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  "body": "\n## Summary\n\nThe Nevanlinna class $N$ is the space of [[holomorphic functions|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] whose positive part of $\\log \\abs{f}$ has bounded averages on circles.\nIts elements are exactly the quotients of two bounded holomorphic functions with zero-free denominator, the functions of bounded type.\nIt contains every [[Hardy space|PNNNFJVF6W]] $H^p$, and its nonzero elements still have radial limits almost everywhere and zeros satisfying the Blaschke condition.\n\n## Explanations\n\n### L1 human\n\nHardy spaces control the size of $\\abs{f}^p$ on circles.\nThe Nevanlinna class only controls the size of $\\log \\abs{f}$ where $\\abs{f} \u003e 1$, which is a much weaker condition.\nFor example, $1/(1 - z)$ is not bounded but lies in $N$, since it is $1$ divided by the bounded zero-free function $1 - z$.\nEven $\\exp\\left(\\frac{1}{1 - z}\\right)$ lies in $N$.\nThe quantity $\\frac{1}{2\\pi} \\int \\log^+ \\abs{f(r e^{it})} \\, dt$ is the proximity part of the [[Nevanlinna characteristic|YT6Z0GWS0W]], which explains the name.\n",
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