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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}$$Could someone have an idea to solve the first differential equation? Or give such a function \Psi(z)? Many thanks! 1 # search for a function satisfying some conditions Hi everyone, I would like to find a function$$\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!