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On the stability of the pexiderized trigonometric functional equation. (English) Zbl 1159.39013

Similar 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

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

and also stability of the Jensen type functional inequality

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

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

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)