Additive ρ-functional inequalities in β-homogeneous normed spaces

In this paper, we solve the following additive ρ-functional inequalities ‖ f (x + y) − f (x) − f (y)‖ ≤ ρ (2 f (x + y 2 ) − f (x) + − f (y)) , (1) where ρ is a fixed complex number with |ρ| < 1, and 2 f (x + y 2 ) − f (x) − f (y) ≤ ‖ρ( f (x + y) − f (x) − f (y))‖, (2) where ρ is a fixed complex number with |ρ| < 1 2 , and prove the Hyers-Ulam stability of the additive ρfunctional inequalities (1) and (2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive ρ-functional equations associated with the additive ρ-functional inequalities (1) and (2) in β-homogeneous complex Banach spaces.