Bounded holomorphic functions with given maximum modulus on all circles

We study $\Omega\subset\mathbb{C}^{d}$, a circular, bounded, strictly convex domain with $C^{2}$ boundary. Let $g$ and $h$ be continuous functions on $\partial\Omega$ with $\vert g(z)\vert<h(z)=h(\lambda z)$ for $z\in\partial\Omega$ and $\vert\lambda\vert=1$. First we prove that $h$ can be approximated by the maximum modulus values of $K$ homogeneous polynomials, where $K$ is independent from $h$. Next we construct $f_{1}\in A(\Omega)$ such that

$$\max_{\vert\lambda\vert=1}\vert(g+f_{1})(\lambda z)\vert=h(z)$$

for $z\in\partial\Omega$. Moreover we can choose $f_{2}\in\mathbb{O}(\Omega)$ with $\vert f_{2}^{*}(z)\vert=h(z)$ for almost all $z\in\partial\Omega$ and $\max_{\vert\lambda\vert<1}\vert f_{2}(\lambda z)\vert=h(z)$ for all $z\in\partial\Omega$.