Convexity of Quotients of Theta Functions

TL;DR

This paper proves the convexity of certain quotients of Jacobi theta functions with fixed parameters, complementing previous results on their monotonicity, and enhances understanding of their mathematical properties.

Contribution

It establishes the convexity of specific quotients of Jacobi theta functions, extending prior monotonicity results and providing new insights into their behavior.

Findings

Convexity of $ heta_{2}(u|i\pi t)/ heta_{2}(v|i\pi t)$ on $0<t<\infty$

Convexity of $ heta_{3}(u|i\pi t)/ heta_{3}(v|i\pi t)$ on $0<t<\infty$

Extension of previous monotonicity results to convexity properties

Abstract

For fixed $u$ and $v$ such that $0\leq u<v<1/2$, the monotonicity of the quotients of Jacobi theta functions, namely, $\theta_{j}(u|i\pi t)/\theta_{j}(v|i\pi t)$, $j=1, 2, 3, 4$, on $0<t<\infty$ has been established in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the first author. In the present paper, we show that the quotients $\theta_{2}(u|i\pi t)/\theta_{2}(v|i\pi t)$ and $\theta_{3}(u|i\pi t)/\theta_{3}(v|i\pi t)$ are convex on $0<t<\infty$.