Several Turán-Type Inequalities for the Generalized Mittag- Leffler Function

In this paper, several Turán-type inequalities for the more generalized Mittag-Leffler function are proved. In addition, we also gave affirmative answers to two open problems posed by Mehrez and Sitnik.


Introduction
The Mittag-Leffler function was first defined by Mittag-Leffler in 1903 [1]. In this paper, he defined the function by where Γð·Þ is a classical gamma function. In 1905, Wiman [2] generalized E α ðzÞ as In 1971, Prabhakar [3] introduced the function mechanics, quantum physics, informatics, and signal processing. In 1930, the most known result in this field is an explicit formula for the resolvent of Riemann-Liouville fractional integral proved by E. Hille and J. Tamarkin. Based on these important formulas, many results are based still for solving fractional integral and differential equations. More properties and numerous applications of the Mittag-Leffler function to fractional calculus are collected, for instance, in References [6,7]. In particular, we also refer to References [3,[8][9][10]. On the recent introduction of the Mittag-Leffler function and its generalizations, the reader may see [11,12]. There are further related generalizations of the Mittag-Leffler function.
Recently, Mehrez and Sitnik ([13,14]) obtained some Turán-type inequalities for the Mittag-Leffler function by considering monotonicity for the special ratio of sections for series of the Mittag-Leffler function. In the course of their research, they used a new method. We call it the Mehrez-Sitnik method (the reader can refer to [15][16][17]). And then they applied this method to the Fox-Wright function and got a lot of interesting new results.
Turán-type inequalities which initiated a new field of research on inequalities for special functions were proved by Paul Turán, it states where −1 < x < 1, n ∈ N, and P n ð·Þ stands for the classical Legendre polynomial.
In this paper, we mainly consider a more general generalization In the following, we mainly prove the monotonicity of ratios for sections of series of generalized Mittag-Leffler functions; the result is also closely connected with Turán-type inequalities.

Definition of the k-Gamma Function and Lemmas
In 2007, Díaz and Pariguan [18] defined the k-analogue of the gamma function for k > 0 and x > 0 as where lim k→1 Γ k ðxÞ = ΓðxÞ. Similarly, we may define the kanalogue of the digamma and polygamma functions as It is well known that the k-analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [18]): For more properties of these functions, the reader may see Reference [19].

Main Results
Our results read as follows.
Then, the Turán-type inequality holds true.

Journal of Function Spaces
Proof. Using the formulas we have where On the other hand, we have Taking into account the inequality [21], which holds for all α > 0, β > 0, and m ≥ 3, and clearly, we have A γ,τ m ðk, α, β, δÞ ≤ 0. This in turn implies that inequality (13) holds.
Proof. By taking δ = k = 1 and q = τ in Theorem 3, we easily obtain the above Turán-type inequality. The proof is complete.