## Rocky Mountain Journal of Mathematics

### Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras

#### 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.

#### Note

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.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 2 (2019), 593-607.

Dates
Revised: 10 August 2018
First available in Project Euclid: 23 June 2019

https://projecteuclid.org/euclid.rmjm/1561318395

Digital Object Identifier
doi:10.1216/RMJ-2019-49-2-593

Mathematical Reviews number (MathSciNet)
MR3973242

Zentralblatt MATH identifier
07079986

#### Citation

Park, Choonkil; Jin, Yuanfeng; Zhang, Xiaohong. Bi-additive $s$-functional inequalities and quasi-multipliers on Banach algebras. Rocky Mountain J. Math. 49 (2019), no. 2, 593--607. doi:10.1216/RMJ-2019-49-2-593. https://projecteuclid.org/euclid.rmjm/1561318395

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