# Log-concavity of difference of theta functions

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 $$\theta_3(z,q)\,=\,\sum\limits_{n=-\infty}^{\infty}\,q^{n^2}\,e^{2niz}\,=\, 1 + 2\sum\limits_{n=1}^{\infty}\,q^{n^2}\,\cos(2nz).$$

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

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

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.

• It seems that using the product representation listed here may help: functions.wolfram.com/EllipticFunctions/EllipticTheta3/08 – Suvrit Apr 6 '16 at 19:09
• Thanks. I have tried to use the product representation, but I couldn't come up with a way to turn the difference f(x) into a product, which would make my life much easier. – KostasT Apr 7 '16 at 11:08