The proof of three power-exponential inequalities

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.