Opuscula Mathematica (Jan 2014)
Conjugate functions, L^{p}-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 \((\ast)\)-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.
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