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Additive $$\rho$$-functional inequalities and equations. (English) Zbl 1314.39026
Summary: In this paper, we investigate the additive $$\rho$$-functional inequalities $\biggl\| f\biggl(\sum\limits^k_{j=1}x_j\biggr)-\sum\limits^k_{j=1}f(x_j)\biggr\|\leq\biggl\|\rho\biggl(kf\biggl(\frac{\sum^k_{j=1}x_j}{k}\biggr)-\sum\limits^k_{j=1}f(x_j)\biggr)\biggr\|\eqno{(0.1)}$ and $\biggl\| kf\biggl(\frac{\sum^k_{j=1}x_j}{k}\biggr)-\sum\limits^k_{j=1}f(x_j)\biggr\|\leq\biggl\|\rho\biggl(f\biggl(\sum\limits^k_{j=1}x_j\biggr)-\sum\limits^k_{j=1}f(x_j)\biggr)\biggr\|,\eqno{(0.2)}$ where $$\rho$$ is a fixed complex number with $$|\rho|<1$$.
Furthermore, we prove the Hyers-Ulam stability of the additive $$\rho$$-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of additive $$\rho$$-functional equations associated with the additive $$\rho$$-functional inequalities (0.1) and (0.2) in complex Banach spaces.

##### MSC:
 39B62 Functional inequalities, including subadditivity, convexity, etc. 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
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