# differential inequality about absolute value

I have a question

$$\Psi: z\in\mathbb{R}\rightarrow\mathbb{R}_+$$

with $\Psi(0)=1$.

Could someone give a simple class of the function $\Psi$ such that the following inequality holds:

$$c|\Psi'(z)|\leq |1-z\frac{\Psi'(z)}{\Psi(z)}|,\ \ \forall z\in\mathbb{R}$$

Many thanks

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$\{\Psi:z\mapsto 1\}$. Why are you asking?

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Thank you for your reply. I would like to know whether we can try to give a general class of such functions. We hope this function can be writen in a closed form with some parameters to chose: $$\Psi=\Psi(z,\alpha),\ \ \alpha\in A$$ where $A$ is a set such that the previous inequality holds. –  Higgs88 Aug 20 '12 at 9:35
How can you hope with only a vague bound on the derivative to be able to parametrize these functions?? There must be way too many of them for that. –  Marc van Leeuwen Aug 21 '12 at 8:35