The proof of three power-exponential inequalities

Aníbal Coronel,Fernando Huancas

doi:10.1186/1029-242x-2014-509

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 Cîrtoaje (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 for all positive real numbers a and b and . In addition, three new conjectures are presented.

Highlights

The power-exponential functions have useful applications in mathematical analysis and in other theories like statistics [ ], biology [, ], optimization [ ], ordinary differential equations [ ], and probability [ ]
The first problem is perhaps one of the most ancient and useful problems concerning to power-exponential functions; see for instance [ – ]
In this paper, we are interested in some inequalities conjectured by Cîrtoaje in [, ], which are very close to the second problem

Summary

Introduction

The power-exponential functions have useful applications in mathematical analysis and in other theories like statistics [ ], biology [ , ], optimization [ ], ordinary differential equations [ ], and probability [ ]. The inequality ara + brb + crc ≥ arb + brc + cra holds true for all positive real numbers a, b, c with a ≤ b ≤ c if and only if r ≤ e. The inequality ara + bra ≥ holds for all nonnegative real numbers a and b if and only if r ≤ .

Results

Conclusion