Abstract

Let (Ω,Σ,μ)(\Omega,\Sigma,\mu ) be a measure space with at least two disjoint sets of finite and positive measure, and S+=S+(Ω,Σ,μ)S_{+}=S_{+}(\Omega,\Sigma,\mu ) denote the set of all μ\mu-integrable simple functions x:Ω→R+\mathbf{x}:\Omega \rightarrow \mathbb{R}_{+} having support Ω(x)\Omega \left( \mathbf{x}\right) of positive measure. Then, for an arbitrary bijection φ:(0,∞)→(0,∞)\varphi:\left(0,\infty \right) \rightarrow \left( 0,\infty \right), the functional Pφ:S+→R+\mathbf{P}_{\varphi }:S_{+}\rightarrow \mathbb{R}_{+} given by Pφ(x):=φ−1(∫Ω(x)φ∘xdμ)\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 Pφ\mathbf{P}_{\varphi } implies the convexity of φ\varphi. The case of superadditivity of Pφ\mathbf{P}_{\varphi} is also discussed.

Author information

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Keywords

  • Convex function
  • L-p-norm
  • Minkowski-type inequality
  • Mulholland inequality
  • Gauss-invariant mean

Mathematics Subject Classification

26A51, 26B25, 26D15, 39B62

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Suggested citation

J. Matkowski. “Convexity of Generators of L^p-like Paranorms.” Journal of Convex Analysis 33 (2026), No. 1&2, 293–301. https://doi.org/10.68381/jca33018

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.