θ函数的商的凸性
Convexity of Quotients of Theta Functions
摘要：
对于固定的$u$和$v$，使得$0\leq u<v<1/2$，Jacobi θ函数，即$θ_{j}(u|iπt)/θ_{j}(v|iπt)$，$j=1, 2, 3, 4$，在$0<t<\infty$上的单调性已经在A.Yu的先前工作中建立。Solynin，K. Schiefermayr，Solynin和第一作者。本文证明了等价物$θ_{2}(u|iπt)/θ_{2}(v|iπt)$和$θ_{3}(u|iπt)/θ_{3}(v|iπt)$在$0<t<\infty$上是凸的.
英文摘要：
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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