### Article

Kyungpook Mathematical Journal 2022; 62(2): 229-242

**Published online** June 30, 2022

Copyright © Kyungpook Mathematical Journal.

### Uniformly Close-to-Convex Functions with Respect to Conjugate Points

Syed Zakar Hussain Bukhari∗, Taimoor Salahuddin, Imtiaz Ahmad and Muhammad Ishaq, Shah Muhammad

Department of Mathematics, Mirpur University of Science and Technology(MUST), Mirpur-10250(AJK), Pakistan

e-mail : fatmi@must.edu.pk, taimoor_salahuddin@yahoo.com, drimtiaz.maths@must.edu.pk and ishaq_381@yahoo.com

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

e-mail : shah.maths@must.edu.pk

**Received**: April 3, 2018; **Revised**: November 29, 2018; **Accepted**: December 19, 2018

### Abstract

In this paper, we introduce a new subclass of

**Keywords**: Characterizations, coefficients estimates, distortion bounds, extreme points, radii problems

### 1. Introduction

Let

We define the class

Let

The Möbius function

where

The simplest representation of the conic domain

This domain represents the right half plane for

We denote the class ofclose-to-convex functions by

**Definition 1.1.** Let

For

A normalized analytic function

The class

where

and

These and related classes have been introduced by various authors, for example, see [2, 3, 4, 5, 7, 9, 15, 20].

**Definition 1.2.** A function

where

Here we let

where

### 2. Some Useful Lemmas

We now state some useful lemmas which we may need to establish our results in the sequel.

**Lemma 2.1.** If

**Lemma 2.2.** If

**Lemma 2.3.** ([14]) Let

**Lemma 2.4.** ([14]) Let

### 3. Main Results

We employ the similar techniqueof Aqlan et al. [1] to find the coefficient estimates for the functions in the class

**Theorem 3.1.** A function

for

where

where

To obtain (3.2) or (3.3), in view of Lemma 2.1, we need to prove that

Also for

Again we consider that

or we can write

Keeping in view (3.6),(3.7) and (3.8), we have

For the limiting value of

Again this inequlity alongwith (3.1) implies (3.6). Hence

or

or

and

Using (3.11) and (3.12) in (3.10), we get

We choose

For

Again taking

where

This completes the proof.

As an immediate consequence of the above Theorem 3.1, we have the following theorem for the coefficients estimates of

**Theorem 3.2.** For a function

for

Using Lemma 2.3 in (3.13), we have

This implies that

The equality holds for the function

### 4. Growth and Distortion Problems

Now we solve the growth problems for functions in the class

**Theorem 4.1.**If a function

and

This implies that

Thus from

Similarly for the lower bounds on the left side of (4.1), we can write

The inequalities (4.3) and (4.4) lead to (4.1). On the same lines, we can find the lower and upper bounds of

### 5. Extreme Points

Here we find the extremal points of functions belonging to the newly introduced class.

**Theorem 5.1.** Let

where

This proves that

Since

Therefore, by Defintion 1.2, we have

where

we have

### 6. Radius Problems

**Theorem 6.1.** If

For

For the coefficient condition required by the Theorem 3.1, the inequality (3.1) is true if

or we can write

This completes the proof.

**Theorem 6.2.** If

where

For a starlike function

From the coefficient condition required by the Theorem 3.1, the inequality (6.2) is true if

where

where

**Theorem 6.3.** If

For

From the condition required by the Theorem 3.1, the inequality (6.3) is true if

where

### 7. Integral Means Inequality

In [19], Silverman found that the function

In [18], he also proved his conjecture for the subclasses

**Theorem 7.1.** If

then for

By Lemma 2.4, it is enough to prove that

Suppose that

This can be written as

This completes the proof by Theorem 3.1.

### 8. Concluding Remarks and Observations

In this research, we used the idea associated with the conic domains and introduced a new subclass of

### Acknowlegdments.

The Authors would like to thank Worthy Vice Chancellor MUST, Mirpur, AJK for his untiring efforts for the promotion of research conducive environment at MUST.

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