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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
Zbl 0758.26014
Full Text:
##### References:
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