Generalized Mean-Value Theorem for an Analytic Function and an Algorithm for the Evaluation of the Function of Mean Values

We prove the mean-value theorem for functions analytic in starlike domains, propose an algorithm for finding the function of mean values, and study its analytic continuation. We present a differential equation for the function of mean values and the interpretation of the Lagrange formula for analytic functions in terms of the theory of differential equations. The set of points of the initial values of the function of mean values is analyzed and the upper bound for the radius of expansion of the function of mean values in Taylor’s series is established.