A functional inequality and its applications (Q5906621)

If \(f(x)\) is replaced by \(g(t) = \ln f(\exp t)\) then much of this paper (except where the domain or range of \(f\) may contain 0) deals with functions satisfying, for given \(p > 0\), \(q > 0\) \((p + q = 1)\), the inequality \(g(pt + qu) < pg(t) + qg(u)\) (or the corresponding equation in the case that \(t = u)\). \{The latter functions were called ``\(p\)-convex'' by \textit{N. Kuhn} [General Inequalities 4, Int. Ser. Num. Math. vol. 71, Birkhäuser, Basel, 269-276 (1984; Zbl 0585.39007)]. See also \textit{R. Ger} [General Inequalities 2, Int. Ser. Num. Math. vol. 47, Birkhäuser, Basel, 193-201 (1980; Zbl 0456.26007)], \textit{Z. Daroczy} and \textit{Zs. Pales} [Stochastica 11, No. 1, 5-12 (1987; Zbl 0659.39006)] and \textit{Gy. Maksa}, \textit{K. N. Nikodem} and \textit{Zs. Pales} [C. R. Math. Rep. Acad. Sci., Soc. R. Canada 13, No. 6, 274-278 (1991; Zbl 0749.26007)].\} Of course, \(g\) is (strictly) convex iff this is satisfied for all such \(p,q\) (Theorem 1, ``the first main result'' of the present paper states that then the corresponding \(n\)-term inequality is also satisfied with arbitrary positive coefficients adding up to \(0 \dots)\). Integral analogues and further inequalities are also offered.