We investigate the Banach manifold consisting of complex ${}^r$ functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $zh$, we study various $L^2$ norms for $T_{\varphi }\left(h\right)$, where $T_{\varphi }$ is the Toeplitz operator with symbol $\varphi$. In Theorem , given polynomials $p$ and $q$ we find a symbol $\varphi$ such that $T_{\varphi }\left(p\right)=q$. We extend some of our results to the polydisc.

We study a correspondence L between some classes of functions holomorphic in the unit disc and functions holomorphic in the left halfplane. This correspondence is such that for every f and w ∈ ℍ, exp(L(f)(w)) = f(expw). In particular, we prove that the famous class S of univalent functions on the unit disc is homeomorphic via L to the class S(ℍ) of all univalent functions g on ℍ for which g(w+2πi) = g(w) + 2πi and $li{m}_{Rez\to -\infty }(g\left(w\right)-w)=0$.

The topology of the maximal-ideal space of $H^{\infty }$ is discussed.