Conjugate functions, L^{p}-norm like functionals, the generalized H??lder inequality, Minkowski inequality and subhomogeneity

摘要

For (h:(0,infty )ightarrow mathbb{R}), the function (h^{st }left( tight) :=th(rac{1}{t})) is called ((st))-conjugate to (h). This conjugacy is related to the H??lder and Minkowski inequalities. Several properties of ((st))-conjugacy are proved. If (arphi) and (arphi ^{st }) are bijections of (left(0,infty ight)) then ((arphi ^{-1}) ^{st }=left( left[ left( arphi ^{st }ight) ^{-1}ight] ^{st }ight) ^{-1}). Under some natural rate of growth conditions at (0) and (infty), if (arphi) is increasing, convex, geometrically convex, then (left[ left( arphi^{-1}ight) ^{st }ight] ^{-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 ightarrow mathbb{R}). Given a bijection (arphi :(0,infty )ightarrow (0,infty )), define (mathbf{P}_{arphi }:Sightarrow lbrack 0,infty )) by [mathbf{P}_{arphi }(x):=arphi ^{-1}igg( intlimits_{Omega (x)}arphi circ leftert xightert dmu igg),] where (Omega (x)) is the support of (x). Applying some properties of the ((st)) operation, we prove that if (intlimits_{Omega }xyleq mathbf{P}_{arphi }(x)mathbf{P}_{psi }(y)) where (arphi ^{-1}) and (psi ^{-1}) are conjugate, then (arphi) and (psi) are conjugate power functions. The existence of nonpower bijections (arphi ) and (psi) with conjugate inverse functions (psi =left[ ( arphi ^{-1}) ^{st}ight] ^{-1}) such that (mathbf{P}_{arphi }) and (mathbf{P}_{psi }) are subadditive and subhomogeneous is considered.