Journal of Convex Analysis 33 (2026), No. 1&2, 293--301
Copyright Heldermann Verlag 2026
Convexity of Generators of Lp-like Paranorms
Janusz Matkowski
Institute of Mathematics, University of Zielona Gora, Poland
J.Matkowski@wmie.uz.zgora.pl
[Abstract-pdf]
Let $(\Omega ,\Sigma ,\mu )$ be a measure space with at least two disjoint sets of finite and positive measure, and $S_{+}=S_{+}(\Omega ,\Sigma ,\mu )$ denote the set of all $\mu $-integrable simple functions $\mathbf{x}:\Omega \rightarrow \mathbb{R}_{+}$ having support $\Omega \left( \mathbf{x}\right)$ of positive measure. Then, for an arbitrary bijection $\varphi :\left(0,\infty \right) \rightarrow \left( 0,\infty \right)$, the functional $\mathbf{P}_{\varphi }:S_{+}\rightarrow \mathbb{R}_{+}$ given by $\mathbf{P}_{\varphi }\left( \mathbf{x}\right) :=\varphi ^{-1}\big( \int_{\Omega (\mathbf{x})}\varphi \circ xd\mu \big)$ is well defined. The results presented support the conjecture that subadditivity of $\mathbf{P}_{\varphi }$ implies the convexity of $\varphi$. The case of superadditivity of $\mathbf{P}_{\varphi}$ is also discussed.
Keywords: Convex function, L-p-norm, Minkowski-type inequality, Mulholland inequality, Gauss-invariant mean.
MSC: 26A51, 26B25, 26D15, 39B62.
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