Refinements and extensions of an inequality. II
By using certain known results for absolutely monotonic and absolutely convex functions, as well as classical integral inequalities, the following result is deduced: \[ (b^{x+ k}- a^{x+k})/(b^x- a^x)\geq {x+ k\over x} \Biggl({a+ b\over 2}\Biggr)^k, \] where \(k\geq 1\), \(x\geq 1\), \(0<a<b\) are real numbers. For \(0< k< 1\) or \(0< x< 1\), this inequality is not generally true. [See also Part III, J. Math. Anal. Appl. 227, No. 2, 439-448 (1998; following review); Part I has not yet been received by Zbl ].
absolutely monotonic and absolutely convex functions
scientific article; zbMATH DE number 1053443