Convexity of Generators of L^p-like Paranorms

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.