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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

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

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

$$\Psi(0)=1$$

Many thanks for your help!

share|improve this question
    
This question comes from a model SVI volatility surface, we want to find out the functions satisfying the previous inequalities in order to eliminate the static arbitrage –  Higgs88 Aug 13 '12 at 13:39
    
Putting $\Psi\equiv1$ gives many examples since any smooth and positive function $\varphi$ would fit. –  Andrew Aug 13 '12 at 15:51
    
Thanks for Andrew –  Higgs88 Aug 14 '12 at 8:05

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