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    <p>The italic style of the theorem paragraph should not be applied to the math tag.</p>

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      <p><span class="theorem_label">Theorem</span><span>. </span>Let <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Δ</mi><mo>⟶</mo><mi>Δ</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">f : \Delta \longrightarrow \Delta,</annotation></semantics></math> where <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo>&lt;</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Delta=\{z\in\mathbb{C}: \vert z \vert \lt 1\}</annotation></semantics></math>, be analytic with <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">a \in \Delta</annotation></semantics></math>. Then</p>
<div class="maruku-equation"><math class="maruku-mathml" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo>|</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>|</mo></mrow><mo>≤</mo><mrow><mo>|</mo><mfrac><mrow><mi>z</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>1</mn><mo>−</mo><mover><mi>a</mi><mo>¯</mo></mover><mi>z</mi></mrow></mfrac><mo>|</mo></mrow></mrow><annotation encoding="application/x-tex"> 
\left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert\le \left\vert\frac{z-a}{1-\overline{a}z}\right\vert 

</annotation></semantics></math></div>
<p>for all <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\vert z \vert \le 1</annotation></semantics></math> and</p>
<div class="maruku-equation"><math class="maruku-mathml" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mo stretchy="false">|</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow></mfrac><mo>≤</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">|</mo><mi>a</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">
\frac{\vert f'(a)\vert}{1-\vert f(a)\vert^2}\le \frac{1}{1-\vert a \vert^2}.

</annotation></semantics></math></div>
<p>Furthermore, equality holds iff <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> realizes a conformal mapping of <math class="maruku-mathml" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> onto itself.</p>
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