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Refinements of Hölder’s inequality derived from functions \(\psi _{p,q,\lambda }\) and \(\varphi _{ p,q,\lambda }\). (English) Zbl 1252.46016

Summary: We investigate a convex function \(\psi_{p,q,\lambda} = \max\{\psi_p, \lambda\psi_q\}\;(1 \leq q < p \leq \infty \)), and its corresponding absolute normalized norm \(\| . \|_{\psi_{p,q,\lambda}} \). We determine a dual norm and use it for getting refinements of the classical Hölder inequality. Also, we consider a related concave function \(\varphi_{ p,q,\lambda} = \min\{\psi_p, \lambda \psi_q\}\;(0 < p < q \leq 1\)).

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
26D15 Inequalities for sums, series and integrals
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