Zuin Phragmen-Lindelofsches Piinzip bei quasiholomorphen und pseudoanalytischen Funktionenf

Abstract

Quasiholomorphic or quasiregular functions w = u+iv are the solutions of a Beltrami equation Wz = v(z)Wz with v(z) ≤K<1. The relation i(l + v)/(l-v) = G/F defines generating functions Fand G in the sense of L. Berss and the functions W = Fu+Gv are called pseudoanalytic; they are the solutions of the elliptic system Wz = AW+BW with suitable A and B. Sometimes pseudoanalytic functions are useful in proving theorems for quasiholomorphic functions. Here some theorems of Phragmen-Lindelof type are proved. If W{z) is defined in the right half-plane H and |W(z)| is bounded by 1 on the imaginary axis, we have the classical results on the existence of the limits of (log M(r))r, m(r)r, and (log |W(reiγ)r cos γ where M(r) is the supremum on Z= r, m(r) is the mean value of log+ W(z) over = r in H. But if (log M(r))tends to zero we have not the classical result that ∣W(z) is bounded in H. We get only a bound for |W(z)|which depends on the coefficients A and B (or v) and which may increase as rλ for each λ in [0,1). So the Phragmen-Lindelof theorem looses its "gap"charactor.