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

Let [inline-graphic not available: see fulltext] be the set of positive real numbers, [inline-graphic not available: see fulltext] a Banach space, [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext] with [inline-graphic not available: see fulltext]. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality [inline-graphic not available: see fulltext] in restricted domains of the form [inline-graphic not available: see fulltext] for fixed [inline-graphic not available: see fulltext] with [inline-graphic not available: see fulltext] or [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext]. As consequences of the results we obtain asymptotic behaviors of the inequality as [inline-graphic not available: see fulltext].