A Unitary Treatment of Certain Inequalities Involving Means

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DOI: 10.46793/KgJMat2102.181A


The aim of this paper is to state and prove certain inequalities that involve means (e.g., the arithmetic, geometric, logarithmic means) using a particular result. First of all we recall useful properties of a real-valued convex function that will be used in the proof of our inequalities. Further, we present three inequalities, the first involving the logarithmic mean, the second involving the classical arithmetical and geometrical means and in the last we introduce a new mean. Finally, we give alternate proofs to the Schweitzer’s inequality and Khanin’s inequality.


Generalized mean, logarithmic mean, convex function, maximum point.


[1]   E. F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1961.

[2]   W. W. Breckner and T. Trif, Convex Functions and Related Functional Equations, Presa Universitară Clujeană, 2008.

[3]   P. S. Bullen, A Dictionary of Inequalities, Kluwer Academic Publishers, Dordrecht, 2003.

[4]   P. Cerrone and S. S. Dragomir, Mathematical Inequalities, A Perspective, CRC Press, Boca Raton, London, New York, 2011.

[5]   I. Maruşciac, Programare Geometrică şi Aplicaţii, Dacia, Cluj-Napoca, 1978.

[6]   D. S. Mitrinović, J. E. Pećarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[7]   C. Niculescu and L. E. Persson, Convex Functions and their Applications, Springer-Verlag, New York, 2006.

[8]   A. Roberts and D. Varberg, Convex Functions, Academic Press, New York, 1973.