Advances in Difference Equations
Volume 2010 (2010), Article ID 432796, 13 pages
doi:10.1155/2010/432796
Research Article

Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors

Jae-Young Chung

Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea

Received 28 June 2010; Revised 30 October 2010; Accepted 25 December 2010

Academic Editor: Roderick Melnik

Copyright © 2010 Jae-Young Chung. 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

Let ℝ + be the set of positive real numbers, 𝐵 a Banach space, 𝑓 ∶ ℝ + → 𝐵 , and 𝜖 > 0 , 𝑝 , 𝑞 , 𝑃 , 𝑄 ∈ ℝ with 𝑝 𝑞 𝑃 𝑄 ≠ 0 . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality ‖ 𝑓 ( 𝑥 𝑝 𝑦 𝑞 ) − 𝑃 𝑓 ( 𝑥 ) − 𝑄 𝑓 ( 𝑦 ) ‖ ≤ 𝜖 in restricted domains of the form { ( 𝑥 , 𝑦 ) ∶ 𝑥 > 0 , 𝑦 > 0 , 𝑥 𝑘 𝑦 𝑠 ≥ 𝑑 } for fixed 𝑘 , 𝑠 ∈ ℝ with 𝑘 ≠ 0 or 𝑠 ≠ 0 and 𝑑 > 0 . As consequences of the results we obtain asymptotic behaviors of the inequality as 𝑥 𝑘 𝑦 𝑠 → ∞ .