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Let $f=g_K^{-1}$. The reflection at infinity used in part (a) gives analytic expansions there for both inverse maps. In particular,uniformly for large $w\in \mathbb{H}$, including approach close to the real axis.

Fix $R<\infty$. If $g_K$ were unbounded on $D\cap \{|z|\le R\}$, there would be points $z_n$ there with $w_n=g_K(z_n)$ tending to infinity in modulus. The inverse expansion would imply $z_n=f(w_n)=w_n+O(1/w_n)$, contradicting boundedness of $z_n$. Thus $g_K$ is bounded on every bounded portion of its domain, including points arbitrarily near a rough hull boundary. This is the inverse-at-infinity criterion for local boundedness of a mapping-out function.

On the region $|z|>R$ for sufficiently large $R$, the expansion of $g_K$ gives $g_K(z)-z=O(1/z)$. On the remaining bounded region, the preceding bound for $g_K$ and the bound for $z$ give a finite bound for their difference. ThereforeThe argument uses reflection near infinity only. It does not assume that the real part of the map extends continuously at every point of an arbitrary hull; the sharp displacement bound for a compact H-hull is a further quantitative version of this boundedness.

Write $g=g_K$ and $f=g^{-1}$. We use the real boundary bounds for a unit-disc H-hull from part (c), including their reflected version. SetStrict monotonicity and those bounds give $-2\le \alpha \le -1$ and $1\le \beta \le 2$. The inverse $f$ extends across $\mathbb{R}\setminus [\alpha ,\beta ]$ and maps these intervals onto $(-\infty ,-1)$ and $(1,\infty )$.

Consider the holomorphic function $F(w)=w-f(w)$ on $\mathbb{H}$. If $w$ approaches a real $u$ outside $[\alpha ,\beta ]$, its inverse tends to a real $x$ with $|x|>1$. Part (c) givesIf $u\in [\alpha ,\beta ]$, every cluster point of $f(w)$ as $w\to u$ lies in the closed unit disc. To justify this without a boundary regularity assumption, first note that $f(w)$ cannot tend to infinity for bounded $w$, since $g(z)=z+O(1/z)$ there. An interior cluster point in $H$ would be mapped to the real number $u$, impossible for $g:H\to \mathbb{H}$. A real cluster point $x$ with $|x|>1$ would, by the reflected extension and strict monotonicity, have $g(x)$ outside $[\alpha ,\beta ]$, also impossible. All remaining finite boundary points lie in $K$ or in $[-1,1]$, hence in the unit disc.

It follows for every such $u$ thatFinally $f(w)=w+O(1/w)$ at infinity, so $F(w)\to 0$ there. Apply the maximum modulus principle on large upper half-discs, using the boundary limsup just obtained, to conclude $|F(w)|\le 3$ everywhere in $\mathbb{H}$. Taking $w=g(z)$ proves the sharp displacement bound for a compact H-hull, including irregular hulls: