Abstract
Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results.
Highlights
Lin and Fukushima [1] introduced the concept of higher order strongly convex functions and used it in the study of mathematical program with equilibrium constraints
We introduce some new classes of higher order strongly convex functions and higher order strongly affine convex functions with respect to the bifunction ξ (·, ·)
We have introduced and studied a new class of convex functions, which are called higher order strongly convex functions
Summary
Lin and Fukushima [1] introduced the concept of higher order strongly convex functions and used it in the study of mathematical program with equilibrium constraints. Characterizations of the higher order strongly convex functions discussed in Lin and Fukushima [1] are not correct. We have deduced the weakly parallelogram laws for the L p -spaces, which have been discussed in References [19,20,21,22,23] from the concept of higher order strongly affine convex functions. This fact can be viewed as an elegant and interesting application of the higher order strongly convex functions. It is expected that the ideas and techniques of this paper may stimulate further research in this field
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