On some inequalities of Brenner and Alzer for concave functions (Q1916783)

The authors offer simple proofs and generalizations of inequalities published by \textit{J. L. Brenner} and \textit{H. Alzer} [Proc. R. Soc. Edinb., Sect. A 118, No. 1/2, 173-192 (1991; Zbl 0736.26008)]. One proof uses the following nice result of \textit{L. Fejér} [Math. Naturwiss. Anz. Ungar. Akad. Wiss. 24, 369-390 (1906)]. Let \(g\) be positive and integrable on \([a, b]\), symmetric with respect to \((a+ b)/2\) and \(f\) concave on \([a, b]\). Then the weighted integral mean \(\int^b_a f(t) g(t) dt/\int^b_a g(t) dt\) always lies between \({f(a)+ f(b)\over 2}\) and \(f({a+ b\over 2})\).