Monotonicity of quotients of theta functions related to an extremal problem on harmonic measure

We prove that for fixed u and v such that u, v ∈ [0, 1/2), the quotients θj (u|iπt)/θj (v|iπt), j = 1, 2, 3, 4, of the theta functions are monotone on 0 < t <∞. The case v = 0 has been used by the second author to study a generalization of Gonchar’s problem on harmonic measure of radial slits. Published by Elsevier Inc