Norm inequalities of Cebysev type for power series in Banach algebras

Abstract

Let [InlineEquation not available: see fulltext.] be a function defined by power series with complex coefficients and convergent on the open disk [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], a Banach algebra, with [InlineEquation not available: see fulltext.]. In this paper we establish some upper bounds for the norm of the Cebysev type difference[InlineEquation not available: see fulltext.], provided that the complex number [lambda] and the vectors [InlineEquation not available: see fulltext.] are such that the series in the above expression are convergent. Applications for some fundamental functions such as the exponential function and the resolvent function are provided as well.

MSC: 47A63, 47A99.