Abstract
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 space H p . Some of our results improve the results in the literature.
Highlights
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
It was introduced by Wiman [2] and was named as Mittag–Leffler type function
We investigate some geometric properties of function Eα,κ (z) with real parameters α and κ
Summary
Convex of order μ, where μ ∈ [0, 1), is not of the form g (z) =. Let μ ∈ [0, 1), (1 − μ) κ 3 + (6μ − 8) κ 2 + (7 − 7μ) κ + (8 − 6μ) > 0. The function Eα,κ (z) is not of the form of l + dz 1 − zeiγ (for μ 6= 1/2) and l + d log 1 − zeiγ (for μ = 1/2). By part (iv) of Theorem 9, Eα,κ (z) ∈ C (μ). By using Lemma 8, we have the required result. The function Eα,κz(z) is complete; h (z) is complete. This implies that h (z) is bounded. If g ∈ R (1/2), Eα,κ (z) ∗ g ∈ R (0)
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