Hi everyone, I would like to find a function

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

satisfying the following conditions:

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

$$\Psi(z)^2+\frac{4}{\theta}\Psi(z)\leq\frac{z^2}{4\theta s}$$

where $\theta$, $s$ are given positive constants.

The second inequality yields

$$0<\Psi(z)\leq\sqrt{\frac{z^2}{4\theta $0\leq\Psi(z)\leq\sqrt{\frac{z^2}{4\theta s}+\frac{4}{\theta^2}}-\frac{2}{\theta}$$

Could someone have an idea to solve the first differential equation? Or give such a function $\Psi(z)$? Many thanks!

Hi everyone, I would like to find a function

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

satisfying the following conditions:

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

$$\Psi(z)^2+\frac{4}{\theta}\Psi(z)\leq\frac{z^2}{4\theta s}$$

where $\theta$, $s$ are given positive constants.

The second inequality yields

$$0<\Psi(z)\leq\sqrt{\frac{z^2}{4\theta s}+\frac{4}{\theta^2}}-\frac{2}{\theta}$$

Could someone have an idea to solve the first differential equation? Or give such a function $\Psi(z)$? Many thanks!