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A Remark on Schur-convexity of the mean of a convex function

In this note the new result and some remarks have been made about proving convexity and Schur-convexity of the mean of a convex function L : [0, 1 ] → R associated with the HermitHadamard inequality which is considered in literature [4] and [5]: L(t ) : = 1/ 2 (b − a ) \int_[ab] [ f(ta + (1 − t )x) + f(tb + (1 − t )x) ]dx, where f : I ⊆ R → R and a, b ∈ I, a < b. [4] HUAN-NAN S HI, Schur-convex functions related to Hadamard-type inequalities, Journal of Mathematical Inequalities 1, 1 (2007), 127– 136. [5] HUAN-NAN S HI, DA-MAO LI AND CHUN GU, The Schur-convexity of the mean of a convex function, Applied Mathematics Letter 22, 6 (2009), 932–937.

Schur-convex function ; integral arithmetic means

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