A remark over the converse of Holder inequality

H. H. CUENYA; F. E. LEVIS

JOURNAL OF INEQUALITIES AND APPLICATIONS

Hindawi Publishing Corporation

Lugar: New York; Año: 1999 vol. 3 p. 127 - 135

1025-5834

Let $(f,mathcal{A},mu)$ be a measure space and $mathcal{L}$ bethe set of measurable nonnegative real functions defined on$Omega$. Let $F:mathcal{L} o  [0,infty]$ be a positivehomogenous functional. Suppose that there are two sets $A, B inmathcal{A}$ such that $ 0 < F(chi_A) 0} subset {d^k: k in mathbb{Z}}$and $$F(xy) le phi^{-1}(F(phi circ x))psi^{-1}(F(psi circy)) $$ for all $x, y in {achi_C : F(chi_C) < infty ,a inmathbb{R}}$, then $phi$ and  $psi$ must be conjugate powerfunctions. In addition, we prove that if there exists a realnumber $d> 0$ such that ${F(chi_C): C in mathcal{A}, F(chi_C)> 0}subset {d^k: k in mathbb{Z}}$ then there are nonpowercontinuous bijective functions $phi$ and $psi$ which the aboveinequality. Also we give an example which shows that the conditionthat  $phi$ and $psi$ are continuous functions is essential.