On the stability of the pexiderized trigonometric functional equation.

*(English)*Zbl 1159.39013Similar to his papers [Adv. Difference Equ. 2007, Article ID 90405 (2007; Zbl 1148.39024)] and [Banach J. Math. Anal. 1, No. 2, 227–236, electronic only (2007; Zbl 1129.39013)], the author investigates the superstability of the pexiderized trigonometric functional inequality

$$\left|f\right(x+y)+f(x-y)-2g(x\left)h\right(y\left)\right|\le \varphi \left(x\right)\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}\varphi \left(y\right)$$

and also stability of the Jensen type functional inequality

$$\left|f\right(x+y)-f(x-y)-2f(y\left)\right|\le \psi (x,y)$$

for each $x,y\in G$, where $(G,+)$ is an abelian group and the mappings $f,g,h:G\to \u2102$ are unknown and $\varphi :G\to \mathbb{R}$ and $\psi :G\times G\to \mathbb{R}$ are known. He also extends his results to the Banach algebras.

Reviewer: Maryam Amyari (Mashhad)

##### MSC:

39B82 | Stability, separation, extension, and related topics |

39B52 | Functional equations for functions with more general domains and/or ranges |

39B62 | Functional inequalities, including subadditivity, convexity, etc. (functional equations) |