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Let Θ3(z):=∑n∈Zexp(iπn2z)be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane H:={z∈C|Imz>0}, and takes positive values along iR>0, the positive imaginary axis, where R>0:=(0,+∞). We define its logarithm logΘ3(z)which is uniquely determined by the requirements that it should be holomorphic in Hand real-valued on iR>0. We derive an integral representation of logΘ3(z)when zbelongs to the hyperbolic quadrilateral F□||:={z∈C|Imz>0,-1≤Rez≤1,|2z-1|>1,|2z+1|>1}.Since every point of His equivalent to at least one point in F□||under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of Hvia the identity recorded by Berndt. The logarithms of the related Jacobi theta functions Θ4and Θ2may be conveniently expressed in terms of logΘ3via functional equations, and hence get controlled as well. Our approach is based on a study of the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This has connections with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.

Generalized Polya frequency functions are introduced through inverse Mellin transformations of the reciprocals of real entire functions with all zeros in sectors Aϕ and - Aϕ for 0 ≤ ϕ ≤ π/4, where Aϕ := {z ∈ C||arg z| ≤ ϕ}. It is shown that the constant π/4 is best possible in this context.