# characterization of a differential inequality

Hi everyone, does someone have an idea to describe the class of functions $\Psi(z)$ satisfying

$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R}_+$$

$$1-\frac{z\Psi'(z)}{\Psi(z)}+8s\Psi''(z)\geq 0$$

and

$$\Psi(0)=1$$

where $s$ is a given positive constant.

Many thanks! (A subset of this class is ok)

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Again, you may get rid of the constant $s$ setting $t=z/\sqrt{8s}$ and $\Psi(z)=\Phi(z/\sqrt{8s})$. – Ilya Bogdanov Aug 27 '12 at 16:34