Abstract

In this paper, we introduce and solve the following additive (<TEX>${\rho}_1$</TEX>, <TEX>${\rho}_2$</TEX>)-functional inequality <TEX>$${\Large{\parallel}}2f(\frac{x+y}{2})-f(x)-f(y){\Large{\parallel}}{\leq}{\parallel}{\rho}_1(f(x+y)+f(x-y)-2f(x)){\parallel}+{\parallel}{\rho}_2(f(x+y)-f(x)-f(y)){\parallel}$$</TEX> where <TEX>${\rho}_1$</TEX> and <TEX>${\rho}_2$</TEX> are fixed nonzero complex numbers with <TEX>$\sqrt{2}{\mid}{\rho}_1{\mid}+{\mid}{\rho}_2{\mid}<1$</TEX>. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (<TEX>${\rho}_1$</TEX>, <TEX>${\rho}_2$</TEX>)-functional inequality (1) in complex Banach spaces.