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Unifying inequalities of Hardy, Copson, and others

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Abstract

In order to provide an alternative proof to an inequality of Hilbert, Hardy proved

$$\sum_{n=0}^{\infty} \left((n+1)^{-1} \sum_{k=0}^{n}x_k\right)^p \leq\left(\frac{p}{p-1} \right)^p\sum_{n=0}^\infty x_n^p. $$

This inequality and its relatives triggered research activity concerning norm inequalities including new inequalities by Copson and Levinson. In this paper, three theorems are established from which related inequalities of Hardy, Copson, and Levinson and various extensions are deduced as corollaries.

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Authors and Affiliations

  1. Department of Mathematics, New York City College of Technology-CUNY, Brooklyn, NY, 11201, USA

    H. Carley

  2. Department of Math and Statistics, Auburn University, Auburn, AL, 36849, USA

    P. D. Johnson Jr.

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    R. N. Mohapatra

Authors
  1. H. Carley
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  2. P. D. Johnson Jr.
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  3. R. N. Mohapatra
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Correspondence to H. Carley.

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Carley, H., Johnson, P.D. & Mohapatra, R.N. Unifying inequalities of Hardy, Copson, and others. Aequat. Math. 89, 497–510 (2015). https://doi.org/10.1007/s00010-013-0230-x

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  • DOI: https://doi.org/10.1007/s00010-013-0230-x

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