Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function

By using the polylogarithm function, a new integral operator is introduced. Strong differential subordination and superordination properties are determined for some families of univalent functions in the open unit disk which are associated with new integral operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained. 1. Introduction Let denote the class of analytic function in the open unit disk . For a positive integer and , let and let . We also denote by the subclass of , with the usual normalization . Let and be formal Maclaurin series. Then, the Hadamard product or convolution of and is defined by the power series . Let the functions and in ; then we say that is subordinate to in , and write , if there exists a Schwarz function in with and such that in . Furthermore, if the function is univalent in , then and (cf [1–3]). Let denote the well-known generalization of the Riemann zeta and polylogarithm functions, or simply the th order polylogarithm function, given by where any term with is excluded; see Lerch [4] and also [5, Sections 1.10 and 1.12]. Using the definition of the Gamma function [5, page 27], a simply transformation produces the integral formula Note that is Koebe function. For more details about polylogarithms in theory of univalent functions, see Ponnusamy and Sabapathy [6] and Ponnusamy [7]. Now, for , we defined the following integral operator: where , and . We also note that the operator defined by (4) can be expressed by the series expansion as follows: Obviously, we have, for , Moreover, from (5), it follows that We note that,(i)for and ( is any integer), the multiplier transformation was studied by Flett [8] and S？l？gean [9];(ii)for and ( ), the differential operator was studied by S？l？gean [9];(iii)for and ( is any integer), the operator was studied by Uralegaddi and Somanatha [10];(iv)for , the multiplier transformation was studied by Jung et al. [11];(v)for , the integral operator was studied by Komatu [12].To prove our results, we need the following definition and theorems considered by Antonino and Romaguera [13], Antonino [14], G. I. Oros and G. Oros [15], and Oros [16]. Definition 1 (see [13] cf [14, 15]). Let be analytic in and let be analytic and univalent in . Then, the function is said to be strongly subordinate to , or is said to be strongly superordinate to , written as , if, for , as the function of is subordinate to . We note that if and only if and . Definition 2 ([15] cf [1]). Let and let be univalent in . If is analytic in and satisfies