Boundary variation, method of

A method of studying univalent functions (cf. Univalent function), based on the study of the variation of a function $ w = f (z) $ that is univalent in a domain of the $ z $- plane, the variation of the function being determined by the appropriate variations of the boundary of the image of this domain.

The fundamental lemma of the method of boundary variation. Let $ D $ be a domain in the $ w $- plane and let the complement $ \Delta $ of $ D $ in the extended plane consist of a number of continua. Let $ \Gamma $ be a continuum in $ \Delta $ and let there exist on $ \Gamma $ an analytic function $ s (w) \neq 0 $ such that for any point $ w _ {0} \in \Gamma $ and for any function $ F (w) $ that can be represented as

$$ \tag{* } F (w) = \ w + A _ {0} + \frac{A _ {1} \rho ^ {2} }{w - w _ {0} } + O ( \rho ^ {3} ) $$

and that is univalent in $ D $, the inequality

$$ \mathop{\rm Re} \{ A _ {1} s (w _ {0} ) \} + O ( \rho ) \geq 0 $$

be valid, and suppose that the estimate of the residual term in (*) is uniform in all closed subdomains of $ D $. $ \Gamma $ will then be an analytic curve that may parametrically be represented by means of the function $ w = w (t) $ of the real parameter $ t $. This parameter may be so chosen that $ \Gamma $ satisfies the differential equation

$$ \left ( \frac{dw }{dt } \right ) ^ {2} s (w) + 1 = 0. $$

This result makes the important role played by quadratic differentials (cf. Quadratic differential) in the solution of extremal problems in the theory of univalent functions clear, since $ s (w) $ proves to be a meromorphic function in many applications. In certain cases it follows from the conditions of the problem that the appropriate poles of $ s (w) $ belong to the boundary of the extremal domain, and it is shown by the fundamental lemma of the method of boundary variation that the boundary of this domain belongs to the union of the closures of the critical trajectories of the quadratic differential

$$ Q (w) dw ^ {2} = \ - s (w) dw ^ {2} . $$

In a number of extremal problems, the fundamental lemma not only yields qualitative results, but also gives sufficient information for the determination of the boundary of the extremal domain, and hence for the complete solution of the problem.

The following results were obtained by means of the method of boundary variation: Qualitative results in the coefficient problem for the class $ S $; in the problem of the maximum of the $ n $- th diameter in a family of continua of a given capacity; the solution of a number of extremal problems of univalent conformal mappings of doubly-connected domains; distortion theorems for multiply-connected domains, which at the same prove existence theorems of univalent conformal mappings of a given multiply-connected domain onto canonical domains, etc.

References

[1]	M. Schiffer, "A method of variation within the family of simple functions" Proc. London Math. Soc., Ser. 2 , 44 (1938) pp. 432–449
[2]	M. Schiffer, "Some recent developments in the theory of conformal mappings" R. Courant (ed.) , Dirichlet's principle. Conformal mapping and minimal surfaces , Interscience (1950)
[3]	M. Schiffer, "Applications of variational methods in the theory of conformal mapping" , Calculus of variations and its applications , Proc. Symp. Appl. Math. , 8 , Amer. Math. Soc. & McGraw-Hill (1958) pp. 93–113

Comments

The "fundamental lemma of the method of boundary variation" is also known as Schiffer's theorem.

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 10