A function /(z) = z+2¡S=2 a¡tz* analytic in D is said to be typically real ... Tic) the collection of all functions /(z) = z+cz2+2£l3 .... maX|2|=r |/(z)| ..... Deutsch. Math.-Verein. 40 (1931), 33-35. 9. -,. Ьber positive harmonische Entwicklungen und ...
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 40, Number 1, September 1973
SOME REGULARITY THEOREMS FOR TYPICALLY REAL FUNCTIONS1 GEORGE B. LEEMAN, JR.
Abstract. This paper studies the class of typically real functions and the subclass of typically real functions with fixed second coefficient. For each of these classes the author proves a few regularity theorems, which describe the coefficients' asymptotic behavior.
Let 5 be the collection of functions /(z)=z+2"=2 a„zn analytic and univalent in the unit disk -D= {z| |z|oo 2«
f uc(0)da(0), JO
by Lebesgue's dominated convergence theorem. The statements of equality follow easily from the definition of ue. Similarly,
(6)
lim a%n+l = [ u0(6) da(0), n^oo 2n + 1
Jo
with the indicated cases of equality. Consequently, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Theorem
1 follows.
1973]
REGULARITY THEOREMS FOR TYPICALLY REAL FUNCTIONS
193
Corollary 1. If fe T, then 0^limr^x_(l -r)2f(r)^l. Equality holds on the left unless/(z)=A[z/(l -z)2] + (l -X)g(z), where 0
