Turán type inequalities for generalized Mittag-Leffler function

In this paper, we show several Turán type inequalities for a generalized Mittag-Leffler function with four parameters via the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,k)$\end{document}(p,k)-gamma function.


Introduction and main results
In , Turán established a remarkable inequality in the special function theory, P n+ (r)  > P n (r)P n+ (r) for all r ∈ (-, ) and n ∈ N, where P n is the Legendre polynomial, that is, P n (r) = F -n, n + ; ; r  .
Here, for given complex numbers a, b and c with c = , -, -, . . . , the Gaussian hypergeometric function is the analytic continuation to the slit place C \ [, ∞) of the series F(a, b; c; z) =  F  (a, b; c; z) = ∞ n= (a, n)(b, n) (c, n) z n n! , |z| < .
Here, (a, ) =  for a = , and (a, n) is the shifted factorial function or the Appell symbol (a, n) = a(a + )(a + ) · · · (a + n -) for n ∈ Z + ; see [, ]. There is an extensive topic dealing with Turán type inequalities, and it has been generalized in many directions for various orthogonal, polynomial and special functions. The Mittag-Leffler function is defined by where (·) is a classical gamma function. The Mittag-Leffler function plays an important role in several branches of mathematics and engineering sciences, such as statistics, chemistry, mechanics, quantum physics, informatics and others. In particular, it is involved in the explicit formula for the resolvent of Riemann-Liouville fractional integrals by Hille and Tamarkin Motivated by [, ], we consider the following generalized Mittag-Leffler function with four parameters: It is easily seen that the functions (.) and (.) are special cases of Wright-Fox functions in the Wright series representation (or multi-index Mittag-Leffler functions) in [].
It is well known that lim p→∞ p, x >  are k-gamma and gamma functions, respectively. These formulas and more properties can be found in [].
The logarithmic derivative of the (p, k)-gamma function is known as a generalized digamma function. Its derivatives ψ (n) p,k (x) are known as generalized polygamma functions.
Our results read as follows.

Proofs of main results
Proof of Theorem . Simple computation yields where we apply that the function ψ p,k (x) is concave on R. Therefore, we find that the function β → p,k (β) p,k (αk+β) is strictly log-convex on (, ∞). Using the fact that the sum of log-convex functions is also log-convex, we see that the function f is strictly log-convex on (, ∞).
Due to inequality (.), we easily derive That is, Using the definition of p,k (x), we easily obtain The proof of Theorem . is complete.