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If $f$ is a complex-valued function analytic on $\{z:\vert z \vert \leqq r \}$, then for $\vert z \vert <r $,

$$ \vert f(z) \vert \leqq \frac{2\vert z \vert}{2-\vert z \vert} \sup\{\Re f(w): \vert w \vert=\vert z \vert\}+\frac{r+\vert z \vert}{r-\vert z \vert}\vert f(0) \vert. $$

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The $2-|z|$ in the denominator after the inequality sign looks like a typo to me. Could you check it, please? There are LaTeX glitches to put right as well; in particular, sup and Re shouldn't be italic. –  John Bentin Dec 20 '12 at 20:39

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