Hölder inequality in Banach space (Q2732208)

Let \(p_1,p_2,\dots, p_n\) be positive real numbers such that \(\sum{1\over p_i}= 1\) and \(f_1,f_2,\dots, f_n\) be integrable functions on some measure space; the classical Hölder inequality applied to the functions \((f_i)^{1/p_i}\) (if they are defined) can be written: \(\|\prod (f_i)^{1/p_i}\|_1\leq \prod\|f_i\|^{1/p_i}_1\).NEWLINENEWLINENEWLINEIn this paper the authors give a generalization of this inequality replacing the space \(L^1\) by \(L^\infty\) or a space of continuous functions and the norm \(\|\;\|_1\) by the norm \(\|\;\|_\infty\) (which is a consequence of the convexity of \(x\mapsto- \ln x)\).NEWLINENEWLINENEWLINENotice that, due to many misprints, this paper is quite difficult to read.