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On an Exponential Functional Inequality and its Distributional Version
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 1 / 01 March 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 30-43
- Print publication:
- 01 March 2015
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- Article
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Let
$G$ be a group and 
$\mathbb{K}\,=\,\mathbb{C}\,\text{or}\,\mathbb{R}$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions 
$f\,:\,G\,\to \,\mathbb{K}$ satisfying the inequality
$$\left| f\left( \sum\limits_{k=1}^{n}{{{x}_{k}}} \right)\,-\,\underset{k=1}{\overset{n}{\mathop{\Pi }}}\,f\left( {{x}_{k}} \right) \right|\,\,\le \phi \left( {{x}_{2}},\,.\,.\,.\,,{{x}_{n}} \right),\,\,\,\,\,\,\forall {{x}_{1}},\,.\,.\,.\,,{{x}_{n}}\,\in \,G,$$Where
$\phi :\,{{G}^{n-1}}\,\to \,[0,\,\infty )$. Also as a a distributional version of the above inequality we consider the stability of the functional equation
$$u\,\circ \,S\,-\,\overbrace{u\,\otimes \,.\,.\,.\,\otimes \,u}^{n-\text{times}}\,=\,0,$$where
$u$ is a Schwartz distribution or Gelfand hyperfunction, 
$\circ$ and 
$\otimes$ are the pullback and tensor product of distributions, respectively, and 
$S\left( {{x}_{1}},\,.\,.\,.\,,{{x}_{n}} \right)\,=\,{{x}_{1}}\,+\,.\,.\,.\,+\,{{x}_{n}}$.
