most recent 30 from http://mathoverflow.net 2013-05-23T02:05:00Z http://mathoverflow.net/feeds/question/115760 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115760/the-bound-on-derivatives-of-a-function Adel Ahmadyan 2012-12-08T00:19:50Z 2012-12-08T00:19:50Z

<p>Let $f(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Is there any way to find an upper bound (any bound, not necessary supremum) on </p> <ul> <li>The $n^{th}$ derivative of $f$ in the interval $[a, b]$. So $| \frac{dn}{dx^n} f | \leq M$</li> <li>All derivatives of $f$ in interval $[a,b]$ so $1\leq n &lt; \infty : | \frac{dn}{dx^n} f | \leq M$. </li> </ul> <p>Moreover, is there special class of functions that have these bounds? for example, $sin(x)$ is and all it's derivatives are bounded within interval [-1,1].</p>