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Zeitschrift für Analysis und ihre Anwendungen


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Volume 14, Issue 1, 1995, pp. 175–183
DOI: 10.4171/ZAA/669

Published online: 1995-03-31

Gauss’ and Related Inequalities

S. Varošanec[1] and J. Pečarić[2]

(1) University of Zagreb, Croatia
(2) University of Zagreb, Croatia

Let $g : [a,b] \to \mathbb R$ be a non-negative increasing differentiable function and $f : [a,b] \to \mathbb R$ a non-negative function such that the quotient $f/g’$ is non-decreasing. Then the function $$Q(r) = (r+ 1) \int^b_a g(x)^r f(x)dx$$ is log-concave. If $g(a) = 0, b \in (a, \infty]$ and the quotient $f/g’$ is non-increasing, then the function $Q$ is log-convex.

Keywords: Gauss’ inequality, Popoviciu ’s inequality, Hölder’s inequality, log-concave functions, log-convex functions

Varošanec S., Pečarić J.: Gauss’ and Related Inequalities. Z. Anal. Anwend. 14 (1995), 175-183. doi: 10.4171/ZAA/669