engleski

On refining of inequalities for convex functions by the concept of superquadracity

In 2004 S. Abramovich, G. Jameson and G. Sinnamon introduced a new interesting class of functions: the class of superquadratic functions. We say that the function ϕ is superquadratic if for any x≥0 there exists C(x)∈R such that ϕ(y)≥ϕ(x)+C(x)(y-x)+ϕ(|y-x|), ∀y≥0. In 2007 S. Abramovich, S. Banić and M. Matić generalized this concept for the functions in several variables. The class of superquadratic functions is strongly related to the class of convex functions: it can be proved that any nonnegative superquadratic function is convex. Using some previously proved characterizations and properties of this new class we establish "superquadratic variants" of several well known inequalities for convex functions. The refinements of many important inequalities for convex functions easily follow as special cases when considered superquadratic functions are nonnegative.

convex functions; superquadratic functions; Jensen's inequality; Hölder's inequality; Slater's inequality; Hermite-Hadamard inequalities

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