Abstract
Let be the set of positive real numbers, a Banach space, and , with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality in restricted domains of the form for fixed with or and . As consequences of the results we obtain asymptotic behaviors of the inequality as .
Highlights
The stability problems of functional equations have been originated by Ulam in 1940 see 1
In 1950-1951 this result was generalized by the authors Aoki 3 and Bourgin 4, 5
The stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the inequality 1.1 in a restricted domain 16
Summary
The stability problems of functional equations have been originated by Ulam in 1940 see 1. For more precise descriptions of the Hyers-Ulam stability and related results, we refer the reader to the paper of Moszner 17. For fixed d > 0, if f : X → Y satisfies the functional inequalities such as that of Cauchy, Jensen and Jensen type, etc. We refer the reader to 18–26 for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality f xpyq − Pf x − Qf y ≤. We prove that if the inequality 1.4 holds for all x, y ∈ Uk,s, there exists a unique function L : R → B satisfying. We obtain asymptotic behaviors of the inequalities 1.4 and 1.9 as xkys → ∞ and kx sy → ∞, respectively
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