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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!

  • 0
    Putting $\Psi\equiv1$ gives many examples since any smooth and positive function $\varphi$ would fit.2012-08-13

0 Answers 0