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 $0<t<\infty$.