Volume 2022, Issue 1 2621811

Research Article

Open Access

The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

Azzh Saad Alshehry

orcid.org/0000-0002-2893-576X

Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia pnu.edu.sa

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Rasool Shah,

Corresponding Author

Rasool Shah

[email protected]

orcid.org/0000-0003-4306-8489

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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Abdul Bariq,

Corresponding Author

Abdul Bariq

[email protected]

orcid.org/0000-0003-3902-1944

Department of Mathematics, Laghman University, Mehtarlam, Laghman, Afghanistan lu.edu.af

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Azzh Saad Alshehry,

Azzh Saad Alshehry

orcid.org/0000-0002-2893-576X

Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia pnu.edu.sa

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Rasool Shah,

Corresponding Author

Rasool Shah

[email protected]

orcid.org/0000-0003-4306-8489

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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Abdul Bariq,

Corresponding Author

Abdul Bariq

[email protected]

orcid.org/0000-0003-3902-1944

Department of Mathematics, Laghman University, Mehtarlam, Laghman, Afghanistan lu.edu.af

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First published: 28 September 2022

https://doi.org/10.1155/2022/2621811

Academic Editor: Sarfraz Nawaz Malik

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Abstract

In this particular research article, we take an analytic function , which makes a four-leaf-shaped image domain. Using this specific function, two subclasses, and , of starlike and convex functions will be defined. For these classes, our aim is to find some sharp bounds of inequalities that consist of logarithmic coefficients. Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant.

1. Introduction and Definitions

To properly comprehend the findings provided in the paper, certain important literature on geometric function theory must first be discussed. In this regard, the letters

and

stand for the normalized univalent (or schlicht) functions class and the normalized holomorphic (or analytic) functions class, respectively. These primary notions are defined in the disc

(1)

where

expresses holomorphic functions class, and

(2)

This class evolved as the foundational component of cutting-edge research in this area. In his paper [1], Koebe established the presence of a “covering constant” ζ, demonstrating that if F is holomorphic and Schlicht in with F^′(0) = 1 and F(0) = 0, then . Many mathematicians were intrigued by this beautiful result. Within a few years, the wonderful article by Bieberbach [2], which gave rise to the renowned coefficient hypothesis, was published.

The below expression provided the coefficients λ_n of logarithmic function J_F(z) for

(3)

The above coefficients have a considerable impact on the theory of Schlicht functions in many estimations. De Branges [3] achieved that n ≥ 1 in1985,

(4)

and equality will be achieved if F has the form

for some φ ∈ ℝ. It is obvious that this inequality provides the most general version of the well-known Bieberbach-Robertson-Milin conjectures concerning the Taylor coefficients of

. We quote [4–6] for further information on the demonstration of de Brange’s conclusion. By taking into account, the logarithmic coefficients, in 2005, Kayumov [7] established Brennan’s conjecture for conformal mappings. The major contributions to study the bounds of logarithmic coefficients for various holomorphic univalent functions are due to Alimohammadi et al. [8], Obradović et al. [9], Ye [10], Deng [11], Girela [12], Roth [13], and Andreev and Duren [14].

For the prescribed functions

, the relation of subordination between Q₁ and Q₂ is as follows (mathematically as Q₁≺Q₂), if an holomorphic function u comes in

with the limitation |u(z)| < 1 and u(0) = 0 in a manner that Q₁(z) = Q₂(u(z)) satisfy. Consequently, the following relation applies if

(5)

if and only if

(6)

By applying the notion of subordination, Ma and Minda [15] proposed a consolidated version of the set

in 1992, and the following is a description of it:

(7)

with the Schlicht function ψ that satisfies

(8)

Various subclasses of the set

have been examined in the past few years as particular choices for family

. For instance,

(i)
(see [16]) and (see [17])
(ii)
(see [18, 19])
(iii)
(see [20, 21]) and (see [22])
(iv)
with ψ(z) = (1 + z/1 − z)^ξ and 0 < ξ ≤ 1 (see [23])
(v)
(see [24]) and (see [25, 26])

For given q, n ∈ ℕ = {1, 2, ⋯}, b₁ = 1, and

with the series representation (1), the Hankel determinant H_q,n(F) is expressed by

(9)

and it was established by Pommerenke and Pommerenke [27, 28]. For several subcollections of Schlicht functions, the determinant H_q,n(F) has been examined. In specific, the sharp estimate of the functional

for sets

(convexfunctions),

(starlikefunctions), and

(boundedturningfunctions) were determined in [29, 30]. However, for the class of close-to-convex functions, the exact bounds of this determinant remain open [31]. The researchers were inspired by the works of Babalola [32], Bansal, et al. [33], Zaprawa [34], Kwon et al. [35], Kowalczyk et al. [36], and Lecko et al. [37].

It is easy to deduce from equation (2) that, for

, the logarithmic coefficients are computed by

(10)

(11)

(12)

(13)

Currently, Lecko and Kowalczyk and Kowalczyk and Lecko [38, 39] studied the following Hankel determinant H_q,n(J_F/2) of logarithmic coefficients

(14)

It has been noted that

(15)

By the virtue of the function

, we define the following classes:

(16)

(17)

Alternatively,

if and only if an analytic function q occurs that satisfies

in such that

(18)

By taking

in (18), we achieve the following function, which serves as an extremal in many of the class

problems.

(19)

The following Alexander-type connection-related two classes were mentioned above. The above two families are interlinked by the following Alexander-type relation

(20)

From (19) and (20), we easily obtain the following extremal functions in various problems of the class

(21)

Clearly, g₀(z), g₀(z²), g₀(z³), and g₀(z⁴) belong to the class

. That is,

(22)

In the present paper, our core objective is to find the sharp coefficient type problems of logarithmic functions for the families and Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant H_2,1(J_F/2).

2. A Set of Lemmas

We must first create the class

in the below set-builder form in order to declare the Lemmas that are employed in our primary findings.

(23)

That is, if

, then q has the below series expansion

(24)

The following Lemma consists of the widely used e₂ formula [40], the e₃ formula [41], and the e₄ formula illustrated in [42].

Lemma 1. Let be given in the form (24), then for

Lemma 2. Let be of the form (24), then for

(25)

(26)

(27)

Lemma 3. Let and has the expansion (24). Then,

(28)

(29)

(30)

The inequalities (28)–(30) are taken from [40, 43] and [26, 44, 45], respectively.

Lemma 4 (see [40].)If has the representation (24), then

(31)

Lemma 5 [46]. Let γ, τ, ψ and ς satisfy that τ, ς ∈ (0, 1) and

(32)

has the expansion (24), then

(33)

3. Coefficient Inequalities for the Class

We start by establishing out the class ’s initial coefficient bounds.

Theorem 6. Let F be the series form (1) and if then

(34)

These bounds are sharp.

Proof. Let Then, Schwarz function u may therefore be used to express (16) as

(35)

From the use of Schwarz function u and if , we have

(36)

and by simple computation, we get

(37)

Using (1), we attain

(38)

By some calculation and using the series expansion of (37), we get

(39)

Now, by comparing (38) and (39), we get

(40)

(41)

(42)

(43)

Utilizing (40) and (10), (11), (12), and (13), we have

(44)

(45)

(46)

(47)

From (44), using triangle inequality and (29), we get

(48)

Also, from (45), application (30), and triangle inequality, we get

(49)

By rearranging (46), we have

(50)

By Lemma 4 and triangle inequality, we obtain

(51)

By rearranging (47), we have

(52)

Comparing the equation of (52) right side with

(53)

we get γ = 1/8, ς = 1/2, τ = 1/2, ψ = 1/2, and

(54)

Thus, Lemma 5’s requirements are all met. Hence,

(55)

These are sharp outcomes. Equality is determined by using (10)–(13) and

(56)

Theorem 7. If then

(57)

The above stated inequality is best possible.

Proof. By utilizing (44) and (45), we have

(58)

Implementation of (28) and triangle inequality, we get

(59)

Equality is determined by using (10), (11), and

(60)

Corollary 8. If then

(61)

This inequality is sharp and can be obtained by using (10), (11), and

(62)

Theorem 9. Let F be the expansion (1) and if then

(63)

The above stated result is the best possible.

Proof. From (44)–(46), we easily attain

(64)

By using Lemma 4 and triangle inequality, we obtain

(65)

Equality is determined by using (10), (11), (12), and

(66)

Theorem 10. Let F be the expansion (1) and if then

(67)

The last stated inequality is the finest.

Proof. From the use (45) and (47), we get

(68)

Comparing the right side of (68) with

(69)

we get γ = 17/96, ς = 17/24, τ = 1/2, ψ = 23/36, and

(70)

Thus, Lemma 5’s requirements are all met. Hence,

(71)

Equality is determined by using (11), (13), and

(72)

Theorem 11. Let be the representation (1). Then,

(73)

This result is sharp.

Proof. We can write the H_2,1(J_F/2) as

(74)

From (44)–(46), we have

(75)

Using (25) and (26) to express e₂ and e₃ in terms of e₁ and also e₁ = e, with 0 ≤ e ≤ 2, we obtain

(76)

By changing |δ| ≤ 1 and |x| = c, where c ≤ 1 and utilizing triangle inequality and pickings e ∈ [0, 2], so

(77)

Differentiate with respect to c, we have

(78)

It is easy exercise to show that Ξ^/(e, c) ≥ 0 on [0, 1], so that Ξ(e, c) ≤ Ξ(e, 1). Putting c = 1, we get

(79)

As Θ^/(e) ≤ 0, so Θ(e) is a decreasing function, so that it gives a maximum value at e = 0

(80)

Equality is determined by using (10), (11), (12), and

(81)

4. Coefficient Inequalities for the Class

For the function of class , we start this portion by determining the absolute values of the first four initial logarithmic coefficients.

Theorem 12. Let F be given by (1) and if then

(82)

These bounds are sharp.

Proof. Let Then, (17) can be written in the form of Schwarz function as

(83)

Using (1), we obtain

(84)

Now, by comparing (84) and (39), we get

(85)

Utilizing (85) and (10), (11), (12), and (13) we have

(86)

(87)

(88)

(89)

From (86), using triangle inequality and (29), we get

(90)

Also, from (87), application (30), and triangle inequality, we get

(91)

By rearranging (88), we have

(92)

By Lemma 4 and triangle inequality, we obtain

(93)

By rearranging (89), we have

(94)

Comparing the right side of (94) with

(95)

we get γ = 13109/248832, ς = 11/27, τ = 19/48, and ψ = 2353/7776. Thus, all the conditions of Lemma 5 are satisfied. Hence, we have

(96)

These are sharp outcomes. Equality is determined by using (10), (11), (12), and (13) along with (22).

Theorem 13. Let be the series form (1). Then,

(97)

This inequality is sharp.

Proof. By utilizing (86) and (87), we have

(98)

Implementation of (28) and triangle inequality, we get

(99)

Equality is determined by using (10), (11), and (22).

For λ = 1, we get the below corollary.

Corollary 14. Let , and it has the form (1). Then,

(100)

This inequality is sharp and can be obtained by using (10), (11), and (22).

Theorem 15. Let F be the form (1) and if then

(101)

This result is sharp.

Proof. By using (86)–(88), we obtain

(102)

By using Lemma 4 and triangle inequality, we obtain

(103)

Equality is determined by using (10), (11), (12), and (22).

Theorem 16. Let F be the form (1) and Then,

(104)

This result is sharp.

Proof. By using (87) and (89), we obtain

(105)

Comparing the right side of (68) with

(106)

we get γ = 35243/497664, ς = 113/216, τ = 19/24, ψ = 707/1944, and

(107)

Thus, all the conditions of Lemma 5 are satisfied. Hence, we have

(108)

Equality is determined by using (11), (13), and (22).

Theorem 17. Let F be given the form (1) and Then,

(109)

This result is sharp.

Proof. We can write the H_2,1(F_F/2) as;

(110)

From (86)–(88), we have

(111)

Using (25) and (26) to express e₂ and e₃ in terms of e₁ and also e₁ = e, with 0 ≤ e ≤ 2, we obtain

(112)

By replacing |δ| ≤ 1 and |x| = c, where c ≤ 1 and using triangle inequality and taking e ∈ [0, 2], so

(113)

Differentiate with respect to c, we have

(114)

It is a simple exercise to show that Ω^′(e, c) ≥ 0 on [0, 1], so that Ω(e, c) ≤ Ω(e, 1). Putting c = 1 gives

(115)

As Θ^′(e) ≤ 0, so Θ(e) is a decreasing function, so that it gives a maximum value at e = 0

(116)

Equality is determined by using (10), (11), (12), and (22).

Conflicts of Interest

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

Acknowledgments

This research was supported by the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R183), Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.

Open Research

Data Availability

The numerical data used to support the findings of this study are included within the article.

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The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

Abstract

1. Introduction and Definitions

2. A Set of Lemmas

3. Coefficient Inequalities for the Class

4. Coefficient Inequalities for the Class

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

Abstract

1. Introduction and Definitions

2. A Set of Lemmas

3. Coefficient Inequalities for the Class

4. Coefficient Inequalities for the Class

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

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