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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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