The monotonicity and convexity of a function involving digamma one and their applications

TL;DR

This paper studies the monotonicity and convexity of a function involving the digamma function and a logarithmic expression, providing new bounds and applications for harmonic numbers and the Euler-Mascheroni constant.

Contribution

It introduces a new function involving the digamma function, analyzes its properties, and determines optimal parameters for tight bounds and accurate approximations.

Findings

Established monotonicity and convexity conditions for the function.

Derived sharp bounds for the digamma function and harmonic numbers.

Constructed sequences that approximate the Euler-Mascheroni constant with high accuracy.

Abstract

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left( 4/15,\infty \right) $ or $\left( 0,\infty \right) \times \left( 1/15,\infty \right) $ by the formula% \begin{equation*} \mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}% \right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{% 15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and convexity of the function $x\rightarrow F_{a}\left( x\right) =\psi \left( x+1\right) -\mathcal{L}(x,a)$, where $\psi $ denotes the Psi function. And, we determine the best parameter $a$ such that the inequality $\psi \left( x+1\right) <\left( >\right) \mathcal{L}% (x,a) $ holds for $x\in \left( -1,\infty \right) $ or $\left( 0,\infty \right) $, and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence $\left( l_{n}\left( a\right) \right) $ defined by $l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right) $, which gives extremely accurate values for $\gamma $.