### AIMS Mathematics

2020, Issue 5: 4482-4493. doi: 10.3934/math.2020288
Research article

# Homomorphism-derivation functional inequalities in C*-algebras

• Received: 23 January 2020 Accepted: 18 May 2020 Published: 20 May 2020
• MSC : 47B47, 11E20, 17B40, 39B52, 46L05, 39B72

• In this paper, we introduce and solve the following additive-additive $(s, t)$-functional inequality $\begin{eqnarray} && \left\|g\left(x+y\right) -g(x) -g(y)\right\| + \left\|2 h\left(\frac{x+y}{2}\right) - h(x) - h(y) \right\| \\ && \le \left\| s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \|t ( h(x+y)-h(x)-h(y))\| , \end{eqnarray}$ where $s$ and $t$ are fixed nonzero complex numbers with $|s| \lt 1$ and $|t| \lt 1$. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital $C^*$-algebras, associated to the additive-additive $(s, t)$-functional inequality (0.1) under some extra condition. Moreover, we introduce and solve the following additive-additive $(s, t)$-functional inequality $\begin{eqnarray} && \|g\left(x+y+z\right) -g(x) -g(y)-g(z)\| +\left\|3h\left(\frac{x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \right\| \\ && \le \left\|s\left( 3 g\left(\frac{x+y+z}{3}\right)-g(x)-g(y)-g(z)\right)\right\| \\ && + \left\|t \left( h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right\| , \end{eqnarray}$ where $s$ and $t$ are fixed nonzero complex numbers with $|s| \lt 1$ and $|t| \lt 1$. Furthermore, we investigate $C^*$-ternary derivations and $C^*$-ternary homomorphisms in $C^*$-ternary algebras, associated to the additive-additive $(s, t)$-functional inequality (0.2) under some extra condition.

Citation: Choonkil Park, XiaoYing Wu. Homomorphism-derivation functional inequalities in C*-algebras[J]. AIMS Mathematics, 2020, 5(5): 4482-4493. doi: 10.3934/math.2020288

### Related Papers:

• In this paper, we introduce and solve the following additive-additive $(s, t)$-functional inequality $\begin{eqnarray} && \left\|g\left(x+y\right) -g(x) -g(y)\right\| + \left\|2 h\left(\frac{x+y}{2}\right) - h(x) - h(y) \right\| \\ && \le \left\| s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \|t ( h(x+y)-h(x)-h(y))\| , \end{eqnarray}$ where $s$ and $t$ are fixed nonzero complex numbers with $|s| \lt 1$ and $|t| \lt 1$. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital $C^*$-algebras, associated to the additive-additive $(s, t)$-functional inequality (0.1) under some extra condition. Moreover, we introduce and solve the following additive-additive $(s, t)$-functional inequality $\begin{eqnarray} && \|g\left(x+y+z\right) -g(x) -g(y)-g(z)\| +\left\|3h\left(\frac{x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \right\| \\ && \le \left\|s\left( 3 g\left(\frac{x+y+z}{3}\right)-g(x)-g(y)-g(z)\right)\right\| \\ && + \left\|t \left( h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right\| , \end{eqnarray}$ where $s$ and $t$ are fixed nonzero complex numbers with $|s| \lt 1$ and $|t| \lt 1$. Furthermore, we investigate $C^*$-ternary derivations and $C^*$-ternary homomorphisms in $C^*$-ternary algebras, associated to the additive-additive $(s, t)$-functional inequality (0.2) under some extra condition.

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