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 $f\in\mathscr{C}^2([0,1])\implies |f\,'|\le{\small\displaystyle\int_0^1} (4|f|+|f\,''|)$
05-04-2018, 04:35 PM
Post: #1
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
$f\in\mathscr{C}^2([0,1])\implies |f\,'|\le{\small\displaystyle\int_0^1} (4|f|+|f\,''|)$
Proof. Let$\underset{\,}{\,\;}(m,M)=(\min,\max)|f'|{\small([0,1]),\;}|f'|(\alpha)=m,\;|f'|(\beta)=M.$
${\qquad}$By mean value TH.$,\,f(x)=f(0)+x\cdot f'(\xi)\;(\underset{\,}{\xi}=\theta x,\;0<\theta_x<1).\;\;$
$\underset{\,}{\therefore\quad}\;$If$\;f\ge 0\,$on$\,(0,1),\,$then$\,|f|\ge mx.\,$hence${\small\displaystyle\int_0^1}|f|\ge{\small\displaystyle\int_0^1}mx dx=\large\frac{m}{2}$
$\underset{\,}{\qquad}$If$\,f< 0\,$on$\,(0,1),\,$then$\,-f(1-x)=-f(1)+xf'(\xi)\ge\,mx$
$\underset{\,}{\qquad}$and again we have$\,{\small\displaystyle\int_0^1}|f|={\small -\displaystyle {\int_0^1}}f(1-x)dx\ge\large\frac{m}{2}.$
$\underset{\,}{\qquad}$If$\,f(\eta)=0,\,$for some$\,\eta\in(0,1),\;\;$then$\;|f(x)|=|(x-\eta)f'(\xi)|\;$thus
$\underset{\,}{\qquad}{\small\displaystyle{\int_0^1}}|f|\ge{\small\displaystyle{​\int_0^{\eta}}}m(\eta-x)+{\small\displaystyle\int_{\eta}^1}m(x-\eta)=m((\eta-\frac{1}{2})^2+\frac{1}{4})\ge\large\frac{m}{4}.$
$\therefore\;\;\;{\small\displaystyle\int_0^1}(4|f|+|f''|)\ge m+\big|{\small\displaystyle\int_{\alpha}^{\beta}}f''(x)dx\big|\ge{\small m+M-m}\ge |f'(x)|.\;\square$
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