A NOTE ABOUT AN AHLFORS INEQUALITY AND INNER RADIUS OF UNIVALENCE

Let f (z) be a holomorphic function defined on unit disc U: {z: lrl < 1} and.Sy: (f " lf') '- LU " lf') ' be its Schwarzian derivative. In 1975, Ahtfors [i] showed that the inequality (1) (2) together with u-+ oo for lzl---+ I arrd u21,, l0 is sufficient to imply the existence of a quasiconformal extension of f (z). Writing |u instead of u, (L) becomes If f (z) is defined on the upper half plane H: \, : Im(z)> O) , it is easy to seethat (2) togetherwith u + oo for Im(z)--+0 and,r/o2 * 0 alsoisasufficient condition for quasiconformal extension of f (z). Let A be any simply connected domain of hyperbolic type in C. We define lhe Poincarö density pa of A by ln'Q)l '^- 7-lhQ)r' where h(z) is any conformal mapping of,4, onto the unit disc [/. For complex-valued functions $ on A we set the norm lldlla::åm Let F(z) be any meromorphic function on,4. Lehto [2] has defined the inner radius of univaJence o1(A) as the supremum of the numbers a) 0 with the property fhat F(z) is injective whenever llSrlla < o. 352 Delin Ta,n Ler g(z) be defined in U (or in ä) and.4: S(U) (or 9(ä)) be a quasidisc. Let o1(A) be the inner radius of univalence for.4. Assume fhat f (z) is any mero-morphic function on U (or in ä). It is clear that the inner radius of univalence "r(A) is also the supremum of the numbers o) 0 with the property that /(z) isinjective whenever llSl- Srllu S a (or llS;- Srll11 < ").In this note we want to show that the Ahlfors inequality is a very powerful tool for investigating ar(.A). Some special choices of u can yield valuable lower bounds for o7(r4.) including some well-known results. In fact we obtain the following results: Theorem L. Let g(z) beholomorphicinU and A:S(U). The