On superquadratic and logarithmically superquadratic functions. (English) Zbl 1522.26016
Summary: In this paper, we develop a concept of a logarithmically superquadratic function. Such a class of functions is defined via superquadratic functions. We first establish some new properties of superquadratic functions. In particular, we derive the corresponding superadditivity relation and its reverse, as well as the external form of the Jensen inequality and its reverse. Then, as a direct consequence of the established results, we obtain the corresponding properties for logarithmically superquadratic functions. Further, we show that logarithmically superquadratic functions with values greater than or equal to one are convex and logarithmically superadditive. In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26A51 | Convexity of real functions in one variable, generalizations |
Keywords:
Jensen inequality; logarithmically superquadratic function; convexity; superadditivity; refinementReferences:
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