A LINEAR OPERATOR ASSOCIATED WITH THE MITTAG-LEFFLER FUNCTION AND RELATED CONFORMAL MAPPINGS

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.

The above-defined functions E α (z) and E α,β (z), as well as their various further generalizations, arise naturally in the solution of fractional differential equations and fractional integro-differential equations which are associated with (for example) the kinetic equation, random walks, Lévy flights, super-diffusive transport problems and in the study of complex systems. In particular, the Mittag-Leffler function is an explicit formula for the resolvent of Riemann-Liouville fractional integrals by Hille and Tamarkin. Several properties of the Mittag-Leffler functions E α (z) and E α,β (z), together with their generalizations, can be found in a number of recent works (see [1][2][3] and [7][8][9][10][11]).
Let Σ(p) denote the class of functions of the form which are analytic in the punctured open unit disk The class Σ(p) is closed under the Hadamard product (or convolution): For f ∈ Σ(p), we consider the following operator T α,β : Σ(p) → Σ(p) associated with the Mittag-Leffler function: where z, α, β ∈ C and (α) > 0. Let P be the class of functions h with h(0) = 1, which are analytic and convex univalent in the open unit disk U = U 0 ∪ {0}.
For functions f and g analytic in U, we say that f is subordinate to g, written f ≺ g, if g is univalent in U, f (0) = g(0) and f (U) ⊂ g(U). Now we introduce the following new subclass of Σ(p).
Let A be the class of functions of the form: for some γ (γ < 1). When 0 γ < 1, S * (γ) is the class of starlike functions of order γ in U. A function f ∈ A is said to be prestarlike of order γ in U if (1.8) We denote this class by R(γ) (see [6]). It is obvious that a function f ∈ A is in the class R(0) if and only if f is convex univalent in U and R 1 2 = S * 1 2 . The study of the Mittag-Leffler functions E α (z) and E α,β (z) is a recent interesting topic in geometric function theory. In the present paper we shall make a further contribution to the subject by showing some convolution properties for meromorphic functions involving the Mittag-Leffler functions.
The following lemmas will be used in our investigation.

Hadamard product properties
In this section we shall derive several Hadamard product properties for functions in the class M α,β (λ; h).
Proof. For f ∈ M α,β (λ; h) and g ∈ Σ(p), we have In view of the conditions of Theorem 2.1, the function z p g(z) has the Herglotz representation: where µ(x) is a probability measure defined on the unit circle |x| = 1 and |x|=1 dµ(x) = 1. Since the function h is convex univalent in U, it follows from (2.1) to (2.3) that This shows that f * g ∈ M α,β (λ; h). The proof of Theorem 2.1 is completed.
Proof. From (2.1) we can write where the function ψ is defined as in (2.2).

(2.9)
The bound γ is sharp when B 1 = B 2 = −1. Finally, for the case when λ = 0, the proof of Theorem 2.3 is simple, and we choose to omit the details involved. Now the proof of Theorem 2.3 is completed.