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A generalization of Hölder and Minkowski inequalities. (English) Zbl 1259.46006
Summary: We give a generalization of Hölder and Minkowski inequalities to normal sequence algebras with absolutely monotone seminorm. Our main results are Theorem 2.1 and Theorem 2.2 which state these extensions. Taking \( F=\ell_{1}\) and \(\|\cdot\|_F=\|\cdot\|_{1}\) in both these theorems, we obtain classical versions of these inequalities. Also, using these generalizations we construct the vector-valued sequence space \( F(X,\lambda,p)\) as a paranormed space which is a most general form of the space \(c_{0}(X,\lambda,p)\) investigated in [J. K. Srivastava and B. K. Srivastava, Indian J. Pure Appl. Math. 27, No. 1, 73–84 (1996; Zbl 0844.46004)].

MSC:
46A45 Sequence spaces (including Köthe sequence spaces)
26D15 Inequalities for sums, series and integrals
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