New solutions to Mulholland inequality. (English) Zbl 1332.26030
R. M. Tardiff [Aequationes Math. 27, 308–316 (1984; Zbl 0549.26013)] states: Let \(f:\mathbb{R}_0^+ \to \mathbb{R}_0^+\) be a differentiable increasing bijection. If both \(f\) and \(\log\circ f' \circ \exp\) are convex, then \(f\in \mathrm{MI}\), where \(\mathrm{MI}\) is the set of all increasing bijections that solve Mulholland’s inequality. In this paper, answers are found to the following open problems:
- 1.
- Is Mulholland’s conditions presented in the above statement also necessary, i.e., is \(\mathrm{MC}=\mathrm{MI}\) or is \(\mathrm{MC} \subseteq \mathrm{MI}\)?, where \(\mathrm{MC}\) denotes the set of all bijections that comply with the assumptions of the above statement.
- 2.
- Is the set of functions solving Mulholland’s inequality closed with respect to their compositions?
Reviewer: James Adedayo Oguntuase (Abeokuta)
MSC:
26D07 | Inequalities involving other types of functions |
39B72 | Systems of functional equations and inequalities |
26D15 | Inequalities for sums, series and integrals |
54E70 | Probabilistic metric spaces |
26A51 | Convexity of real functions in one variable, generalizations |
03E72 | Theory of fuzzy sets, etc. |
Keywords:
convex function; dominance of strict triangular norms; geometrically convex function; Minkowski inequality; Mulholland inequality; probabilistic metric spacesCitations:
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