Some inequalities of Levinson’s type for Lipschitzian mappings between two Banach spaces

In our previous papers [J. Math. Inequal. 9 (2015), 1271-1285. ; J. Inequal. Appl. 2015 (2015), 1-15. ; J. Micic, manuscript] we observe Levinson's operator inequality and some mappings related to this inequality and its converses. As a continuation of this research, some inequalities of Levinson's type for Lipschitzian mappings between two Banach spaces are considered. To obtain these results, appropriate results related to Jensen's operator inequality are given. Applications for functions of norms in Banach spaces and functions defined by power series in Banach algebras are given.