example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:51:55Z http://mathoverflow.net/feeds/question/115378 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115378/example-of-a-concave-function-with-lim-x-to-0-fracgx-x-ln-x-infty-w example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition FF 2012-12-04T09:13:05Z 2012-12-04T13:51:48Z <p>I'm looking for the example of a concave function $g\colon [0,1]\mapsto \mathbb{R}$, with $g(0)=0$, for which</p> <ol> <li>$\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ and</li> <li>$\lim\limits_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}=1$ for every $\lambda>1$</li> </ol> http://mathoverflow.net/questions/115378/example-of-a-concave-function-with-lim-x-to-0-fracgx-x-ln-x-infty-w/115397#115397 Answer by Alexandre Eremenko for example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition Alexandre Eremenko 2012-12-04T13:51:48Z 2012-12-04T13:51:48Z <p>Take $g_1(x)=x\log^2x$. Properties 1,2 are evidently satisfied, and computation of the second derivative shows that it is negative for $0&lt; x&lt;1/e$. Now rescale: $g(x)=g_1(x/e)$.</p>