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  "body": "\n## Definition\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nFor $x \u003e 0$ let $\\log^+ x = \\max(\\log x, 0)$, and let $\\log^+ 0 = 0$.\nThe *Nevanlinna class* $N$ is the set of [[holomorphic functions|0DG0Z72CQD]] $f \\colon \\mathbb{D} \\to \\C$ such that\n$$\\sup_{0 \u003c r \u003c 1} \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log^+ \\abs{f(r e^{it})} \\, dt \u003c \\infty.$$\n\n## Remarks\n\nA holomorphic function on $\\mathbb{D}$ lies in $N$ if and only if $f = g/h$ with $g$ and $h$ bounded holomorphic on $\\mathbb{D}$ and $h$ without zeros.\nSuch functions are said to be of bounded type.\nSince $\\log^+ x \\le x^p / p$ for $p \u003e 0$, every Hardy space $H^p$ is contained in $N$.\n\nExample: $f(z) = \\exp\\left(\\frac{1}{1 - z}\\right)$.\nFor $z = r e^{it}$ one has $\\log \\abs{f(z)} = \\operatorname{Re} \\frac{1}{1 - z} = \\frac{1 - r \\cos t}{1 - 2r \\cos t + r^2} \u003e 0$.\nThis equals $\\frac{1}{2}\\left(1 + \\frac{1 - r^2}{1 - 2r \\cos t + r^2}\\right)$, and the Poisson kernel has mean $1$, so the mean of $\\log^+ \\abs{f}$ on each circle is exactly $1$.\nSo $f \\in N$, although $f$ lies in no $H^p$.\n\nNon-example: $f(z) = \\sin\\left(\\frac{\\pi}{1 - z}\\right)$ is holomorphic on $\\mathbb{D}$ but not in $N$.\nIts zeros are $z_k = 1 - 1/k$ for $k \\ge 1$, and $\\sum_k (1 - \\abs{z_k}) = \\sum_k 1/k$ diverges.\nBy Jensen's formula, the zeros of a function in $N$ that is not identically $0$ satisfy the Blaschke condition $\\sum (1 - \\abs{z_k}) \u003c \\infty$.\n",
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