Maximizing the Ratio of a Generalized Sigmoid to its Argument
Abstract
The ratio f(x)/x, where f is a real-valued, univariate S-shaped function, is shown to be quasi-concave, and to always have a unique global maximizer, which can be identified graphically. The analysis is strictly based on geometrical properties derived from the sigmoidal shape. It imposes no specific algebraic functional form on the sigmoid. The function f is defined over the non-negative real numbers, is increasing and continuously differentiable, "starts out" convex at the origin, and smoothly transitions to concave as it approaches 1 asymptotically.
 This optimization is particularly relevant to a transmitter with a limited supply of energy optimally choosing its transmission power for data communication over a wireless medium in the presence of interference. But the conclusions and/or development herein may interest students of neural networks, and of many dynamical systems in which sigmoidal functions play a fundamental role.