search for a function satisfying some conditions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:56:41Z http://mathoverflow.net/feeds/question/105626 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105626/search-for-a-function-satisfying-some-conditions search for a function satisfying some conditions Higgs88 2012-08-27T13:42:48Z 2012-08-27T16:48:41Z <p>Hi everyone, I would like to find a function </p> <p>$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$</p> <p>satisfying the following conditions:</p> <p>$$1-\frac{z\Psi'(z)}{\Psi(z)}+8s\Psi''(z)\geq 0$$</p> <p>$$\Psi(z)^2+\frac{4}{\theta}\Psi(z)\leq\frac{z^2}{4\theta s}$$</p> <p>where $\theta$, $s$ are given positive constants.</p> <p>The second inequality yields </p> <p>$$0\leq\Psi(z)\leq\sqrt{\frac{z^2}{4\theta s}+\frac{4}{\theta^2}}-\frac{2}{\theta}$$</p> <p>Could someone have an idea to solve the first differential equation? Or give such a function $\Psi(z)$? Many thanks!</p>