Convexity of Quotients of Theta Functions

Abstract

For fixed $u$ and $v$ such that $0\le u<v<1/2$, 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 $0<t<\infty$.

arXiv comment17 pages, 6 figures