H.M.S.; Writing—Review and editing, H.M.S.; supervision, H.M.S. H.H.S. revised the article as per suggestions from Referees.

Funding

The third author is partially supported by UKM grant: GUP-2017-064.

Conflicts of Interest

The authors declare no conflict of interest.

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Srivastava, H.M.; Ahmad, Q.Z.; Darus, M.; Khan, N.; Khan, B.; Zaman, N.; Shah, H.H. Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli. Mathematics 2019, 7, 848. https://doi.org/10.3390/math7090848

AMA Style

Srivastava HM, Ahmad QZ, Darus M, Khan N, Khan B, Zaman N, Shah HH. Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli. Mathematics. 2019; 7(9):848. https://doi.org/10.3390/math7090848

Chicago/Turabian Style

Srivastava, Hari M., Qazi Zahoor Ahmad, Maslina Darus, Nazar Khan, Bilal Khan, Naveed Zaman, and Hasrat Hussain Shah. 2019. "Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli" Mathematics 7, no. 9: 848. https://doi.org/10.3390/math7090848

APA Style

Srivastava, H. M., Ahmad, Q. Z., Darus, M., Khan, N., Khan, B., Zaman, N., & Shah, H. H. (2019). Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli. Mathematics, 7(9), 848. https://doi.org/10.3390/math7090848

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Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli

Abstract

1. Introduction

2. A Set of Lemmas

3. Main Results and Their Demonstrations

4. Conclusions

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