The proof of three power-exponential inequalities

TL;DR

This paper proves three open power-exponential inequalities for positive real numbers, confirming conjectures and introducing new inequalities and conjectures, advancing the understanding of exponential inequalities.

Contribution

The paper provides the first proofs of three open conjectures in power-exponential inequalities and introduces new related conjectures, expanding the theoretical framework.

Findings

Confirmed three open conjectures in power-exponential inequalities

Provided a new proof of a specific exponential inequality for positive reals

Proposed three new conjectures related to exponential inequalities

Abstract

In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\^i}rtoaje in the following two works: "{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented.