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Gao, Peng

On a result of Levin and Stečkin. (English) Zbl 1264.26026

Int. J. Math. Math. Sci. 2011, Article ID 534391, 15 p. (2011).
Summary: The following inequality for \(0<p<1\) and \(a_n\geq 0\) originates from a study of G. H. Hardy, J. E. Littlewood and G. Pólya [Inequalities. Cambridge: At the University Press (1988; Zbl 0634.26008)]: \[ \sum_{n-1}^\infty \left(\frac 1n \sum_{k-n}^n a_k\right)^p\geq c_p\sum_{n-1}^\infty a_n^p. \] V. I. Levin and S. B. Stechkin [Am. Math. Soc., Transl., II. Ser 14, 1–29 (1960; Zbl 0100.04503)] proved the previous inequality with the best constant \(c_p = (p/(1-p)^p)\) for \(0 < p\leq 1/3\). In this paper, we extend the result of Levin and Stechkin to \(0 <p \leq 0.346\).

Cited in 3 Documents

MSC:

26D15 Inequalities for sums, series and integrals

Citations:

Zbl 0100.04503; Zbl 0634.26008; Zbl 0010.10703
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References:

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