Fonctions Holomorphes

Je n'y arrive pas !
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<BR>On se place sur la bande <SPAN CLASS="MATH"><IMG WIDTH="202" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img1.png&quot; ALT="$ \Omega=\{z\in\mathbb{C}:0<Re(z)<1\}$"></SPAN>, on a une fontion <SPAN CLASS="MATH"><IMG WIDTH="74" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img2.png&quot; ALT="$ f:\overline{\Omega}\rightarrow\mathbb{C}$"></SPAN> holomorphe sur <SPAN CLASS="MATH"><IMG WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img3.png&quot; ALT="$ \Omega$"></SPAN> avec <SPAN CLASS="MATH"><IMG WIDTH="137" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img4.png&quot; ALT="$ sup_{z\in\overline{\Omega}}\mid f(z)\mid <\infty$"></SPAN>, on note <SPAN CLASS="MATH"><IMG WIDTH="198" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img5.png&quot; ALT="$ M(x)=sup_{y\in\mathbb{R}}\mid f(x+iy)\mid$"></SPAN>
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<BR>On veut montrer que <SPAN CLASS="MATH"><IMG WIDTH="170" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82799/cv/img1.png&quot; ALT="$ M(x)\leq M(0)^{1-x}M(1)^{x}$"></SPAN> , <SPAN CLASS="MATH"><IMG WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img7.png&quot; ALT="$ \forall x\in [0,1]$"></SPAN>
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<BR>Indication fournie : considérer <SPAN CLASS="MATH"><IMG WIDTH="199" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img8.png&quot; ALT="$ g(z)=M(0)^{z-1}M(1)^{-z}f(z)$"></SPAN> pour <SPAN CLASS="MATH"><IMG WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img9.png&quot; ALT="$ z\in\overline{\Omega}$"></SPAN>
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<BR>Je pense que <SPAN CLASS="MATH"><IMG WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img10.png&quot; ALT="$ g$"></SPAN> est holomorphe sur <SPAN CLASS="MATH"><IMG WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img3.png&quot; ALT="$ \Omega$"></SPAN>, sinon j'ai essayé d'écrire <SPAN CLASS="MATH"><IMG WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img10.png&quot; ALT="$ g$"></SPAN> en dévellopant avec <SPAN CLASS="MATH"><IMG WIDTH="75" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img11.png&quot; ALT="$ z=x+iy$"></SPAN>, mais rien, et on sait que <SPAN CLASS="MATH"><IMG WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img12.png&quot; ALT="$ f$"></SPAN> atteint son max sur la frontière de <SPAN CLASS="MATH"><IMG WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img3.png&quot; ALT="$ \Omega$"></SPAN> c'est à dire <SPAN CLASS="MATH"><IMG WIDTH="94" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="http://www.les-mathematiques.net/phorum/2006/03/18/82798/cv/img13.png&quot; ALT="$ i\mathbb{R}\cup (1+i\mathbb{R})$"></SPAN>.<BR><BR><BR>
[Corrigé selon ton indication. AD]