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The Schur-convexity of the mean of a convex function. (English) Zbl 1180.26009
The authors establish the Schur-convexity at the upper and lower limits of the integral for the mean of a convex function. Furthermore, a new proof of the inequality
\[ f\bigg(\frac{a+b}{2}\bigg)=H(0) \leq H(t) \leq H(1)= \frac1{b-a}\int^ b_ a f(x)\,dx \] obtained by S. S. Dragomir [J. Math. Anal. Appl. 167, No. 1, 49–56 (1992; Zbl 0758.26014)] is given, where
\[ H(t)=\frac1{b-a}\int^ b_ a f\bigg(tx+(1-t)\frac{a+b}{2}\bigg)\,dx \]
and \(f:[a,b] \to \mathbb R\) is a convex function.

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
Citations:
Zbl 0758.26014
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References:
[1] Dragomir, S.S., Two mappings in connection to hadamard’s inequalities, J. math. anal. appl., 167, 1, 49-56, (1992) · Zbl 0758.26014
[2] Yang, G.S.; Hong, M.C., A note on hadamard’s inequality, Tamkang J. math., 28, 1, 33-37, (1997) · Zbl 0880.26019
[3] Elezovic, N.; Pecaric, J., A note on Schur-convex functions, Rocky mountain J. math., 30, 3, 853-856, (2000) · Zbl 0978.26013
[4] Wang, B.Y., Foundations of majorization inequalities, (1990), Beijing Normal Univ. Press Beijing, China, (in Chinese)
[5] Kuang, J.C., Applied inequalities, (2002), Shandong Press of Science and Technology Jinan, China, (in Chinese)
[6] Ostle, B.; Terwilliger, H.L., A companion of two means, Proc. montana acad. sci., 17, 1, 69-70, (1957)
[7] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its application, (1979), Academies Press New York · Zbl 0437.26007
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