Convexity of quotients of theta functions

For fixed u and v such that 0 ⩽ u < v < 1 / 2 , the monotonicity of the quotients of Jacobi theta functions, namely, θ j ( u | i π t ) / θ j ( v | i π t ) , j = 1 , 2 , 3 , 4 , on 0 < t < ∞ 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 | i π t ) / θ 2 ( v | i π t ) and θ 3 ( u | i π t ) / θ 3 ( v | i π t ) are convex on 0 < t < ∞ .