Conjugate functions, Lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity

Abstract

For $h:(0,\infty )\rightarrow \mathbb{R}$, the function $h^{\ast }\left( t\right) :=th(\frac{1}{t})$ is called $(∗)$-conjugate to $h$. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of $(∗)$-conjugacy are proved. If $\varphi$ and $\varphi ^{\ast }$ are bijections of $\left(0,\infty \right)$ then $(\varphi ^{-1}) ^{\ast }=\left( \left[ \left( \varphi ^{\ast }\right) ^{-1}\right] ^{\ast }\right) ^{-1}$. Under some natural rate of growth conditions at $0$ and $infty$, if $\varphi$ is increasing, convex, geometrically convex, then $\left[ \left( \varphi^{-1}\right) ^{\ast }\right] ^{-1}$ has the same properties. We show that the Young conjugate functions do not have this property. For a measure space $(\Omega ,\Sigma ,\mu )$ denote by $S=S(\Omega ,\Sigma ,\mu )$ the space of all $\mu$-integrable simple functions $x:\Omega \rightarrow \mathbb{R}$. Given a bijection $\varphi :(0,\infty )\rightarrow (0,\infty )$, define $\mathbf{P}{\varphi }:S\rightarrow \lbrack 0,\infty )$ by $\mathbf{P}{\varphi }(x):=\varphi ^{-1}\bigg( \int\limits_{\Omega (x)}\varphi \circ \left\vert x\right\vert d\mu \bigg),$ where $\Omega(x)$ is the support of $x$. Applying some properties of the $(∗)$ operation, we prove that if $\int\limits_{\Omega }xy\leq \mathbf{P}{\varphi }(x)\mathbf{P}{\psi }(y)$ where $\varphi ^{-1}$ and $\psi ^{-1}$ are conjugate, then $\varphi$ and $\psi$ are conjugate power functions. The existence of nonpower bijections $\varphi$ and $\psi$ with conjugate inverse functions $\psi =\left[ ( \varphi ^{-1}) ^{\ast}\right] ^{-1}$ such that $\mathbf{P}{\varphi }$ and $\mathbf{P}{\psi }$ are subadditive and subhomogeneous is considered.