# Quadratic $\rho$-functional inequalities in $\beta$-homogeneous normed spaces

Volume 10, Issue 1, pp 104--110 Publication Date: January 26, 2017       Article History
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### Authors

Yuanfeng Park - Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea. Yinhua Lu - Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P. R. China. - Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China. Gang Cui - Department of Mathematics, Yanbian University, Yanji 133001, P. R. China. Choonkil Jin - Department of Mathematics, Yanbian University, Yanji 133001, P. R. China.

### Abstract

In this paper, we solve the quadratic $\rho$-functional inequalities $\|f(x+y)+f(x-y)-2f(x)-2f(y)\|\leq\|\rho(4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y))\|,$ where $\rho$ is a fixed complex number with $|\rho|<1$, and$\|4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y)\|\leq\|\rho(f(x+y)+f(x-y)-2f(x)-2f(y))\|,$ where $\rho$ is a fixed complex number with $|\rho|<1$. Using the direct method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (1) and (2) in $\beta$- homogeneous complex Banach spaces.

### Keywords

• Hyers-Ulam stability
• $\beta$-homogeneous space
• quadratic $\rho$-functional inequality.

•  39B62
•  39B72
•  39B52
•  39B82

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