Suppose $f$ is holomorphic on $D_{1}(0)$ the open unit disc. Let $\Gamma_{1} = \{z : |z| = 1, x>0, y>0\}$ where $z = x+iy$ and define $\Gamma_{2}, \Gamma_{3}, \Gamma_{4}$ similarly. On $\Gamma_{i}$, $|f(z)| \le M_{i}$. How can we show $|f(0)|\le (M_{1}M_{2}M_{3}M_{4})^{\frac{1}{4}}$?
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Hint: Consider $g(x) = f(x) f(ix) f(-x) f(-ix)$. |
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Jensen's formula states that either f(0)=0 or $$\log|f(0)|=\frac{1}{2\pi}\int_0^{2\pi}\log|f(re^{i\theta})|\,d\theta-\sum_{k=1}^n\log(\frac{r}{|a_k|})$$ for any function $f$ analytic on a region containing $\overline{B(0;r)}$ with zeros $a_1, a_2, \ldots, a_n$ in $B(0;r)$, repeated according to multiplicity. Therefore, for any $0<r<1$ we have $$\log|f(0)|\leq\frac{1}{2\pi}\int_0^{2\pi}\log|f(re^{i\theta})|\,d\theta\leq\frac{1}{4}(\log(m_1)+\log(m_2)+\log(m_3)+\log(m_4),$$ where $m_1=\sup_{0<\theta<\pi/2}\{|f(re^{i\theta}|\}$, $m_2=\sup_{\pi/2<\theta<\pi}\{|f(re^{i\theta}|\}$, etc. Letting $r\rightarrow1$, we have $$\log|f(0)|\leq\frac{1}{4}(\log(M_1)+\log(M_2)+\log(M_3)+\log(M_4))$$ $$|f(0)|\leq(M_1M_2M_3M_4)^\frac{1}{4}$$ |
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