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 formwhich 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 ifwhere 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 ifWe 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 thatwhere , 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 satisfywherethen, is in class .
Proof. Suppose that inequality (14) holds, and denote From the definition, we can verify that if then . In fact, denotingwe haveHere, 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 formwhose 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 thatwhereand 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 haveFromwe haveThis implies that inequality (25) holds.
Using the same argument, we obtain the following inequality.
Lemma 8. If , then there exists such thatwhereand 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 , definewhere is defined by (15). Then if and only if it can be expressed in the form where for all , and .
Proof. Suppose thatThen,Thus, it follows from Theorem 2 that .
Conversely, suppose that . Since we denoteAnd . 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 thatIf 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 thatwhich 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).