On the mean convergence of the Taylor and Laurent series

Abstract

For analytic functions in the domain ${\Delta=D\bigcup D_\infty}$ we find conditions expressed in terms of Laurent coefficients, necessary for convergence in the average of its Laurent series. For functions analytic in a circle ~ $D$, necessary and sufficient conditions of convergence in the mean Taylor series are obtained, expressed in terms of the coefficients of the Taylor series, and also the boundedness in the metric of the space $L_1$ of partial sums of the Taylor series