Geodesic
New Norm Inequalities of Čebyšev Type for Power Series in Banach Algebras
Kragujevac Journal of Mathematics, Tome 39 (2015) no. 1, p. 41

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Let $f(\lambda)=\sum_{n=0}^{\infty }\alpha_{n}\lambda^{n}$ be a function defined by power series with complex coefficients and convergent on the open disk $D(0,R)\subset\Bbb{C}$, $R>0$ and $x,y\in\cal{B}$, a Banach algebra, with $xy=yx$. In this paper we establish some new upper bounds for the norm of the \emph{Čebyšev type difference} \[ f(ambda)f(ambda xy)-f(ambda x)f(ambda y), \] providing that the complex number $\lambda$ and the vectors $x,y\in\cal{B}$ are such that the series in the above expression are convergent. These results complement the earlier resuls obtained by the authors. Applications for some fundamental functions such as the \emph{exponential function} and the \emph{resolvent function} are provided as well.
Classification : 47A63 47A99
Keywords: Banach algebras, power series, exponential function, resolvent function, norm inequalities 
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     author = {S. S. Dragomir and M. V. Boldea and M. Megan},
     title = {New {Norm} {Inequalities} of {\v{C}eby\v{s}ev} {Type} for {Power} {Series} in {Banach} {Algebras}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {41 },
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S. S. Dragomir; M. V. Boldea; M. Megan. New Norm Inequalities of Čebyšev Type for Power Series in Banach Algebras. Kragujevac Journal of Mathematics, Tome 39 (2015) no. 1, p. 41 . http://geodesic.mathdoc.fr/item/KJM_2015_39_1_a4/