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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