Pólya-type inequalities for arbitrary functions (Q2771478)

The inequality NEWLINE\[NEWLINE\int_a^b \mathfrak G_2'(g,h)f\geq \mathfrak G_2\Biggl(\int_a^bg'f,\int_a^bh'f \Biggr)NEWLINE\]NEWLINE is an audacious generalization by \textit{H. Alzer} [Bul. Inst. Politeh. Iaşi, Secţ. I 36, No. 1-4, 17-18 (1990; Zbl 0765.26004)], of an inequality by Pólya to which it reduces on taking \(a=0, b=1, g(x) = x^{2p+1}, h(x) = x^{2q+1}\). The present paper presents several further generalizations. The first replaces the two geometric means in Alzer's result with two different generalized quasi-arithmetic means that are dominated by the arithmetic mean; geometric means are of course examples of such means. Other generalizations use two different symmetric, or completely symmetric, means. In addition, the results are stated for a set of \(n\) functions, rather than two functions as in the original. The proofs are based on an inequality in which the two geometric means are replaced by two functions which have the very simple property: if \(\underline w\) is a real \(n\)-tuple then \(F\in W( \underline w)\) if \(F( \underline x) \leq \sum_{i=1}^nw_ix_i\). The paper ends by extending the results to positive bounded linear operators on a Hilbert space.