Glasnik Matematicki, Vol. 48, No. 2 (2013), 335-356.

3-CONVEX FUNCTIONS AND GENERALIZATIONS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA

Sadia Khalid, Josip Pečarić and Marjan Praljak

Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan
and
Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia
e-mail: pecaric@element.hr

Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan
and
Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
e-mail: mpraljak@pbf.hr

Abstract.   In this paper, we present some generalizations of an inequality of Hardy-Littlewood-Pólya. We give the n-exponential convexity and log-convexity of the functions associated with the linear functionals defined as the non-negative differences of the generalized inequalities and prove the monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the results can be applied.

2010 Mathematics Subject Classification.   26A24, 26A48, 26A51, 26D15.

Key words and phrases.   Non-increasing sequence in weighted mean, convex function, 3-convex function, n-exponential and logarithmic convexity, mean value theorems, divided difference.

DOI: 10.3336/gm.48.2.08

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