On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of $\vartheta_4$

TL;DR

This paper proves a conjecture by Faulhuber and Steinerberger that the logarithmic derivative of the theta function _4 exhibits strictly decreasing and convex behavior when scaled by y^2, confirming a specific property of this special function.

Contribution

The paper provides a rigorous proof confirming the conjectured properties of the logarithmic derivative of _4, advancing understanding of its behavior.

Findings

Confirmed that y^2 _4'(y)/_4(y) is strictly decreasing

Established the strict convexity of y^2 _4'(y)/_4(y)

Validated a conjecture by Faulhuber and Steinerberger

Abstract

Faulhuber and Steinerberger conjectured that the logarithmic derivative of $\vartheta_4$ has the property that $y^2\,\vartheta_4'(y)/\vartheta_4(y)$ is strictly decreasing and strictly convex. In this small note, we prove this conjecture.