The Nevanlinna class on the disc

Assume countable choice. Write $\mathbb{T}$, $m$ and $\mathbb{D}$ for the torus with its normalized Haar measure and the unit disc (The one-dimensional torus and its normalized Haar integral), and put ${\log }^{+}x:=\max \{\log x,0\}$ for $x>0$, with ${\log }^{+}0:=0$.

A holomorphic function $f:\mathbb{D}\to \mathbb{C}$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions) belongs to the Nevanlinna class $N(\mathbb{D})$ if the subharmonic function ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }$ has a harmonic majorant on $\mathbb{D}$, that is, if there is a harmonic function $h$ on $\mathbb{D}$ with ${\log }^{+}\mathrm{\mid }f(z)\mathrm{\mid }\le h(z)(z\in \mathbb{D})\mathrm{.}$ Since ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }\ge 0$, any such majorant satisfies $h\ge 0$ on $\mathbb{D}$; the function ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }$ is subharmonic: it is zero if $f\equiv 0$, and otherwise $\log \mathrm{\mid }f\mathrm{\mid }$ is subharmonic and ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }=\max (\log \mathrm{\mid }f\mathrm{\mid },0)$ is a finite maximum of subharmonic functions, with $\log \mathrm{\mid }f\mathrm{\mid }=-\mathrm{\infty }$ at the zeros of $f$ (The logarithm of the modulus of a holomorphic function is subharmonic, Subharmonic functions on plane domains). Constant functions are harmonic, so every bounded holomorphic function lies in $N(\mathbb{D})$.

Equivalent sup-mean form. A holomorphic $f$ belongs to $N(\mathbb{D})$ if and only if ${\sup }_{0<r<1}{\int }_{\mathbb{T}}{\log }^{+}\mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )<+\mathrm{\infty }\mathrm{.}$ The forward direction is the mean value property of a harmonic majorant: if ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }\le h$ with $h$ harmonic, then ${\int }_{\mathbb{T}}{\log }^{+}\mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )\le {\int }_{\mathbb{T}}h(r\zeta )dm(\zeta )=h(0)$ for every $0<r<1$ (Plane harmonic functions satisfy the mean-value property); the converse is the Poisson-modification and increasing-Harnack construction of A harmonic majorant of log^+|F| exists exactly when the radial log^+ means are bounded, which is quoted here as the well-definedness statement for the two equivalent forms. Both forms are used in this pair: the majorant form in The Nevanlinna class is a bounded quotient class and Boundary values and log-integrability of Nevanlinna-class functions, the sup-mean form in Blaschke factorization of a Nevanlinna-class function.

The Hardy classes are contained in $\mathbf{N}(\mathbb{D})$. Every $H^p(\mathbb{D})$, $0<p\le \mathrm{\infty }$, is contained in $N(\mathbb{D})$ (Analytic Hardy spaces on the unit disc). For $0<p<\mathrm{\infty }$ one has ${\log }^{+}x\le x^p\mathrm{/}p$ for every $x\ge 0$: the inequality is trivial for $x\le 1$, and for $x\ge 1$ it follows from $\frac{d}{dx}(x^p\mathrm{/}p-\log x)=x^{p-1}-1\mathrm{/}x\ge 0$ and its value at $x=1$ is $1\mathrm{/}p>0$. Hence ${\int }_{\mathbb{T}}{\log }^{+}\mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )\le \frac{1}{p}{\int }_{\mathbb{T}}\mathrm{\mid }f(r\zeta ){\mathrm{\mid }}^{p}dm(\zeta )\le \frac{1}{p}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}^p<+\mathrm{\infty }$ for every $0<r<1$, so the sup-mean form of membership holds and $f\in N(\mathbb{D})$. For $p=\mathrm{\infty }$ one has ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }\le {\log }^{+}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}$ pointwise, because $\mathrm{\mid }f(z)\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}$ and ${\log }^{+}$ is nondecreasing; the constant function $h:={\log }^{+}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}$ is harmonic with ${\log }^{+}\mathrm{\mid }f\mathrm{\mid }\le h$, so $f\in N(\mathbb{D})$ directly by the majorant form. (For $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}<1$ the constant majorant is $0$, which is why the bound is written with ${\log }^{+}$ on both sides.)

This is the disc Nevanlinna class of holomorphic functions. No result from Nevanlinna value-distribution theory for meromorphic functions on $\mathbb{C}$ is used in this pair. No choice principle beyond countable choice is used.