Some Inclusion Relationships of Certain Subclasses of -Valent Functions Associated with a Family of Integral Operators

By making use of the new integral operator , we introduce and investigate several new subclasses of -valent starlike, -valent convex, -valent close-to-convex, and -valent quasi-convex functions. In particular, we establish some inclusion relationships associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of those given in earlier works. 1. Introduction Let denote the class of functions of the form which are analytic and -valent in the unit disc , and and let . A function is said to be in the class of -valent starlike functions of order in if and only if The class was introduced by Patil and Thakare [1]. Owa [2] introduced the class of -valent convex of order in if and only if It is easy to observe from (2) and (3) that We denote by and where and are the classes of -valently starlike functions and -valently convex functions, respectively, (see Goodman [3]). For a function , we say that if there exists a function such that Functions in the class are called -valent close-to-convex functions of order and type . The class was studied by Aouf [4] and the class was studied by Libera [5]. Noor [6, 7] introduced and studied the classes and as follows. A function is said to be in the class of quasi-convex functions of order and type if there exists a function such that It follows from (5) and (6) that For functions given by (1) and given by the Hadamard product (or convolution) of and is given by For the function , we introduced the operator as follows: From (10), it is easy to verify that Remark 1. Consider (i) For , where the operator was introduced and studied by Liu and Owa [8], and , where the operator was introduced and studied by Jung et al. [9]; (ii) For and , where is the familiar integral operator, which was defined by Cho and Kim [10]. The operator was introduced by Bernardi [11] and we note that was introduced and studied by Libera [12] and Livingston [13]. The main object of this paper is to investigate the various inclusion relationships for each of the following subclasses of the normalized analytic function class which are defined here by means of the operator given by (10). Definition 2. In conjunction with (2) and (10), Definition 3. In conjunction with (3) and (10), Definition 4. In conjunction with (5) and (10), Definition 5. In conjunction with (6) and (10), Remark 6. Consider (I) For , in the above definitions, we have (II) For and , in the above definitions, we have (III) For , in the above definitions, we have where the classes , , , and were introduced and studied by