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Is it true that the following function $$\frac{\pi ^2 \left(t^2-4 (-1+t) \text{cos}\left[\frac{\pi }{m}\right]^2\right) \text{csc}\left[\frac{\pi (-2+t)}{m}\right]^2}{m^2}, t\in[0,1]$$ attains its maximum in 0 and 1. Here $m>3$.

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migrated from mathoverflow.net Jun 29 '13 at 16:54

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@Ruslan my bad, looking at the wrong picture –  Greg Martin Jun 29 '13 at 18:28

2 Answers 2

The function $\sin((2-t)\pi/m)$ can be linearized by a function $at +b$ attaining the same values in endpoints $0,1$. Then $at +b\le \sin((2-t)\pi/m)$ implying that our function $f(t)$ is smaller than a rational function $h(t)=(At^2+B t+C)/(at + b)^2$. Now we can easily see that $h$ has a single extreme point $x\in[0,1]$ which is a local minimum. Q.E.D.

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Let us denote $$f(t) := \left(t^2-4(-1+t)\cos^2\left(\frac \pi m \right)\right)\csc^2 \left(\frac {\pi(-2+t)} m \right) .$$ We start from the verification $$f(0)=f(1)= \csc^2\left( \frac \pi m \right) . $$ Next, in order to prove the statement under consideration, it is enough to prove the convexity of $f(t)$ on $(0,1)$. Because $f(t) \in C^2(0,1)$, the convexity is equivalent to $f''(t) \ge 0$ there. After calculating $$f''(t)=-2\, \left( \csc \left( {\frac {\pi \, \left( t-2 \right) }{m}} \right) \right) ^{2}$$ $$ \left(12\pi^{2} \left( \cos \left( \frac {\pi} {m} \right) \right)^{2} \left( \cot \left( \frac {\pi \left( t-2 \right) }{m} \right) \right) ^{2}t-\right.$$ $$12\,{\pi }^{2} \left( \cos \left( {\frac {\pi }{m}} \right) \right) ^{2} \left( \cot \left( {\frac {\pi \, \left( t-2 \right) }{m}} \right) \right) ^ {2}-$$ $$3\,{\pi }^{2} \left( \cot \left( {\frac {\pi \, \left( t-2 \right) }{m}} \right) \right) ^{2}{t}^{2}+4\,{\pi }^{2} \left( \cos \left( {\frac {\pi }{m}} \right) \right) ^{2}t-$$ $$8\,\pi \, \left( \cos \left( {\frac {\pi }{m}} \right) \right) ^{2}\cot \left( {\frac { \pi \, \left( t-2 \right) }{m}} \right) m-4\,{\pi }^{2} \left( \cos \left( {\frac {\pi }{m}} \right) \right) ^{2}-$$ $$\left.{\pi }^{2}{t}^{2}+4\, \pi \,\cot \left( {\frac {\pi \, \left( t-2 \right) }{m}} \right) mt-{ m}^{2} \right) {m}^{-2} $$ we find its asymtotics in $m$ with help of Maple: $$ f''(t)=\frac 2 3 {\frac {{\pi }^{2} \left( {t}^{4}-8\,{t}^{3}+24\,{t}^{2}-20\,t+28 \right) }{ \left( t-2 \right) ^{4}}} +O\left(\frac 1 {m^2}\right), m \to \infty. $$ This implies the convexity of $f(x)$ on $(0,1)$ for big values of $m$.

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I have a problem with compiling of a part of the formula for $f''(t)$. My WinEdt produces it OK. –  user64494 Jul 1 '13 at 4:33
    
One can see the calculations with Maple in a PDF file, downloading it from RapidShare. –  user64494 Jul 1 '13 at 5:24
    
I "fixed" your math display problem. But I would highly suggest you use a variety of brackets, in particular [ and {, to help make the expression clearer. –  Willie Wong Jul 1 '13 at 11:02
    
@ Willie Wong : It is kind of you! Many thanks! The reason is that I use the latex command of Maple –  user64494 Jul 1 '13 at 12:19

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