Notes on three conjectures involving the digamma and generalized digamma functions

Ladislav Matejíčka

doi:10.1186/s13660-018-1936-z

Ladislav Matejíčka

Open Access

https://doi.org/10.1186/s13660-018-1936-z

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Abstract

In the paper, we solve one conjecture on an inequality involving digamma function, an open problem, and a conjecture on monotonicity of functions involving generalized digamma function. We also prove a new inequality for digamma function.

Highlights

In the last years, the (p, k)-analogue of the gamma and polygamma functions has been studied intensively by a lot of authors
We proved a new inequality (Theorem 1) for the digamma function
Authors’ contributions The author completed the paper and approved the final manuscripts

Summary

Introduction

The (p, k)-analogue of the gamma and polygamma functions has been studied intensively by a lot of authors. The digamma function [11,12,13, 24] is defined by 1∞ x ψ(x) = Γpk(x) = x(x + k) · · · (x + pk) for k > 0 and x > 0, p ≥ 0, p ∈ N , and the (p, k)-digamma function ψpk(x) = Conjecture 1 ([19]) For p > 0 and k ≥ 1, the function φpk(x) = ψpk(x) + ln e1 x

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