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# On an Exponential Functional Inequality and its Distributional Version

Published:2014-04-03
Printed: Mar 2015
• Jaeyoung Chung,
Department of Mathematics, Kunsan National University, Kunsan, 573-701 Korea
 Format: LaTeX MathJax PDF

## Abstract

Let $G$ be a group and $\mathbb K=\mathbb C$ 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 $f\colon G\to \mathbb K$ satisfying the inequality $\Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G,$ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$.
 Keywords: distribution, exponential functional equation, Gelfand hyperfunction, stability
 MSC Classifications: 46F99 - None of the above, but in this section 39B82 - Stability, separation, extension, and related topics [See also 46A22]

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