Additive-Quadratic ρ-Functional Equations in β-Homogeneous Normed Spaces

Abstract

Let M1f(x,y):=34f(x+y)−14f(−x−y)+14f(x−y)+14f(y−x)−f(x)−f(y) and M2f(x,y):=2f(x+y2)+f(x−y2)+f(y−x2)−f(x)−f(y). We solve the additive-quadratic ρ-functional inequalities ∥M1f(x,y)∥≤∥ρM2f(x,y)∥, (1) where ρ is a fixed complex number with |ρ|<12, and ∥M2f(x,y)∥≤∥ρM1f(x,y)∥, (2) where ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers–Ulam stability of the additive-quadratic ρ-functional inequalities (1) and (2) in β-homogeneous complex Banach spaces.