Monotonicity, concavity, and inequalities related to the generalized digamma function

Li Yin,Yong-Li Wang,Li-Guo Huang,Xiu-Li Lin

doi:10.1186/s13662-018-1695-7

Abstract

In this paper, we establish a concave theorem and some inequalities for the generalized digamma function. Hence, we give complete monotonicity property of a determinant function involving all kinds of derivatives of the generalized digamma function.

Highlights

It is well known that the Euler gamma function is defined by ∞(x) = tx–1e–t dt, x > 0.The logarithmic derivative of (x) is called the psi or digamma function
The gamma, digamma, and polygamma functions play an important role in the theory of special function, and have many applications in many other branches such as statistics, fractional differential equations, mathematical physics, and theory of infinite series
It is natural to ask if one can generalize these results to the generalized digamma function with single parameters

Summary

Introduction

1 Introduction It is well known that the Euler gamma function is defined by The logarithmic derivative of (x) is called the psi or digamma function. In [6], the k-analogue of the gamma function is defined for k > 0 and x > 0 as follows: k(x) = It is natural that the k-analogue of the digamma function is defined for x > 0 by d ψk(x) = dx log k(x) = It is worth noting that Nantomah et al gave (p, k)-analogue of the gamma and the digamma functions in [15].

Results

Conclusion