A note on the main theorem for absolutely monotonic functions

TL;DR

This paper critically examines a key theorem for absolutely monotonic functions, demonstrating that it is invalid without extra conditions, thus clarifying the theorem's proper scope and limitations.

Contribution

It identifies the necessity of additional restrictions for the main theorem on absolutely monotonic functions to hold true, correcting a longstanding assumption.

Findings

The main theorem is not universally valid for all absolutely monotonic functions.

Additional restrictions are required for the theorem to be applicable.

The paper clarifies the conditions under which the theorem is valid.

Abstract

We prove that the main theorem for absolutely monotonic functions from the book Mitrinovi\'{c}~D.S., Pe\v{c}ari\'{c}~J.E., Fink~A.M. "Classical And New Inequalities In Analysis", Kluwer Academic Publishers, 1993, is not valid without additional restrictions.