On the Ulam stability of F(z) + F(2z) = 0

Abstract

In this paper it is shown that the complex functional equation F(z) + F(2z) = 0, z ∈ Ω := C\(−∞, 0], is stable in the strong sense of Ulam. It means that given an analytic and bounded function f (z) on Ω satisfying | f (z) + f (2z)| < δ, z ∈ Ω, for some δ > 0, there exists an analytic solution F(z) of the above functional equation such that | f (z) − F(z)| < K(δ) on each compact A of Ω, where K(δ) is a positive real function that tends to 0 as δ → 0. This result is extended to analytic functions f (z) on Ω satisfying | f (z)+ f (2z)| < δ, z ∈ Ω, for some δ > 0, not necessarily bounded on Ω.