Refinements and extensions of an inequality. III (Q1281557)

The authors give a complete characterization of the set \((x,y)\subset (-\infty,\infty)\times(-\infty,\infty)\) such that \[ (b^{x+ y}-a^{x+y})/(b^x- a^x)\geq {x+ y\over x} \Biggl({a+ b\over 2}\Biggr)^y, \] where \(a,b>0\) are given real numbers. In the proof the theory of extended mean values \(E(r,s; x,y)\) introduced by \textit{E. B. Leach} and \textit{M. C. Sholander} [J. Math. Anal. Appl. 92, 207-223 (1983; Zbl 0517.26007)] is applied. This method proves the importance of general means and their inequalities. [For Part II see J. Math. Anal. Appl. 211, No. 2, 616-620 (1997; preceding review].