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The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established. MSC: Primary 26D15; 05E05; 26B25

The Schur-convexity, the Schur-geometric convexity and the Schurharmonic convexity of two mappings which related to Hadamard-type integral inequalities are researched. And three refinements of Hadamard-type integral inequality are obtained, as applications, some inequalities related to the arithmetic mean, the logarithmic mean and the power mean are established.

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.

a b s t r a c tThe Schur-convexity and the Schur-geometric convexity with variables (x, y) ∈ R 2 ++ for fixed (s, t) of Gini means G(r, s; x, y) are discussed. Some new inequalities are obtained.