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A remark on Schur-convexity of the mean of a convex function. (English) Zbl 1329.26053

Summary: 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 \(L:[0,1] \to \mathbb{R}\) associated with the Hermite-Hadamard inequality which is considered in literature [H.-N. Shi, J. Math. Inequal. 1, No. 1, 127–136 (2007; Zbl 1131.26017); H.-N. Shi et al., Appl. Math. Lett. 22, No. 6, 932–937 (2009; Zbl 1180.26009)]:
\[ L(t):= \frac1{2(b-a)}\int_a^b[ f (ta + (1-t)x)+f(tb+(1-t)x)]dx, \]
where \(f : I \subseteq \mathbb{R}\to\mathbb{R}\) and \(a,b \in I\), \(a < b\).

MSC:

26E60 Means
26D15 Inequalities for sums, series and integrals
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