Some extension on the Kellogg theorem (Q2732398)

Let \(f(z)\) be analytic and univalent in the open unit disk \(\mathbb{D}\). Suppose that \(f\) maps \(\mathbb{D}\) onto a domain bounded by a smooth closed Jordan curve. Then the classical theorem of Kellogg gives a sufficient condition for \(f'(z)\) to extend continuously to the closure of \(\mathbb{D}\). In the paper under review, the author uses the concept of modulus of continuity to formulate and prove Kellogg's theorem.