schlicht functions

Definition.

The class of univalent functions on the open unit disc in the complex planesuch that for any $f$ in the class we have $f(0)=0$ and $f^{\prime }(0)=1$ is calledthe class of schlicht functions. Usually this class is denoted by$𝒮$.

Note that if $g$ is any univalent function on the unit disc, then the function$f$ defined by

$$f(z):=\frac{g(z)-g(0)}{g^{\prime }(0)}$$

belongs to $𝒮$. So to study univalent functions on the unit disc it suffices to study $𝒮$. A basic result on these gives that this set is in fact compact in the space of analytic functions on the unit disc.

Theorem.

Let $\mathrm{\{}{f}_n\mathrm{\}}$ be a sequence of functions in $\mathrm{S}$ and$f_n\mathrm{\to }f$ uniformly on compact subsets of the open unit disc. Then $f$is in $\mathrm{S}$.

Alternatively this theorem can be stated for all univalent functions by the above remark, but there a sequence of univalent functions can converge eitherto a univalent function or to a constant. The requirement that the first derivative is 1 for functions in $𝒮$ prevents this problem.