Convolution and Partial Sums of Certain Multivalent Analytic Functions Involving Srivastava – Tomovski Generalization of the Mittag – Leffler Function

We derive several properties such as convolution and partial sums of multivalent analytic functions associated with an operator involving Srivastava–Tomovski generalization of the Mittag–Leffler function.


Introduction
The Mittag-Leffler function E α (z) [1] and its generalization E α,β (z) [2] are defined by the following series: and respectively.It is known that these functions are extensions of exponential, hyperbolic, and trigonometric functions, since The functions E α (z) and E α,β (z) arise naturally in the resolvent of fractional integro-differential and fractional differential equations which are involved in random walks, super-diffusive transport problems, the kinetic equation, Lévy flights, and in the study of complex systems.In particular, the Mittag-Leffler function is an explicit formula for the solution the Riemann-Liouville fractional integrals that was developed by Hille and Tamarkin.
Let A(p) be the class of functions of the form which are analytic in U.For p = 1, we write A := A(1).The Hadamard product (or convolution) of two functions is given by Let P denote the class of functions ϕ with ϕ(0) = 1.Suppose that f and g are analytic in U.If there exists a Schwarz function w such that f (z) = g(w(z)) for z ∈ U, then we say that the function f is subordinate to g and write f (z) ≺ g(z) for z ∈ U. Furthermore, if g is univalent in U, then the following equivalence holds true: Throughout this paper, we assume that α, β, γ, k ∈ C; Re(α) > max{0, Re(k) − 1} and Re(k) > 0.
We define the function Q γ,k α,β (z) ∈ A(p) associated with the Srivastava-Tomovski generalization of the Mittag-Leffler function by Note that H 1,1 0,β f (z) = f (z).From ( 6), we easily have the following identity: It is noteworthy to mention that the Fox-Wright hypergeometric function q Ψ s is more general than many of the extensions of the Mittag-Leffler function.Now, we introduce a new subclass of A(p) by using the operator H γ,k α,β .

Properties of the Class
where and ρ is given by The bound ρ is sharp when Proof.We consider the case when λ > 0. Since f j ∈ Ω γ,k α,β (λ; ϕ j ), it follows that and Now, if f ∈ A(p) is defined by (11), we find from ( 14) that where Further, by using ( 14) and the Herglotz theorem, we see that Moreover, according to Lemma, we have Thus, it follows from ( 16) to (18) that which proves that f ∈ Ω γ,k α,β (λ; ϕ) for the function ϕ given by (12).In order to show that the bound ρ is sharp, we take the functions f j ∈ A(p) (j = 1, 2) defined by for which we have and Hence, for the function f given by ( 11), we have which shows that the number ρ is the best possible when For the case when λ = 0, the proof of Theorem 1 is simple, and we choose to omit the details involved.Now the proof of Theorem 1 is completed. where and and for z ∈ U.The estimates in (22) and (23) are sharp for each m ∈ N.
Similarly, if we put then we can deduce that which yields (23).
The bound in ( 23) is sharp for each m ∈ N, with the extremal function f given by (26).The proof of Theorem 2 is thus completed.