o, : r,. _{ laul = r,. Conversely, every function satisfying (2) belongs to Bp. For these basic facts of the class .Bo see for example [7]. The function. (4) lo@): +l(E)r''-,].
Annales Academie Scientiarum Fennicrc Series A. I. Mathematica Volumefr 7, 1982, 1,65-176
ON THE RAI{GE OF REAL COEFFICIENTS OF FUNCTIONS WITH BOUNDED BOUI{DARY ROTATION HEIKKI HAARIO
Introduction
For k>2 let Bk denote the class of
analytic functions
f(r) - z*arzz+...
(1)
which map the unit disc onto a domain whose boundary rotation is at most kz. Each function of .Bo can be obtained as a solution of the equation (2)
where
p is a real Borel measure on /:(-
(3)
_{
o,
:
r,
_{
n,
frj
laul
=
satisfying
r,.
Conversely, every function satisfying (2) belongs to Bp. For these basic facts of the for example [7].
class .Bo see
The function
(4)
lo@):
+l(E)r''-,]
: ยง.n.s,'s,.
is the solution of (2) when p is a measure concentrated on the points e:0, E:n. The coefficient conjecture for the class,Bo was la,l"2.
Ecluality holcls only
if
_ k(li2-2) -4
l'(w)
-r l-*(rw) w,ith z - *
I.
Hrxrr Haanlo
176
References
[1] AnnnoNov, D., and S. FnrnouNp: On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation. - Ann. Acad. Sci. Fenn. Ser. A I Math. 524, 1972, l*14. [2] BnaNNnN, D. A., J. G. CruNn, and W. E. Krnq,l,N: On the coefficient problem for functions of bounded boundary rotation. - Ibid. 523, 1973, l-18. [3j Haanro, H.: On the coefrcient bodies of univalent functions. - Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 22, 1978, l-49. [4] Krnw,4N, W. E., and G. Scnosrn: Inverse coefficients for function of bounded boundary rotation. - J. Analyse Math. 36, 1979, 167-178. [5] P5r,ucrn, A.: Functions of bounded boundary rotation and convexity. - Ibid. 30, 1976,
437-451. [6] Scrrrrun, M., and O. Tauur: On the fourth coefficient of univalent functions with bounded boundary rotation. - Ann. Acad. Sci. Fenn. Ser. A I Math.396' 1967, l-26. [7] Tnrrlrr,rr, O.: Extremum problems for bounded univalent functions. - Lecture Notes in Mathematics 646, Springer-Verlag, Berlin-Heidelberg-New York, 1978.
University of Helsinki Department of Mathematics
SF-00100 Helsinki l0 Finland
Received 16 October 1981
