Conformal invariants and function-theoretic null-sets
Lars Valerian Ahlfors, Arne Beurling · Acta Mathematica · 1950
The abbreviations M~, 21f,~(Zo) or M~($2) will be used when no misunderstanding can result.It will be assumed that,(s2) is not empty.The class ~ is said to be mo~wtonic if s s implies ~($2)< ~($2').By (I) we have then(2) (s0, $2) =< (s0, $23.Suppose now that z'= h(z) defines a one to one conformal mapping of $2 onto a region $2', and set Zo ~ h(z0).We shall say that the class ~ is con formally) for all such mappings.For a conformally invariant class we have evidentlyThis can be written in the more symmetric formand it is seen that the differential(5) M,~ (~, $2) I ~.Zz I defines a conformally invariant metric in $2. M~ is itself a relative conformal invariant, and this is the type of invariant we shall be mainly concerned with.Absolute invariants can be introduced either as ~he quotient of two relative invariants or by forming the curvature of the metric (5).If ~ is both monotonic and conformally invariant we can combine (2) and (4) to obtain (6) M (eo, $2')'ld gl =< $2)" Ide0l whenever z'= h(z) maps $2 conformally and one to one onto a subregion of $2'.We shall refer to (6) as the weak monotonic property of M~.A ~tronger result is obtained if ~ is analytically invariant.By this we mean that f(z') E ~ (s implies f(h (z)) E ~ ($2) whenever h (z') is single-valued and analytic in $2 with values in $2', regardless of whether h(z) is univalent or not.Since analytic invariance implies conformal invariance the metric (5) will have the same invariance property as before.An analytically invariant class is eo ipso monotonic.Hence (6) is valid, but the stronger assumption implies that (6) holds not only for one to one mappings, but for arbitrary analytie mappings of $2 into $2'.In this ease we shall say that M,~ has the strong monotonic property.