A harmonic majorant of log^+|F| exists exactly when the radial log^+ means are bounded

Forward implication. Suppose $h$ is a harmonic majorant of $u$ on $\mathbb{D}$, i.e. $h$ is harmonic and $h\ge u$. Fix $0<r<1$; $h$ is harmonic on a neighbourhood of $\overline{D(0,r)}$, so integrating $u(r\zeta )\le h(r\zeta )$ over $\mathbb{T}$ and applying the mean-value property in torus form gives ${\int }_{\mathbb{T}}u(r\zeta )dm(\zeta )\le {\int }_{\mathbb{T}}h(r\zeta )dm(\zeta )=h(0)<+\mathrm{\infty }\mathrm{.}$ Taking the supremum over $r$ gives the stated bound, so the forward implication holds.

If $F\equiv 0$, then $u=0$, its radial means are zero and the zero harmonic function is a majorant, so both conditions hold. Assume henceforth $F\equiv ̸0$. The function $u={\log }^{+}\mathrm{\mid }F\mathrm{\mid }$ is subharmonic and nonnegative by [L1]. It is also continuous: $F$ is continuous and $x\mapsto {\log }^{+}x$, with value $0$ at $x=0$, is continuous on $[0,\mathrm{\infty })$.

The modifications and their values at the origin. Since $u$ is continuous by step 1.2, in [L2] choose the specific boundary approximants ${\varphi }_n=u{\mathrm{\mid }}_{\mathrm{\partial }D(0,r)}+1\mathrm{/}n$. Their extensions are $h_n=P_r[u]+1\mathrm{/}n$, by uniqueness and linearity of the Dirichlet solution. Thus $H_r=P_{D(0,r)}u=P_r[u]$, continuous on the closed radius-$r$ disc, harmonic inside, with boundary value $u$ and $H_r\ge u$ by [L3]. Its mean value at the origin is $H_r(0)={\int }_{\mathbb{T}}u(r\zeta )dm(\zeta )$ by [L4].

Monotonicity in the radius. Let $0<r<\rho <1$. On $\mathrm{\partial }D(0,r)$, $H_r=u$ by step 2.1 whereas $H_{\rho }\ge u$ by [L3]. Both are harmonic on $D(0,r)$ and continuous on its closure, so their difference is nonnegative there by the minimum principle [L6]. Hence $H_{\rho }\ge H_r$ on $D(0,r)$.

Harnack bounds for the converse. Suppose $C<\mathrm{\infty }$. For $\mathrm{\mid }z\mathrm{\mid }<R<r<1$, apply [L7] to $H_r+\epsilon$ on the radius-$R$ disc and let $\epsilon \downarrow 0$. Using step 2.1 gives $0\le H_r(z)\le (R+\mathrm{\mid }z\mathrm{\mid })(R-\mathrm{\mid }z\mathrm{\mid }{)}^{-1}C$. Letting $R\uparrow r$ yields $H_r(z)\le (r+\mathrm{\mid }z\mathrm{\mid })(r-\mathrm{\mid }z\mathrm{\mid }{)}^{-1}C$. In particular, for any fixed $\mathrm{\mid }z\mathrm{\mid }<r_0<1$, these values are uniformly bounded for $r\ge r_0$, by $(r_0+\mathrm{\mid }z\mathrm{\mid })(r_0-\mathrm{\mid }z\mathrm{\mid }{)}^{-1}C$.

Construction of the harmonic majorant. Put $r_n:=1-1\mathrm{/}(n+1)$ for $n\ge 1$ and $H_n:=H_{r_n}$. Fix $R<1$. For all $n$ with $r_n>R$, the function $H_n$ is harmonic on a neighbourhood of $\overline{D(0,R)}$ (namely on $D(0,r_n)$), and by step 3.1 the sequence $(H_n{)}_{n\ge n_R}$ is increasing on $D(0,R)$; it is bounded at the origin by $C$ by step 2.1, so the increasing Harnack convergence principle [L7] provides a harmonic function $H^{(R)}$ on $D(0,R)$ with $H_n\to H^{(R)}$ locally uniformly there. For $R<R^{\mathrm{\prime }}<1$ the two limits $H^{(R)}$ and $H^{(R^{\mathrm{\prime }})}$ agree on $D(0,R)$ by uniqueness of pointwise limits of the same sequence, so there is a single harmonic function $H$ on $\mathbb{D}$ with $H_n\to H$ locally uniformly on $\mathbb{D}$. For each fixed $z\in \mathbb{D}$ and all $n$ with $r_n>\mathrm{\mid }z\mathrm{\mid }$ one has $H_n(z)\ge u(z)$ by [L3], and $(H_n(z))$ is eventually nondecreasing by step 3.1, so $H(z)={\lim }_n{H}_n(z)\ge u(z)$. Hence $H$ is a harmonic majorant of $u$ on $\mathbb{D}$.

Assembly. If a harmonic majorant exists, step 1.1 bounds the radial $u$-means by its value at the origin, so the supremum is finite. Conversely, if the supremum is finite and equal to $C$, step 4.1 constructs a harmonic majorant $H$ of $u$ on $\mathbb{D}$; the construction uses the subharmonicity and upper semicontinuity of $u$ from step 1.2 together with the Poisson modifications, so both implications hold.