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 Cirtoaje (J. Inequal. Pure Appl. Math. 10:21, 2009; J. Nonlinear Sci. Appl. 4(2):130-137, 2011). Moreover, we present a new proof of the inequality a(ra) + b(rb) >= a(rb) + b(ra) for all positive real numbers a and b and r is an element of [0, e]. In addition, three new conjectures are presented.