Journal of Function Spaces
Volume 2017 (2017), Article ID 5739196, 6 pages
https://doi.org/10.1155/2017/5739196

Research Article

Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative

Xiaofei Li,¹ Deng Ding,¹ Liping Xu,² Chuan Qin,³ and Songbo Hu⁴

¹Department of Mathematics, University of Macau, Taipa 999078, Macau
²School of Information and Mathematics, Yangtze University, Jingzhou 434000, China
³Engineering and Technology College, Yangtze University, Xueyuan Rd., Jingzhou 434000, China
⁴School of Health Sciences, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Liping Xu; xlp211@126.com

Received 3 September 2016; Accepted 21 November 2016; Published 17 January 2017

Academic Editor: Enrique Llorens-Fuster

Copyright © 2017 Xiaofei Li et al. 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.

In this paper, we define and study some subclasses of multivalent analytic functions of higher order in the unit disc. These classes generalize some classes previously studied. We obtain coefficient inequalities, distortion theorems, extreme points, and integral mean inequalities. We derive some results as special cases.

1. Introduction

Let and denote as the class of multivalent functions of the form

which are analytic in the open unit disk:

For two parameters and , function is said to be in class of -valent -uniformly star-like functions of order in , if and only if

where denotes taking the real part of argument. On the other hand, function is said to be in class of -valent -uniformly convex functions of order in , if and only if

We note from (3) and (4) that

The classes and were introduced recently by Khairnar and More [1]. Various subclasses of analytic and univalent or multivalent functions were studied in many papers (see, e.g., [2–4]). Recently, Nishiwaki and Owa in [5] introduced two classes consisting of all functions , which satisfy

and consisting of all functions , which satisfy

where and . We notice from definitions of these classes that

For each , it is easily seen upon differentiating both sides of (1) times with respect to that

where , and denotes -permutations of objects; that is,

Let and such that , and assume that

Srivastava et al. in [6] introduced a subclass of the -valent function class consisting of functions of form (1), which satisfy the following analytic criterion: for all ,

Recently, many papers have discussed such aspects of analytic univalent or multivalent functions. For example, Aouf et al. in [7, 8] discussed a subclass of -valent analytic function with negative coefficient by using higher-order derivatives and investigated many properties of distortion theorems, closure theorems, modified Hadamard products, and radii of close-to-convexity, starlikeness, and convexity. K. I. Noor and M. A. Noor in [9] defined some subclasses of analytic functions related to -uniformly close-to-convex functions of higher order and studied the following: rate of growth of coefficients, inclusion relations, radius problems, and necessary conditions for univalency. Seoudy in [10] defined a class of -valent functions defined by certain linear operator and obtained subordination and inclusion properties.

Following Srivastava et al. [6] and Nishiwaki and Owa [5], we define a new subclass of multivalent functions involving higher-order derivative.

Definition 1. Let and such that , and and , two real numbers, satisfy that and . Function , if and only if and satisfies the following inequality:

From the above definition, it is clear that . In this paper, we obtain several properties including the coefficient inequalities, distortion theorems, extreme points, and integral means inequalities for this subclass of multivalent functions involving higher-order derivative.

2. Coefficient Inequalities

We derive sufficient conditions for which are given by using coefficient inequalities.

Theorem 2. Let be a function of form (1). If the coefficients of satisfy

where

then, is in class .

Proof. Suppose that inequality (14) holds, and denote

From the definition, we can verify that if

then . In fact, denoting

we have

Here, we use technology . If (14) satisfies, we drive that the last expression above is bounded by 1 which implies . Thus, the proof of Theorem 2 is completed.

Example 3. Function given by

belongs to class for and with .

As a special case of Theorem 2, as in [2], we can obtain the following corollary.

Corollary 4. Function is in class , if

In view of Theorem 2, we introduce subclass which consists of functions of the form

whose Taylor-Maclaurin coefficients are nonnegative and satisfy inequality (14). By the coefficient inequalities for classes , we have the following theorem.

Theorem 5. If , then

Since , we get the following corollary, which is a theorem in [2].

Corollary 6. If , then

3. Distortion Theorems

Lemma 7. If , then there exists such that

where

and is given in (15).

Proof. From the definition of , there exists such that function is increasing with respect to when . According to Theorem 2, we have

From

we have

This implies that inequality (25) holds.

Using the same argument, we obtain the following inequality.

Lemma 8. If , then there exists such that

where

and is defined by (15).

Theorem 9. Let be a function in class . Then, for ,

where and are given in Lemmas 7 and 8, respectively.

Proof. Let be a function of form (22). For , using Lemma 7, we have

Using the same argument, we can prove the following result.

Theorem 10. Let be a function in class . Then, for ,

where and are given in Lemmas 7 and 8, respectively.

4. Extreme Points

Theorem 11. Let and, for each , define

where is defined by (15). Then if and only if it can be expressed in the form

where for all , and .

Proof. Suppose that

Then,

Thus, it follows from Theorem 2 that .
Conversely, suppose that . Since

we denote

And . By a simple calculation, we get .

Corollary 12. The extreme points of are functions and

for each .

5. Integral Means Inequalities

Assume that two functions and are analytic in . We say that is subordinate to , written as , if there exists an analytic function in with and such that .

Lemma 13 (see [11]). If and are analytic in with , then, for and ,

Theorem 14. Let and be given by (35). Suppose that

If there exists function , , that satisfied the condition

then, for , we have

Proof. In order to obtain the result, it is necessary to prove the following inequality:

From Lemma 13, it is sufficient to verify the subordination:

Thus, there exists an analytic function in such that

We find that

which readily yields and

This means that the hypotheses of are satisfied and the theorem is proved.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work is partially supported University of Macau, with the Research Grant (MYRG068(Y1-L2)-FST13-DD), the research fund from Engineering and Technology College Yangtze University (no. 15J0802), and research fund from Wuhan University (no. 2042016KF0029).

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