Which functions have all derivatives everywhere positive? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:04:56Z http://mathoverflow.net/feeds/question/86738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86738/which-functions-have-all-derivatives-everywhere-positive Which functions have all derivatives everywhere positive? Will Sawin 2012-01-26T17:41:17Z 2012-01-26T18:14:58Z <p>Consider the class of functions from \$\mathbb R\$ to \$\mathbb R\$, such that the function is positive everywhere and its \$n\$th derivative is positive everywhere for all \$n\$.</p> <p>The only examples I can construct are the functions \$ae^{bx}+c\$ for \$a,b,c>0\$.</p> <p>Are these functions the only examples?</p> <p>If not, for which nonlinear functions \$g\$ does \$e^{g(x)}\$ have this property?</p> http://mathoverflow.net/questions/86738/which-functions-have-all-derivatives-everywhere-positive/86741#86741 Answer by Stefan Waldmann for Which functions have all derivatives everywhere positive? Stefan Waldmann 2012-01-26T17:51:14Z 2012-01-26T17:51:14Z <p>Well, there are certainly more. If you look at the chain rule then you see that the \$n\$-th derivative is a linear combinations of products of derivatives of the two functions you compose with positive coefficients. Thus if you have two functions with your property, then their composition will again have only positive derivatives. So you can go on...</p> http://mathoverflow.net/questions/86738/which-functions-have-all-derivatives-everywhere-positive/86743#86743 Answer by Gerald Edgar for Which functions have all derivatives everywhere positive? Gerald Edgar 2012-01-26T18:13:37Z 2012-01-26T18:13:37Z <p>See <strong>completely monotonic</strong> in the literature. Function \$f(x)\$ is completely monotonic if and only if \$f(-x)\$ is one of yours.</p> <p>S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679.</p> <p><a href="http://mathworld.wolfram.com/CompletelyMonotonicFunction.html" rel="nofollow">http://mathworld.wolfram.com/CompletelyMonotonicFunction.html</a></p> http://mathoverflow.net/questions/86738/which-functions-have-all-derivatives-everywhere-positive/86744#86744 Answer by Liviu Nicolaescu for Which functions have all derivatives everywhere positive? Liviu Nicolaescu 2012-01-26T18:14:58Z 2012-01-26T18:14:58Z <p>If \$f(x)\$ is a function with positive derivatives, then the function \$f(-x)\$ is completely monotonic. The completely monotonic functions are clasified by Bochner's theorem</p> <p>Nimza (mathoverflow.net/users/17896), On the generalisation of Bernstein's theorem on monotone functions, <a href="http://mathoverflow.net/questions/86524" rel="nofollow">http://mathoverflow.net/questions/86524</a> (version: 2012-01-24)</p>