A generalization of the Hyers-Ulam-Rassias stability of the beta functional equation (Q2770599)

A functional equation \(E[h]=0\) is Hyers-Ulam-Rassias(-Găvruta)-stable if, given a function \(\phi,\) there exists a function \(\Phi\) such that \(|E[f]|\leq\phi\) implies the existence of a unique \(g\) for which \(E[g]=0\) and \(|f-g|\leq\Phi\); cf. \textit{D. H. Hyers} [Proc. Nat. Acad. Sci. U.S.A. 27, 222-224 (1941; Zbl 0061.26403)], \textit{S. M. Ulam} [Problems in Modern Mathematics, Ch. VI, Wiley, New York, (1964; Zbl 0137.24201)], \textit{Th. M. Rassias} [Proc. Am. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)], \textit{P. Găvruta} [J. Math. Anal. Appl. 184, No. 3, 431-436 (1994; Zbl 0818.46043)]. Here the result is offered that the equation \(H(x+1,y+1)-[(1/y)+(1/x)](x+y+1)H(x,y)=0,\) satisfied by the reciprocal of the beta function, is stable in that sense and so is the functional equation \(h(x+1)-xh(x)=0\) of the gamma function.