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2019 Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras
Choonkil Park, Yuanfeng Jin, Xiaohong Zhang
Rocky Mountain J. Math. 49(2): 593-607 (2019). DOI: 10.1216/RMJ-2019-49-2-593

Abstract

In this paper, we solve the following bi-additive $s$-functional inequalities: $$ \displaylines { \, \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \hfill \cr \hfill \quad \le \Bigl \|s \Bigl (2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) + 2f\Bigl (\frac {x-y}{2}, z+w\Bigr ) - 2f(x,z )+ 2 f(y, w)\Bigr )\Bigr \| ,\! \cr \, \Bigl \|2f\Bigl (\frac {x+y}{2}, z-w\Bigr ) +2 f\Bigl (\frac {x-y}{2}, z+w\Bigr ) -2 f(x,z )+2 f(y, w)\Bigr \| \hfill \cr \hfill \le \|s ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) )\|, \cr \, } $$ where $s$ is a fixed nonzero complex number with $|s |\lt 1$. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital $C^*$-algebras associated with the bi-additive $s$-functional inequalities above.

Funding Statement

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology, grant No. NRF-2017R1D1A1B04032937. The second author was supported by the National Natural Science Foundation of China, grant No. 11761074.

Citation

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Choonkil Park. Yuanfeng Jin. Xiaohong Zhang. "Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras." Rocky Mountain J. Math. 49 (2) 593 - 607, 2019. https://doi.org/10.1216/RMJ-2019-49-2-593

Information

Received: 3 March 2018; Revised: 10 August 2018; Published: 2019
First available in Project Euclid: 23 June 2019

zbMATH: 07079986
MathSciNet: MR3973242
Digital Object Identifier: 10.1216/RMJ-2019-49-2-593

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Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 2 • 2019
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