Notes on Certain Multivalent Analytic Functions Associated with a Linear Operator

The purpose of the present paper is to investigate some properties of multivalent analytic functions. 1. Introduction Let denote the class of the functions of the form which are analytic in the open unit disk . Also let the Hadamard product (or convolution) of two functions be given by In [1], Liu introduced the following generalized Srivastava-Attiya operator : where It is not difficult to see from (5) and (6) that When , the operator is the well-known Srivastava-Attiya operator [2]. The generalized Srivastava-Attiya operator has been studied by several authors (see [1–5]). In this investigation, we focus on certain inequalities consisting of the following differential operator: Our results generalize the recent results obtained by Irmak et al. [6]. In order to prove our main results, we need the following lemmas. Lemma 1 (see [7]). Let and suppose that the function satisfies for all , , and . If is analytic in and for all , then . Lemma 2 (see [7]). Let and suppose that the function satisfies for all , , and . If is analytic in and for all , then . 2. Main Results We now state and then prove each of our main results given by Theorems 3 and 4 below. Theorem 3. Let with for all , and also let and . If where , then Proof. Put Then the function is analytic in with . A simple computation shows that Now letting we obtain that for all . Further, for any , , and , since , we also have which shows that . Therefore, according to Lemma 1, we obtain . This completes the proof of Theorem 3. Theorem 4. Let with for , and also let and . If where then Proof. Suppose that Then is analytic in . It is easily seen from (18) that Further, since it leads to for . Also, for any , and , we have that is, . Finally, by Lemma 2, we obtain that . The proof of Theorem 4 is completed. Remark 5. For and , Theorems 3 and 4 reduce to the results obtained by Irmak et al. [6].