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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

  • The $n^{th}$ derivative of $f$ in the interval $[a, b]$. So $| \frac{dn}{dx^n} f | \leq M$
  • All derivatives of $f$ in interval $[a,b]$ so $1\leq n < \infty : | \frac{dn}{dx^n} f | \leq M$.

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].

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 But what does $M$ depend on? You obviously can't find a single universal constant $M$ that will work. And as for a special class of functions on, say, $[-1,1]$, just restrict to all smooth functions on $[-1,1]$ whose derivatives of any order are bounded by $100$. You need to provide more detail on what you're looking for. – Deane Yang Dec 8 at 0:27
The answer may also depend on what you mean by "smooth". – Goldstern Dec 8 at 0:37
Maybe you're asking something roughly like the following question: For which smooth functions $f$ on $\mathbb{R}$ does there exist a constant $M$ (that depends on $f$) such that $|f^{(n)}(x)| \le M$ for any $n \ge 0$ and $x \in [-1,1]$? – Deane Yang Dec 8 at 2:56

closed as not a real question by Alexandre Eremenko, Deane Yang, Anthony Quas, Michael Renardy, Will Jagy Dec 8 at 5:04

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