Convexity of Quotients of Theta Functions

For fixed $u$ and $v$ such that $0\leq u<v<1/2$, the monotonicity of the quotients of Jacobi theta functions, namely, $θ_{j}(u|iπt)/θ_{j}(v|iπt)$, $j=1, 2, 3, 4$, 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 $θ_{2}(u|iπt)/θ_{2}(v|iπt)$ and $θ_{3}(u|iπt)/θ_{3}(v|iπt)$ are convex on $0<t<\infty$.

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