New inequalities for the function $y=t\ln t$

TL;DR

This paper introduces new inequalities for the function y=t ln t, analyzing its local behavior near t=1 and its global behavior over [1,∞) and (0,1], expanding understanding of its properties.

Contribution

It proposes several novel inequalities for y=t ln t, focusing on both local and global behaviors, extending previous results in the literature.

Findings

New inequalities for y=t ln t established

Analysis of local behavior near t=1 conducted

Global behavior on [1,∞) and (0,1] characterized

Abstract

The main aim of this note, which can be viewed as a certain addendum to the paper \cite{2020}, is to propose several new inequalities for the function $y=t\ln t.$ We consider the local behaviour of this function near the point $t=1,$ as well as the global behaviour of this function on the intervals $[1,\infty)$ and $(0,1].$