Hosszú’s Functional Equation

Abstract

In 1967, M. Hosszú introduced the functional equation $f(x+y-xy)=f(x)+f(y)-f(xy)$ in a presentation at a meeting on functional equations held in Zakopane, Poland. In honor of M. Hosszú, this equation is called Hosszú’s functional equation. As one can easily see, Hosszú’s functional equation is a kind of generalized form of the additive Cauchy functional equation. In Section 4.1, it will be proved that Hosszú’s equation is stable in the sense of C. Borelli. We discuss the Hyers–Ulam stability problem of Hosszú’s equation in Section 4.2. In Section 4.3, Hosszú’s functional equation will be generalized, and the stability (in the sense of Borelli) of the generalized equation will be proved. It is surprising that Hosszú’s functional equation is not stable on the unit interval. It will be discussed in Section 4.4. In the final section, we will survey the Hyers–Ulam stability of Hosszú’s functional equation of Pexider type.