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Journal of Function Spaces and Applications
Volume 2012, Article ID 385925, 6 pages
http://dx.doi.org/10.1155/2012/385925
Research Article

On Necessary Condition for the Variable Exponent Hardy Inequality

Faculty of Education, Dicle University, 21280 Diyarbakir, Turkey

Received 2 January 2012; Revised 20 February 2012; Accepted 24 February 2012

Academic Editor: Carlo Sbordone

Copyright © 2012 Aziz Harman. 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

We derive a necessary condition for exponent functions 𝑝 , 𝛽 such that the variable exponent Hardy inequality β€– π‘₯ 𝛽 ( π‘₯ ) βˆ’ 1 ∫ π‘₯ 0 𝑓 ( 𝑑 ) 𝑑 𝑑 β€– 𝐿 𝑝 ( . ) ( 0 , 𝑙 ) ≀ 𝐢 β€– π‘₯ 𝛽 ( π‘₯ ) 𝑓 β€– 𝐿 𝑝 ( . ) ( 0 , 𝑙 ) holds.