Explicit Tauberian bounds for multivariate functions (Q1899963)

The paper is closely related to an earlier paper of \textit{A. M. Odlyzko} [Explicit Tauberian estimates for functions with positive coefficients, J. Comput. Appl. Math. 41, No. 1-2, 187-197 (1992; Zbl 0763.40006)]. Let \((a_n)^\infty_{n = 0}\) be a sequence of nonnegative real numbers. Put \(f(x) = \sum^\infty_{n = 0} a_n x^n\) and consider the summatory function \(\mu (y) = \sum_{0 \leq n \leq y} a_n\). A. M. Odlyzko derived a lower bound for \(\mu (y)\) in terms of \(f\) and its derivatives. In this paper the author gives a generalization of this bound to the case of multivariate generating functions.