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it's related to an inequality wich has a link with Dottie number :

It's simple define the following functions : $f(x)=\frac{\sin(x)}{x}$ and $g_n(x)= \cos(\cos(\cdots(x)\cdots)$ where we compose $n$ times the function $\cos(x)$ with herself and on each interval between two consecutive zeroes of $f^{(n)}$ we have : $$\max |f^{(n)}(x)|> \max\left|\frac{g_n(x)}{x}\right| $$

My question is how to prove the inequality above ?

Edit :I add the condition $x\geq n^2\geq 16$

I think we can implement a recurrent inequality of Redheffer like this :

Theorem : Let $f_k(a_1,\cdots,a_k),g_k(a_1,\cdots,a_k)$ be real-valued functions defined for $a_k$ in the set $D_k$,$1\leq k \leq n $ and for wich there exist real-valued functions $F_k$ such that for all $k$,$1\leq k\leq n$, $$\underset{a_k \in D_k}{\sup}(\mu f_k(a_1,\cdots,a_k)-g_k(a_1,\cdots,a_k))=F_k(\mu)f_{k-1}(a_1,\cdots,a_{k-1})$$ where $f_0=1$.Then $$\sum_{k=1}^{n}\mu_k f_k(a_1,\cdots,a_k)\leq \sum_{k=1}^{n}g_k(a_1,\cdots,a_k)$$provided we can find real numbers $\lambda_k$,$1\leq k\leq n+1$ such that :

$\lambda_1\leq 0$$\quad$$\lambda_{n+1}=0$$\quad$$\lambda_{k}=F_k^{-1}(\lambda_{k})-\lambda_{k+1}$

Where $F_k^{-1}(y)$ denotes any $x$ such that $F_k(x)=y$

Thanks a lot .

  • $\phi_l: f^{(n)}(\phi_l)=0$ is not a definition: the $n$-th derivative of $\text{sinc}(x)$ vanishes at an infinity of real points: which one of them is $\phi_0$ or $\phi_1$? – Jack D'Aurizio Dec 03 '17 at 01:01
  • It is simpler to remove such $\phi_l$s and state that "on each interval between two consecutive zeroes of $f^{(n)}$ we have..." – Jack D'Aurizio Dec 03 '17 at 01:15
  • That is not true, anyway. If we take $n=4$ and the interval $[1,9,5.6]$ the opposite inequality holds. – Jack D'Aurizio Dec 03 '17 at 01:18
  • Do you mean 1.9? – marty cohen Dec 03 '17 at 04:27
  • @Jack d'Aurizio it's true for a sufficient large $x$ that I don't mention in my OP by example for $x\geq n^2$. –  Dec 03 '17 at 10:42
  • @Jack D'Aurizio can I edit my post ? –  Dec 03 '17 at 11:38
  • I think that could be helpful https://math.stackexchange.com/questions/868318/nth-derivative-cardinal-sine-function and https://math.stackexchange.com/questions/227317/explaining-cos-infty and finally https://en.wikipedia.org/wiki/Bessel_function –  Dec 06 '17 at 19:09
  • What is the question? – Alex Provost Dec 06 '17 at 22:06
  • @Alex Provost thanks for your question I have edited . –  Dec 07 '17 at 14:54

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I think you should see at this paper.There is a lot of informations concerning your inequality, and furthermore there is a an interesting representation with $\pi$ in this link.

max8128
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