Functions near some \((\alpha _1,\alpha _2)\)-double Jordan derivations in \(p\)-Banach algebras
\textit{M. Mirzavaziri} and \textit{E. O. Tehrani} [Bull. Iran. Math. Soc. 35, No. 1, 147--154 (2009; Zbl 1189.46058)] introduced an \((\alpha, \beta)\)-double derivation \(\alpha \circ \beta\) in a Banach algebra \(A\) for given linear derivations \(\alpha\) and \(\beta\) in \(A\). The author defines an \((\alpha, \beta)\)-double Jordan derivation \(f= \alpha \circ \beta\) in a Banach algebra \(A\) for given linear derivations \(\alpha\) and \(\beta\) in \(A\) as follows: \[ f(a^2 ) = f(a) a + a f(a) +\alpha(a)\beta(a) + \beta(a)\alpha(a) \] for all \(a\in A\), and proves the Hyers-Ulam stability of \((\alpha, \beta)\)-double Jordan derivations in Banach algebras. But there are some errors in the results: \(r<1\) in Theorem 2.6, \(r>1\) in Corollary 2.7, \(r>1\) in Theorem 2.8 and \(r>1\) in Corollary 2.9 are needed to obtain the correct results. The techniques and the ideas are so simple. The results are useful for beginners in the theory of functional inequalities and functional equations.
scientific article; zbMATH DE number 6802154