Boundedness of approximate trigonometric functions (Q1021858)

\textit{J. A. Baker} [Proc. Am. Math. Soc. 80, 411--416 (1980; Zbl 0448.39003)] and \textit{W. P. Cholewa} [Proc. Am. Math. Soc. 88, 631--634 (1983; Zbl 0547.39003)] investigated the stability of the cosine functional equation and sine functional equation, respectively. In this paper the authors consider the following functional inequality \[ |f(x+y)-f(x-y)-2f(x)f(y)|\leq \varphi(x),\quad x,y\in G, \] where \(f:G \to \mathbb C\) is a function, \(G\) is an abelian group and \(\varphi:G \to \mathbb R\) is a given nonnegative function. They show that such function \(f\) is bounded. They also extend their result when \(G\) is a semisimple commutative Banach algebra.