A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

Abstract

In the article, the sufficient and necessary conditions such that a class of functions which involve the psi function and the ratio (x + t)/ (x + s) are logarithmically completely monotonic are established, the best bounds for the ratio (x + t)/ (x + s) are given, and some comparisons with known results are carried out, where s and t are two real numbers and x > − min{s, t}.

A new upper bound in the second Kershaw's double inequality and its generalizations

“…There also have been a lot of literature, such as [5,11,10,16,19,23] and the related references therein, about the refinements, extensions, and generalizations of the second Kershaw's double inequality (12).…”

“…The aim of this paper is to refine the very right-hand side inequalities in (19) and (16), and then new upper bounds in the second Kershaw's double inequality and for the divided differences of the psi and polygamma functions are established.…”

A new lower bound in the second Kershaw's double inequality

“…There have been a lot of literature about these two double inequalities and their history, background, refinements, extensions, generalizations and applications. For more detailed information, refer to [9,10,14,15] and the references therein.…”

Refinements, extensions and generalizations of the second Kershaw's double inequality

“…For more information, please refer to [5,7,9,10,11,12,14,19,20,21,22,23,30,32,34,35,36,38,39,40,41,49] and the references therein.…”