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?

The author provides answers to the above open problems. In particular, it was shown that there exists a larger set of solutions to Mulholland’s inequality compared to the one delimited by Mulholland’s condition. Furthermore, it was also shown that the set of functions solving Mulholland’s inequality is not closed with respect to compositions. The proofs of the above problems were presented and discussed.