International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 534391, 15 pages
http://dx.doi.org/10.1155/2011/534391
Research Article

On a Result of Levin and Stečkin

Peng Gao

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371

Received 18 August 2010; Accepted 25 January 2011

Academic Editor: Seppo Hassi

Copyright Β© 2011 Peng Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The following inequality for 0 < 𝑝 < 1 and π‘Ž 𝑛 β‰₯ 0 originates from a study of Hardy, Littlewood, and Pólya: βˆ‘ ∞ 𝑛 = 1 βˆ‘ ( ( 1 / 𝑛 ) ∞ π‘˜ = 𝑛 π‘Ž π‘˜ ) 𝑝 β‰₯ 𝑐 𝑝 βˆ‘ ∞ 𝑛 = 1 π‘Ž 𝑝 𝑛 . Levin and Stečkin proved the previous inequality with the best constant 𝑐 𝑝 = ( 𝑝 / ( 1 βˆ’ 𝑝 ) ) 𝑝 for 0 < 𝑝 ≀ 1 / 3 . In this paper, we extend the result of Levin and Stečkin to 0 < 𝑝 ≀ 0 . 3 4 6 .