Some subordination results associated with generalized Srivastava-Attiya operator

The operator Js,b(f) was introduced in (Srivastava and Attiya in Integral Transforms Spec. Funct. 18(3-4): 207-216, 2007), which makes a connection between Geometric Function Theory and Analytic Number Theory. In this paper, we use the techniques of differential subordination to investigate some classes of admissible functions associated with the generalized Srivastava-Attiya operator in the open unit disc U={z∈C:|z|<1}. MSC:30C80, 30C10, 11M35.


Introduction
and * denotes the Hadamard product (or convolution). Furthermore, Srivastava and Attiya [] showed that (.) http://www.advancesindifferenceequations.com/content/2013/1/105 we denote by where * denotes the convolution or Hadamard product. We note that Moreover, let D be the set of analytic functions q(z) and injective onŪ\E(q), where and q (z) =  for ζ ∈ ∂U\E(q). Further, let D a = {q(z) ∈ D : q() = a}. In our investigations, we need the following definitions and theorem.
Definition . Let f (z) and F(z) be analytic functions. The function f (z) is said to be subordinate to F(z), written f (z) ≺ F(z), if there exists a function w(z) analytic in U, with w() =  and |w(z)| ≤ , and such that f (z) = F(w(z)).
Definition . Let : C  × U → C be analytic in domain D, and let h(z) be univalent in U.
If p(z) is analytic in U with (p(z), zp (z)) ∈ D when z ∈ U, then we say that p(z) satisfies a first-order differential subordination if: The univalent function q(z) is called dominant of the differential subordination (.), if p(z) ≺ q(z) for all p(z) satisfies (.), ifq(z) ≺ q(z) for all dominant of (.), then we say thatq(z) is the best dominant of (.).

Some subordination results with J t s,b
Definition . Let be a set in C and q(z) ∈ D ∩ A p . The class of admissible functions [ , q] consists of those functions φ : C  × U → C that satisfy the admissibility condition: where z ∈ U, ζ ∈ ∂U\E(q) and k ≥ p.
Proof Let us define the analytic function p(z) as Using the definition of J t s,b , we can prove that Let us define the parameters u, v and w as Now, we define the transformation by using the relations (.), (.), (.) and (.), we have Therefore, we can rewrite (.) as Then the proof is completed by showing that the admissibility condition for φ ∈ [ , q] is equivalent to the admissibility condition for as given in Definition .. Since If = C is a simply connected domain, then = h(U) for some conformal mapping h(z) of U onto . In this case the class [h(U), q] is written as [h, q].
The following theorem is a direct consequence of Theorem ..
]. If f (z) ∈ A(p) satisfies the following subordination relation: The next corollary is an extension of Theorem . to the case where the behavior of q(z) on ∂U is not known. http://www.advancesindifferenceequations.com/content/2013/1/105

Corollary . Let ⊂ C and let q(z) be univalent in
Proof By using Theorem ., we have J t s+,b ≺ q ρ (z). Then we obtain the result from q ρ (z) ≺ q(z).
Theorem . Let h(z) and q(z) be univalent in U, with q() =  and set q ρ (z) = q(ρz) and h ρ (z) = h(ρz). Let φ : C  × U → C satisfy one of the following conditions: Proof The proof is similar to the proof of [, Theorem .d, p.], therefore, we omitted it.

Theorem . Let h(z) be univalent in U.
Let φ : C  × U → C. Suppose that the differential equation has a solution q(z) with q() =  and satisfies one of the following conditions: and q(z) is the best dominant.
Proof Following the same proof in [, Theorem .e, p.], we deduce from Theorems . and . that q(z) is a dominant of (.). Since q(z) satisfies (.), it is also a solution of (.) and, therefore, q(z) will be dominated by all dominants. Hence, q(z) is the best dominant. http://www.advancesindifferenceequations.com/content/2013/1/105 In the case q(z) = Mz, M >  and in view of the Definition ., the class of admissible functions [ , q] denoted by [ , M] is defined below.

Definition . Let be a set in C and M > . The class of admissible functions [ , M]
consists of those functions φ : C  × U → C that satisfy the admissibility condition where z ∈ U, and Re(Le -iθ ) ≥ (k -)kM for all real θ and k ≥ p.
Proof Let us define the analytic function p(z) as By using (.), we have (.) http://www.advancesindifferenceequations.com/content/2013/1/105 Define the parameters u, v and w as now, we define the transformation by using the relations (.), (.), (.) and (.), we have Therefore, we can rewrite (.) as Then the proof is completed by showing that the admissibility condition for φ ∈  [ , q] is equivalent to the admissibility condition for as given in Definition .. Since The following theorem is a direct consequence of Theorem ..