Open Access Research

Exponential convexity for majorization

Asif R Khan1,2*, Naveed Latif2,3 and Josip Pečarić2,4

Author Affiliations

1 Department of Mathematics, University of Karachi, University Road, Karachi, Pakistan

2 Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

3 Department of Mathematics, GC University, Faisalabad, Pakistan

4 Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia

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Journal of Inequalities and Applications 2012, 2012:105 doi:10.1186/1029-242X-2012-105

Published: 8 May 2012

Abstract

In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the exponential convexity and logarithmic convexity for majorization type results by using class of continuous functions in linear functionals. We also construct positive semi-definite matrices for majorization type results. We will vary on choice of a family of functions in order to construct different examples of exponentially convex functions and construct some means. We also prove the monotonic property.

Mathematics Subject Classification (2000): 26A51; 39B62; 26D15; 26D20; 26D99.

Keywords:
convex function; majorization; second-order divided difference; positive semi-definite matrix; exponential-convexity; log-convexity