PROOFS OF THREE OPEN INEQUALITIES WITH POWER-EXPONENTIAL FUNCTIONS

The main aim of this paper is to give a complete proof to the open inequality with power-exponential functions a^ea+b^eb≥a^eb+b^ea,which holds for all positive real numbers a and b. Notice that this inequality was proved in for only a≥b≥1e and1e≥a≥b.In addition, other two open inequalities with power-exponential functions are proved, and three new conjectures are presented