Conjugate functions, L^p-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity

Janusz Matkowski

Abstract. For \(h:(0,\infty )\rightarrow \mathbb{R}\), the function \(h^{\ast }\left( t\right) :=th(\frac{1}{t})\) is called \((\ast)\)-conjugate to \(h\). This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of \((\ast)\)-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 \((\ast)\) 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.

Keywords: \(L^{p}\)-norm like functional, homogeneity, subhomogeneity, subadditivity, the converses of Minkowski and Hölder inequalities, generalization of the Minkowski and Hölder inequalities, conjugate (complementary) functions, Young conjugate functions, convex function, geometrically convex function, Wright convex function, functional equation.

Mathematics Subject Classification: 26D15, 26A51, 39B22, 39B62, 46B25, 46E30.

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