Approximate gamma–beta type functions

Abstract

We show that every unbounded approximate gamma–beta type function is of gamma–beta type. That is, we obtain the superstability of a gamma–beta type functional equation β ( x , y ) f ( x + y ) = f ( x ) f ( y ) and also investigate the stability in the sense of R . Ger of this equation in the following setting : | β ( x , y ) f ( x + y ) f ( x ) f ( y ) − 1 | ≤ φ ( x , y ) . From these results, we obtain stabilities of a generalized exponential functional equation and Cauchy’s gamma–beta functional equation.