- #1
Euge
Gold Member
MHB
POTW Director
- 2,073
- 244
Here is this week's POTW:
-----
Let $F$ be an entire function for which there exists $t > 0$ such that $\lvert F(z)\rvert = O(\exp(\lvert z\rvert^t))$ as $\lvert z\rvert \to \infty$. Show that there is a constant $M > 0$ such that for all $n$ sufficiently large, $$\lvert F^{(n)}(0)\rvert \le Mn!\left(\frac{et}{n}\right)^{n/t}$$
-----
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
-----
Let $F$ be an entire function for which there exists $t > 0$ such that $\lvert F(z)\rvert = O(\exp(\lvert z\rvert^t))$ as $\lvert z\rvert \to \infty$. Show that there is a constant $M > 0$ such that for all $n$ sufficiently large, $$\lvert F^{(n)}(0)\rvert \le Mn!\left(\frac{et}{n}\right)^{n/t}$$
-----
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!