On Jensen’s Inequality Involving Averages of Convex Functions
DOI:
https://doi.org/10.5644/SJM.08.1.04Keywords:
Jensen's inequality, log-convex function, exponential convexityAbstract
Using Wulbert result from Wulbert we deduce a method for constructing exponentially convex functions. To this date there is no known operative criteria for recognizing exponentially convex functions, so our method is of a special interest since there is a lack of examples of this class of functions. Constructed exponentially convex functions are then used for the construction of two and three parameters Cauchy means, that have the very nice property: they have the monotonicity property in their defining parameters. Particularly, we generalize recent results on Cauchy means from Matlob and Schur.
2000 Mathematics Subject Classification. Primary 26D15, Secondary 26D9
Downloads
References
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh, 1965.
M. Anwar and J. Peˇcari´c New means of Cauchy's type, J. Inequal. Appl., Art. ID 163202, 10 pages, (2008).
T. M. Bisgaard and Z. Sasv´ari, Characteristic Functions and Moment Sequences, Positive De niteness in Probability, Huntington, NY: Nova Science Publishers, 2000.
V. ˇCuljak, I. Franji´c, J. Peˇcari´c and G. Roqia, Schur-convexity of averges of convex functions , J. Inequal. Appl., Art. ID 581918, 25 pages, (2011).
W. Ehm, M. G. Genton and T. Gneiting, Stationary covariances associated with exponentially convex functions, Bernoulli, 9 (4) (2003), 607–615.
E. Issacson and H. B. Keller, Analysis of Numerical methods, Dover Publications, Inc. New York, 1966.
J. Jakˇseti´c and J. Peˇcari´c, Exponential Convexity Method, (submitted).
A. McD. Mercer, Some new inequalities involving elementary mean values, J. Math. Anal. Appl., 229 1999), 677–681.
J. E. Peˇcari´c, F. Proschan, and Y. L.Tong, Convex functions, partial orderings, and statistical applications, Academic Press, Inc, 1992.
J. Peˇcari´c and G. Roqia, Generalization of Stolarsky type means, J. Inequal. Appl., Art. ID 720615, 15 pages, (2010).
J. Peˇcari´c, M. R. Lipanovi´c and H. N. Srivastava, Some mean value theorems of Cauchy type, Fract. Calc. Appl. Anal., 9 (2) (2006), 143–158.
J. Peˇcari´c, I. Peri´c and H. N. Srivastava, A family of Cauchy type mean value theorems, J. Math. Anal. Appl.,36 (2005), 730–739.
J. Peˇcari´c, I. Peri´c and M. R. Lipanovi´c, Integral representations of generalized Whiteley means and related inequalities, Math. Inequal. Appl.,12 (2) (2009), 295–309.
J. Peˇcari´c and V. ˇSimi´c, Stolarsky-Tobey mean in n variables, Math. Inequal. Appl., 2 (3) (1999), 325–341.
J. L. Schiff, The Laplace transform. Theory and applications. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999.
S. Simi´c, An extension of Storalsky means to the multivariable case, Int. J. Math. Mat. Sci., Art. ID 432857, 14 pages, (2009).
S. Simi´c, On certain new inequalities in information theory, Acta Math. Hungar., 124 (4) (2009), 353–361.
D. E. Wulbert, Favard's Inequality on average values of convex functions, Math. Comp. Mod., 30 (3) (2000), 853–856.
