Estimations in Hölder's type inequalities (Q2720278)

Suppose that \(0< m_1< M_1\), \(0< m_2< M_2\), \(\lambda> 0\) and \(p,q\in (1,\infty)\), \(1/p+ 1/q= 1\). The paper under review yields an optimal upper bound, of the form \(nM_1M_2F(\lambda, p,m_1/M_1, m_2/M_2)\), for the function NEWLINE\[NEWLINES_{p,\lambda}: [m_1,M_1]\times [m_2, M_2]\to \mathbb{R},NEWLINE\]NEWLINE given by the formula \(S_{p,\lambda}(a,b)= (\sum a_k)^{1/p}(\sum b_k)^{1/q}\). This upper bound (which is not always the maximum) is a consequence of separate convexity of \(S_{p,\lambda}(a,b)\) in \(a\) and \(b\). The particular case where \(p= q= 2\) and \(\lambda= 1\) were first considered by N. Ozeki in the 70's.