Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 482717, 8 pages doi:10.1155/2010/482717 Research Article Note on q-Nasybullin’s Lemma Associated with the Modified p-Adic q-Euler Measure Taekyun Kim, 1 Young-Hee Kim, 1 Lee-Chae Jang, 2 Seog-Hoon Rim, 3 and Byungje Lee 4 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 2 Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea 3 Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea 4 Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, South Korea Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr Received 1 December 2009; Accepted 14 March 2010 Academic Editor: N. Govil Copyright q 2010 Taekyun Kim et al. 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. We derive the modified p-adic q-measures related to q-Nasybullin’s type lemma. 1. Introduction Let p be a fixed prime number. Throughout this paper, the symbols Z, Z p , Q p , and C p denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively. Let N be the set of natural numbers and Z   N ∪{0}. The p-adic absolute value in C p is normalized in such a way that |p| p  1/p see 1–17. For f ∈ N with f ≡ 1 mod 2, let f f, p be the least common multiple of f and p. We set Z f  lim ← − n Z fp n Z , for n ≥ 0, Z ∗ f  ∪ 0<a< fp a,p1  a  fp Z p  , a  fp n Z p   x ∈ Z f | x ≡ a  mod fp n  , 1.1 where a ∈ Z lies in 0 ≤ a< fp n .