My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by \begin{equation} \theta_3(z,q)\,=\,\sum\limits_{n=-\infty}^{\infty}\,q^{n^2}\,e^{2niz}\,=\, 1 + 2\sum\limits_{n=1}^{\infty}\,q^{n^2}\,\cos(2nz). \end{equation}

For $q\in(0,1)$, $x\in(l,h)\subset\mathbb{R}$, and $\delta =h-l$ let \begin{equation} f(x) = \theta_3\left(\frac{x\pi}{\delta 2},q\right) - \theta_3\left(\frac{(x-2h)\pi}{\delta 2},q\right). \end{equation}

I am trying to prove that $f(x)$ is log-concave, i.e. that \begin{equation} f''(x)\,f(x) \leq f'(x)^2. \end{equation}

A big number of identities related to theta functions can be found here: http://mathworld.wolfram.com/JacobiThetaFunctions.html.

Any help, or lead on existing results that could potentially be useful would be greatly appreciated.