# Parametrization of a class of functions

Could someone gives some examples of the pair $(\varphi(\theta), \Psi(z))$ such that

$$1+f(\theta)F(z)\geq 0,\ \ \forall\theta>0,\ \ z\in\mathbb{R}$$

with

$$f(\theta)=\theta\frac{\varphi'(\theta)}{\varphi(\theta)},\ \ F(z)=z\frac{\Psi'(z)}{\Psi(z)}$$

and

$$\left(1-\frac{z\Psi'(z)}{2\Psi(z)}\right)^2-\frac{\theta\varphi(\theta)^2}{4}\left(\frac{(\Psi') ^{2}(z)}{\Psi(z)}-2\Psi''(z)\right)-\frac{(\theta\varphi(\theta))^2}{16}(\Psi')^{2}(z)\geq 0,\ \ \forall\theta>0,\ \ z\in\mathbb{R}$$

where

$$\varphi\in\mathcal{C}^1: \theta\in\mathbb{R}_+\longrightarrow\varphi(\theta)>0$$

$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\longrightarrow\Psi(z)>0$$

$$\Psi(0)=1$$

Putting $\Psi\equiv1$ gives many examples since any smooth and positive function $\varphi$ would fit. –  Andrew Aug 13 '12 at 15:51