Inequalities for Averages of Convex and Superquadratic Functions.

2016

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We consider the averages An(f) = 1/(n 1) P n 1 r=1 f(r/n) and Bn(f) = 1/(n + 1) P n=0 f(r/n). If f is convex, then An(f) increases with n and Bn(f) decreases. For the class of functions called superquadratic, a lower bound is given for the successive differ- ences in these sequences, in the form of a convex combination of functional values, in all cases at least f(1/3n). Generalizations are formulated in which r/n is replaced by ar/an and 1/n by 1/cn. Inequalities are derived involving the sum P n=1 (2r 1) p .

Subfields: Applied Mathematics, Geometry and Topology

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