Complete monotonicity related to the k-polygamma functions with applications

Li Yin,Jumei Zhang,Xiuli Lin

doi:10.1186/s13662-019-2299-6

Li Yin, Jumei Zhang + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-2299-6

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Abstract

In this paper, we prove complete monotonicity of some functions involving k-polygamma functions. As an application of the main result, we also give new upper and lower bounds of the k-digamma function.

Highlights

The Euler gamma function is defined for all positive real numbers x by ∞Γ (x) = tx–1e–t dt.The logarithmic derivative of Γ (x) is called the psi or digamma function
The polygamma functions ψ(m)(x) for m ∈ N are defined by ψ (m)(x) dm dxm ψ (x)
From the above theorem it follows that completely monotonic functions on [0, ∞) are always strictly completely monotonic unless they are constant

Summary

Introduction

For some of the work as regards origin, history, the complete monotonicity, and inequalities of these special functions one may refer to [1–12, 18–21, 29, 30, 33–39] and the references therein. We may define the k-analog of the digamma and polygamma functions as d ψk(x) = dx ln Γk(x) and ψk(m)(x) dm dxm ψk(x). From the above theorem it follows that completely monotonic functions on [0, ∞) are always strictly completely monotonic unless they are constant (see [34]). In [46], Yin et al gave a concave theorem and some inequalities for the k-digamma function.

Results

Conclusion