Instead of investigating various isolated extremal problems in the theory of schlicht functions, the authors have concentrated their efforts on the investigation of the family of extremal schlicht functions in the large.

Table of Contents

• History of schlicht functions and elementary properties of the nth region
• Variations of schlicht functions
• The critical points of the differential equation
• The} $\Gamma$-structure. Behavior in the large
• Geodesics. Continuity theorem
• Functions which are regular in $|z| < 1$ and satisfy the differential equation
• The length-area principle. Teichmüller's method
• Relations between} $P(w)$ and $Q(z)$
• Löwner curves
• Linear forms and the supporting surface
• The portion of the boundary of} $V_n$ corresponding to single analytic slits
• Parametrization of the boundary of} $V_n$
• The region} $V_3$
• A method for investigating the conjecture} $|a_4|\leq 4$
• The region of values of the derivative of a schlicht function (by Arthur Grad)