= Sharp gradient bound for positive harmonic functions
{title2=$|\nabla u(z)|\le2u(z)/(1-|z|^2)$}
For a positive <harmonic function> on the <unit disc>, choose a global <harmonic conjugate> and apply the <Carathéodory derivative bound for the right half-plane> to $u+iv$. This gives $|\nabla u(0)|\le2u(0)$ by the <Cauchy-Riemann equations>. A <disk automorphism> sending zero to $z$ has derivative modulus $1-|z|^2$, so the <chain rule> gives the displayed bound. Composing a boundary-pole <Poisson kernel> with a <disk automorphism> attains equality, proving the coefficient is optimal at every point.
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