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Given $f\in C^\infty(\mathbb R)$ is a smooth real function, $f(0)=0,f'(0)=1$.

For all $x\in\mathbb R$ and $n\in\mathbb N$, we have $$|f(x)|\le 1,\ \left|\frac{d^n}{dx^n}f(x)\right|\le 1, $$

prove that $f(x)=\sin x$.

This is not a homework problem. Are there any proves or references to this? How strong the condition that derivatives of all orders bounded is? I can only prove that $f$ is analytic and could be seen as an entire function with order $1$ of growth on $\mathbb C$.

cybcat
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  • Someone else asked lots of this type of question a couple of months back. But I don't know how to search for or find those questions... – Adam Rubinson Mar 14 '23 at 19:18
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    @AdamRubinson: You are probably thinking of this https://math.stackexchange.com/q/4630409/42969, this https://math.stackexchange.com/q/4630788/42969, and this https://math.stackexchange.com/q/4631046/42969 from February. – Martin R Mar 14 '23 at 19:20
  • @MartinR yes those are the ones. So this question is also very closely related then: https://math.stackexchange.com/questions/1863239/if-f-is-a-smooth-real-valued-function-on-real-line-such-that-f0-1-and-f – Adam Rubinson Mar 14 '23 at 19:22

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