LAUSR

Abstract Let $X$ and $Y$ be Banach spaces and let $f\,:\,X\,\to \,Y$ be an odd mapping. For any rational number $r\,\ne \,2$ , C. Baak, D. H. Boo, and Th. M. Rassias proved the Hyers–Ulam stability of the functional equation $$rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{x}_{j}}}}{r} \right)\,+\,\sum\limits_{\begin{smallmatrix} i\left( j \right)\,\in \left\{ 0,\,1 \right\} \\ \sum\nolimits_{j=1}^{d}{i\left( j \right)=\ell } \end{smallmatrix}}{rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{\left( -1 \right)}^{i\left( j \right)}}{{x}_{j}}}}{r} \right)}\,=\,\left( C_{d-1}^{\ell }\,-\,C_{d-1}^{\ell -1}\,+\,1 \right)\,\sum\limits_{j=1}^{d}{f\left( {{x}_{j}} \right),}$$ where $d$ and $\ell$ are positive integers so that $1\,<\,\ell \,<\,\frac{d}{2}$ , and $C_{q}^{p}\,:=\frac{q!}{\left( q-p \right)!p!},\,p,\,q\,\in \,\mathbb{N}$ with $p\,\le \,q$ . In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerning the $^{*}$ -homomorphisms and the multipliers between ${{C}^{*}}$ -algebras are also considered.