Geometric Properties of Normalized Mittag – Leffler Functions

The aim of this paper is to investigate certain properties such as convexity of order μ, close-to-convexity of order (1 + μ)/2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag–Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy spaceHp. Some of our results improve the results in the literature.


introduction
The one parameter Mittag-Leffler function E α (z) defined by was introduced by Mittag-Leffler [1].This function of complex variable is entire.The series defined by Equation (1) converges in C when Re(α) > 0. Consider that the function E α,κ (z) which generalizes the function E α (z) is defined by It was introduced by Wiman [2] and was named as Mittag-Leffler type function.The series in Equation (2) converges in C when Re (α) > 0 and Re (κ) > 0. Furthermore, the functions defined in (1) and (2) are entire of order 1/Re (α) and of type 1, for more details, see [3].The function E α,κ (z) and its analysis with its generalizations is increasingly becoming a rich research area in mathematics and its related fields.A number of researchers studied and analyzed the function given in (2) (see Wiman [2,4,5]).One can find this function in the study of kinetic equation of fractional order, Lévy flights, random walks, super-diffusive transport as well as in investigations of complex systems.
In a similar manner, the advanced studies of these functions reflect and highlight many vital properties of these functions.The function E α,κ (z) generalizes many functions such as The interested readers are suggested to go through [6][7][8][9].
Let A be the family of all functions g having the form and are analytic in D = {z : |z| < 1} and S denote the family of univalent functions from A.
The families of functions which are convex, starlike and close-to-convex of order µ , respectively, are defined as: and It is clear that C * (0) = C, S * (0) = S * and K (0) = K.Consider the class H of all analytic functions in D and µ < 1, Baricz [10] introduced the classes For η = 0, we have the classes of analytic functions P 0 (α) and R 0 (α) respectively.Also for η = 0 and α = 0, we have the classes P and R.
For the functions g ∈ A given by (1) and h ∈ A given by then the convolution (Hadamard product) of g and h is defined as: It is clear that the function E α,κ (z) is not in class A. Recently, Bansal and Prajapat [11] considered the normalization of the function E α,κ (z) given as In this article, we investigate some geometric properties of function E α,κ (z) with real parameters α and κ.
We need the following results in our investigations.
, for z ∈ D.
Proof.For the proof of this result, we have to show that where ∀ m ≥ 1, α ≥ 3 2 and κ ≥ 3 2 .Now, to show that {a m } ∞ m=1 is decreasing, we prove that a m + a m+2 ≥ 2a m+1 .Take 2 and κ ≥ 3 2 , which shows that {a m } ∞ m=1 is decreasing and convex sequence.Now, from the Lemma 4, we have which is equivalent to Proof.To show that E α,κ (z) is starlike in D, we prove that {ma m } and {ma m − (m + 1)a m+1 } both are non-increasing in view of Lemma 5. Since a m ≥ 0 for the normalized Mittag-Leffler function under the given conditions, consider Here, A m = mΓ(κ) Γ(α(m−1)+κ) .By taking the same computations as in Theorem 2, we get the proof.
Consider the integral operator F γ : D → C, where γ ∈ C, γ = 0, Here, F γ ∈ A. We prove that F γ ∈ S in D.
Proof.A calculation gives Since E α,κ (z) ∈ A, then by Schwarz Lemma, triangle inequality and ( 6), we obtain By using Lemma 2, F γ ∈ S in D.
This implies that .
This completes the proof.
(iv) We use the inequality . By using (7) and (8), we have This implies that , hence the result.
Substituting µ = 0 in Theorem 9, we obtained the following results.

Hardy Space of Mittag-Leffler Function
Consider the class H of analytic functions in D = {z : |z| < 1} and H ∞ denote the space bounded functions on H. Let g ∈ H, set If M q (r, g) is bounded for r ∈ [0, 1) , then g ∈ H q .It is clear that For some details, see [18].It is also known [18] that, if Re (g (z)) > 0 in D, then g ∈ H p , p < 1, g ∈ H p/(1−p) , 0 < p < 1.