On a class of analytic multivalent functions
Introduction
Let denote the class of functions f of the form:which are analytic in , where and Also let us put For analytic functionsby we denote the Hadamard product or convolution of f and g, defined byWe say that an analytic function f is subordinate to an analytic function g, and write , if and only if there exists a function , analytic in such thatandIn particular, if g is univalent in , we have the following equivalence:
A set is said to be starlike with respect to a point if and only if the linear segment joining to every other point lies entirely in E. A set E is said to be convex if and only if it is starlike with respect to each of its points, that is if and only if the linear segment joining any two points of E lies entirely in E. Let f be analytic and univalent in . Then f maps onto a convex domain if and only ifSuch function f is said to be convex in (or briefly convex). The condition (2) for convexity was first stated by Study [24]. Now let and let f be analytic univalent in . Then f maps onto a starlike domain with respect to if and only ifSuch function f is said to be starlike in with respect to (or briefly starlike). The condition (3) for starlikeness is due to Nevalinna [15]. It is well known that if an analytic function f satisfies (3) and then f is univalent and starlike in .
One can alter the conditions (2), (3) by setting other limitations on the behaviour of and of in . In this way many interesting classes of analytic functions have been defined (see for instance [7]). Robertson introduced in [17] the classes , of starlike and convex functions of order , which are defined by
If , then a function in either of these sets is univalent, if it may fail to be univalent. In particular we denote .
Janowski [8] introduced the classIn this paper we take advantage of to define other class of functions.
Let For complex parameters and , ; the generalized hypergeometric function is defined bywhere is the Pochhammer symbol defined by Definition 1 Let be a operator such thatwhere is given by (6).
This operator is called the Dziok–Srivastava operator [6]. We observe that for a function f of the form (1), we havewhere
The Dziok–Srivastava operator includes various other linear operators which were considered in earlier works. Now we show a few of them. For and and the Dziok–Srivastava operator becomes the Carlson–Shaffer operator :which was introduced by Carlson and Shaffer [2] and has found many applications (the Noor operator [16], the Choi–Saigo–Srivastava operator [3] and others). For and we obtain the linear operator:which was considered by Hohlov [9]. Ruscheweyh [19] introduced an operator such thatwhich implies thatand is called the Ruscheweyh differential operator. Thus we observe from (9), (7) that for Next we recall the generalized Bernardi integral operator defined by (cf. [1])therefore for and we haveand for we haveThe Bernardi operator becomes the Livingston operator [13] or the Libera operator [12] for special choices of and . The Bernardi operator with was considered in [11].
By using the definitions of fractional calculus, Srivastava and Owa [22] (see also [4], [21], [23]) have defined the linear operator bywhereThen it is easily observed that for
A complete account of the research on the other familiar operators could easily occupy a book. Definition 2 Let be the operator given in Definition 1. Also letWe denote by the class of functions f of the form (1) which satisfy the following condition:
For this means that belongs to the class defined in (5). If and , then for , thus by (13) we have that belongs to the class of starlike univalent functions of order . After some calculations we obtainwhere, for convenience,By (14) the condition (13) is equivalent for each , , the following subordination:Therefore we will also denote the class by . The class includes the earlier considered classes. We list two of them. For Kim and Srivastava [10] investigated the class of functions such thatwhere the operator is given by (8). This means that or , .
Dziok and Srivastava [6] (see also [5]) making use of the generalized hypergeometric function, have introduced a class of analytic functions with negative coefficients. They considered the class defined by condition (16) with , where and are positive real and , , , is real and . Coefficients estimates, distortion theorems, extreme points, and the radii of convexity and starlikeness for this class was given.
The main object of this paper is to investigate a inclusion properties between the classes for different operators .
Section snippets
Main results
We begin with a lemma, which will be useful later on. Lemma 1 Let and satisfy eitherwhen , orwhen . If and if is given by (10), then andwhere the function is given byThis result is sharp.[14]
Lemma 1 in the more general case there is in [14, p. 111]. Theorem 1 If f is of the
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