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A class of symmetric functions for multiplicatively convex function. (English) Zbl 1144.26015
Let $$f$$ be a non-negative function defined on the positive semiaxis. The author associates to $$f$$ a new symmetric function and the corresponding symmetric mean. Their properties, including Schur-geometric convexity, are investigated under the hypothesis that $$f$$ is multiplicatively convex. Some inequalities are established, generalizing known results.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations 26D15 Inequalities for sums, series and integrals
##### Keywords:
multiplicatively convex functions; inequalities
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