On the Schwarz reflection principle.

On the Schwarz Reflection Principle

“…Instead of a function f( z), Lohwater has also considered its modulus I f( z) I [6]. In this connection, we shall now define three generalized classes I U I , I U 1-, and I U I + as follows.…”

“…Let EI be the set of all points e i </> on C such that 1imr~ I If(re i </» 1= 1, then the measure I EI 1= 2' 77". Denote by CpU, e i </» the radial cluster set of f( z) at the point e i </>, then clearly we have (6) CpU, e i </» C {K: I K 1= l} for each e i </> EEl· On the other hand, if e i </> E E*, then there is a point e ili on I w 1= 1 for which limr~1 z(re ili ) = e i </>. Let r li be the radius in Dw ending at e ili and let y</> = z(rli), then y</> is an arc lying in Hn and ending at e i </>.…”

“…Introduction. In [6,Theorem 3], Lohwater proved the following result: Letf(z) be meromorphic in the unit disk D = {z: 1 z 1 < I} with bounded characteristic in the sense of Nevanlinna (see [2,p. 38]) and let the radial limits limr~d(rei6) = f(e i6 ) have modulus 1 for almost all points e i6 E A = {e i6 : a < () < P}.…”

On the generalized Seidel class 𝑈

“…In [7,Theorem 3], Lohwater proved that if / G U on A(a, ß) with bounded characteristic in the sense of Nevanlinna (see [1, p. 38]), and if P is a singular point of /on A(a, ß), then every value of modulus 1 which is not in the range of /at P is an asymptotic value of / at some point of each subarc of A(a, ß) containing the point P. He then asked whether this result is still true if / is not of bounded characteristic (see [7, p. 156]). Recently, in [5], we have solved this problem in the affirmative sense as follows.…”

“…As an application of Lemma 2, we shall prove the following theorem of Lohwater [7,Theorem 2] which will be needed in proving Theorem 2.…”