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Majorization for a Class of Analytic Functions Defined by -Differentiation
We introduce a new class of multivalent analytic functions defined by using -differentiation and fractional -calculus operators. Further, we investigate majorization properties for functions belonging to this class. Also, we point out some new and known consequences of our main result.
1. Introduction and Preliminaries
Let denote the class of functions of the form which are analytic and -valent in the open unit disk . For analytic functions and in , we say that the function is majorized by in (see ) and write if there exists a Schwarz function , analytic in , such that
For the convenience of the reader, we now give some basic definitions and related details of -calculus which are used in the sequel.
For any complex number the -shifted factorials are defined as and in terms of the basic analogue of the gamma function where the -gamma function is defined by If , the definition (4) remains meaningful for as a convergent infinite product In view of the relation we observe that the -shifted factorial (4) reduces to the familiar Pochhammer symbol , where .
It may be noted that the -Gauss hypergeometric function (see Gasper and Rahman [2, p.3, eqn. (1.2.14)]) is defined by and as a special case of the above series for , we have Also, the -derivative and -integral of a function on a subset of are, respectively, given by (see [2, pp. 19–22]) Therefore, the -derivative of , where is a positive integer, is given by where and is called the -analogue of . As , we have . Here we list some relations satisfied by :
Recently, many authors have introduced new classes of analytic functions using -calculus operators. For some recent investigations on the classes of analytic functions defined by using -calculus operators and related topics, we refer the reader to [3–13] and the references cited therein. In the following, we define the fractional -calculus operators of a complex-valued function , which were recently studied by Purohit and Raina .
Definition 1 (fractional -integral operator). The fractional -integral operator of a function of order is defined by where is analytic in a simply connected region of the -plane containing the origin and the -binomial function is given by The series is single valued when and (see for details , pp. 104–106); therefore, the function in (15) is single valued when , , and .
Definition 2 (fractional -derivative operator). The fractional -derivative operator of a function of order is defined by where is suitably constrained and the multiplicity of is removed as in Definition 1.
Definition 3 (extended fractional -derivative operator). Under the hypotheses of Definition 2, the fractional -derivative for a function of order is defined by where , and denotes the set of natural numbers.
Remark 4. It follows from Definition 2 that
Using , we define a -differintegral operator , as follows: where in (20) represents, respectively, a fractional -integral of of order when and a fractional -derivative of of order when . It is easy to see from (20) that
Definition 5. A function is said to be in the class of -valent functions of complex order in if and only if
It can be seen that, by specializing the parameters, the class reduces to many known subclasses of analytic functions. For instance, if then (1), the class of starlike functions of complex order (see ),(2), the class of convex functions of complex order (see ),(3), the class of -spiral-like functions (see ),(4), the class of starlike functions of order .
2. Majorization Problem for the Class
We start by proving the following -analogue of the result given by Nehari in .
Lemma 6. If is analytic and bounded in , then
Proof. If is bounded in , then is also bounded in . Clearly vanishes when . Therefore, the function is regular when and also at all other points of . Furthermore, is bounded in . In fact, and for ; hence by maximum principle, throughout . Then from (26) we have
Theorem 7. Let the function be in the class and suppose that . If is majorized by in for , then where
Proof. Let Since , we have and where denotes the well known class of bounded analytic functions in , which satisfy the conditions (cf. ) It follows from (30) and (31) that In view of the identity (22), we have the following inequality from (33) by making some simple calculations: Since is majorized by in , there exists an analytic function such that and . Applying -differentiation with respect to and multiplying by , we have from (35) Using (22), in the above equation, we get Noting that is bounded in and using Lemma 6 we obtain Appling (34) and (38) in (37) we get where the function , defined by takes its maximum value at with given by (29). Furthermore, if where given by (29), then the function increases in the interval , so that does not exceed Therefore, from this fact, (39) gives inequality (28).
Letting , and in Theorem 7, we have the following.
Corollary 8 (see ). Let the function be analytic and univalent in the open unit disk and suppose that , the class of starlike functions of complex order . If is majorized by in , then where
For , Corollary 8 reduces to the following result.
Corollary 9. Let the function be analytic and univalent in the open unit disk and suppose that , the class of -spiral-like functions. If is majorized by in , then for .
Further setting , in Corollary 8 we get the following.
Corollary 10 (see ). Let the function be analytic and univalent in the open unit disk and suppose that is starlike in . If is majorized by in , then
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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