Journal of Function Spaces
Volume 2018 (2018), Article ID 8654835, 6 pages
https://doi.org/10.1155/2018/8654835

Research Article

On a New Subclass of -Valent Close-to-Convex Mappings Defined by Two-Sided Inequality

Yi-Hui Xu¹ and Jin-Lin Liu²

¹Department of Mathematics, Suqian College, Suqian 223800, China
²Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Correspondence should be addressed to Jin-Lin Liu; jlliu@yzu.edu.cn

Received 5 April 2018; Accepted 30 May 2018; Published 28 June 2018

Academic Editor: Henryk Hudzik

Copyright © 2018 Yi-Hui Xu and Jin-Lin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A new subclass of -valent close-to-convex mappings defined by two-sided inequality is introduced. Some sufficient conditions for functions to be in are given.

1. Introduction

Let be the class of functions of the form

which are -valent analytic in the open unit disk . There and in the following, let , and be the sets of positive integers, complex numbers, and real numbers, respectively. A function analytic in is said to be close-to-convex if there is a convex function such that

for all . The concept of close-to-convex was introduced by Kaplan [1] in 1952. A number of results for close-to-convex functions in have been obtained by several authors (see, e.g., [2–14]).

A function is said to be in the class if it satisfies the following two-sided inequality:

for and . Note that if , then is -valent close-to-convex in . Furthermore, if , then (3) becomes

A function is called -valent close-to-convex of order and type in .

Given two functions and , which are analytic in , we say that the function is subordinate to and write or , if there exists a Schwarz function , analytic in with and such that . In particular, if is univalent in , we have the following equivalence:

Throughout this paper, we let

In order to prove our main results, we need the following lemmas.

Lemma 1. The function defined by

is analytic and univalent convex in and

Proof. In view of (6), it is easy to see that and the transformation

maps the convex region

conformally onto the right-half -plane so that corresponding to . Since

maps the right-half -plane onto , from (7), (9), and (11) we find that

maps conformally onto with . The proof of Lemma 1 is completed.

Lemma 2 (see [15]). Let the function be analytic and univalent in and let the functions and be analytic in a domain containing , with . Set

and suppose that

(i) is univalent starlike in ,(ii)

If is analytic in with , , and

then . The function is the best dominant of (14).

Lemma 3 (see [16]). Let be analytic function in of the form

with in . If there exists a point such that

for and

then

where

where and .

In this paper we shall derive some criteria for a function to be in the class .

2. Main Results

Our first result is the following theorem.

Theorem 4. Let and . Also let and . If the function is analytic in with and and satisfies

where

is close-to-convex in , then

The bounds and in (22) are sharp for the function defined by (7).

Proof. We choose

in Lemma 2. By Lemma 1, the function is analytic and univalent convex in and

Clearly, and are analytic in a domain containing and , with when . For

the function given by

is univalent starlike in because

Further, we have

where is given by (21) and so

For , it follows from (27) and (29) that

The other conditions of Lemma 2 are also satisfied. Therefore, we conclude that

and the function is the best dominant of (20). The proof of the theorem is completed.

Theorem 5. If satisfies and

for all , where

then . The bounds and in (32) are the largest numbers such that (3) holds true.

Proof. For satisfying , we define the function by

Then is analytic in with and for all . Taking the logarithmic differentiations in both sides of (34), we have

and

for all . Putting and in Theorem 4 and using (36), we find that if

where

is (close-to-convex) univalent in , then

that is, .
For , , and , we have

and

Now we consider the following two cases.
(i) If

then we deduce from (38), (40), and (41) that

which yields

Let . Then

and it follows from (44) and (45) that

(ii) If , then it follows from (38) to (41) that

and so

Let . Then

and from (48) and (49) we have

Noting that , we deduce from (46) and (50) that properly contains the region in the complex -plane. Therefore, if a function satisfies (32), then the subordination relation (37) holds true. This shows that .
Furthermore, for the function defined by (7), we have

Hence, by using (46) and (50), we conclude that the bounds and in (32) are the best possible ones. The proof of the theorem is completed.

Theorem 6. Let . Also let and . If satisfies

for , then

In particular, if , then or is -valent close-to-convex of order .

Proof. We denote

Then condition (52) becomes

We want to prove that

If there exists a point such that

then, from Lemma 3, we have

where , and

For the case , we have

where

The function

has a negative derivative

Hence

Therefore, (60) becomes

which contradicts (55). Thus

For the case , we have . Applying the same method as the above, we get

This contradicts (55). The proof of the theorem is completed.

Applying the same method as the above we can prove the following theorem.

Theorem 7. Let . Also let and . If satisfies

for , then

In particular, if , then or is -valent close-to-convex of order .

Data Availability

For every result in this paper, detailed proof has been given. Readers can understand this paper by reading these detailed proofs carefully. All results in this paper can be released.

Conflicts of Interest

There are no conflicts of interest among the authors.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11571299), Natural Science Foundation of Jiangsu Province (Grant no. BK20151304), and Natural Science Foundation of Jiangsu Gaoxiao (Grant no. 17KJB110019).

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