Logarithmic convexity of extended mean values

In this article, the logarithmic convexity of the extended mean values are proved and an inequality of mean values is presented. As by-products, two analytic inequalities are derived. Two open problems are proposed.

[1]  K. Stolarsky,et al.  Generalizations of the Logarithmic Mean , 1975 .

[2]  Zsolt Páles,et al.  Inequalities for differences of powers , 1988 .

[3]  E. B. Leach,et al.  Extended mean values II , 1983 .

[4]  Qi Feng,et al.  Refinements and Extensions of an Inequality, II , 1997 .

[5]  Feng Qi (祁锋),et al.  A simple proof of monotonicity for extended mean values , 1998 .

[6]  Feng Qi (祁锋),et al.  A new proof of monotonicity for extended mean values , 1999 .

[7]  Feng Qi,et al.  Generalized Abstracted Mean Values , 2000 .

[8]  Feng Qi,et al.  NEW PROOFS OF WEIGHTED POWER MEAN INEQUALITIES AND MONOTONICITY FOR GENERALIZED WEIGHTED MEAN VALUES , 2000 .

[9]  Feng Qi (祁锋),et al.  TAMKANG JOURNAL OF MATHEMATICS Volume 29, Number 2, Summer 1998 ON A TWO-PARAMETER FAMILY OF NONHOMOGENEOUS MEAN VALUES , 2021 .

[10]  Principal Investigator,et al.  In Cooperation with , 2001 .

[11]  Feng Qi (祁锋),et al.  Refinements and Extensions of an Inequality, III☆ , 1998 .

[12]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[13]  E. B. Leach,et al.  Extended Mean Values , 1978 .

[14]  Feng Qi,et al.  Generalized weighted mean values with two parameters† , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  Feng Qi (祁锋),et al.  Other Proofs of Monotonicity for Generalized Weighted Mean Values , 1999 .

[16]  Feng Qi (祁锋),et al.  Note on monotonicity of generalized weighted mean values , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[18]  Young-Ho Kim,et al.  Refinements and Extensions of an Inequality , 2000 .

[19]  Feng Qi,et al.  The function $(b^x-a^x)/x$: Inequalities and properties , 1998 .