Let $F:\mathbf{C}\to\mathbf{C}$ be an analytic function whose restriction to $\mathbf{R}$ is positive and convex. Also assume
$$\int_{\mathbb{R}}\vert{e^{F(i\theta)}}\vert \;d\theta < \infty\ \ \ \text{ and }\ \ \ \vert{e^{F(i\theta)}}\vert \leq K, \quad K \in \mathbb{R},$$
so that we can take the Fourier inverse
$$\mathcal{F}^{-1} e^{F(i\theta)}(x) = \frac{1}{2\pi}\int_{\mathbb{R}}e^{F(i\theta)}e^{-i\langle\theta,x\rangle}\;d\theta.$$
My question is: for which $F$ is this Fourier inverse everywhere positive?
So far I've tried thinking about this using the method of steepest descent, which I think gives a positive answer to the question for certain $F$ but I'm not sure.