Two Conjectures by Albert Baernstein II

3.9.

TWO CONJECTURES BY ALBERT BAERNSTEIN

II*

In [I] I proved a factorization theorem for zero-free univalent disk ~ F(O) = I .

Let So denote the set of all functions F analytic and I--I in ~

THEOREM lytic in

functions

~

I.

If ~ o

, then, for each ~

,~(0,O

in the unit

with O ~ F ( ~ ) ,

, there exist functions B and Q ana-

such that

where 5 r

, and

laa,~O.14~.

The "Koebe function" for the class So is k(z) = [(I + z)/(1 -- z)] 2 which maps the slit plane Theorem I . Conjecture

[W~'l~Wl4~] I.

9

~

onto

This suggests that it might be possible to let I § I in

If FG~o , then there exist functions B and Q analytic in ~

such that

F(~).= B(;~)Q(~), zeD, where

5 ~ "4,,I/S~H",

and I ~ I < U .

We do not insist that B or Q be univalent, nor that Q(0) = i. However, when the functions are adjusted so that IQ(0) 1 = I, then ilBll~ and llB-Zlloo should be bounded independently of F. Using the fact that QI/2 has positive real coefficients {a n } of Q satisfy Janl ~< 4n, n >i wood's conjecture asserts that this inequality A proof of Conjecture I could possibly tell us wood's conjecture, and this in turn might lead Bieberbach's conjecture.

part, it is easy to show that the power series I, with equality when Q(z) = k(z). Littleis true for coefficients of functions in So. something new about how to attempt Littleto fresh ideas about h o w t o prove (the stronger)

Theorem I is easily deduced from a decomposition theorem obtained by combining results of Helson and Szeg~ [2] and Hunt, Muckenhoupt,

and Wheeden

[3].

Suppose

J-~Li(l') ' and f real

valued Consider the zero-free analytic function F defined by ~ ( ~ ) = e ~ ( ~ ) + ~ ( % ) ) , ~ c 9 where f(z) denotes the harmonic extension of f(e ie) and f the conjugate of f. Also, let S(F) denote the set of all functions obtained by "hyperbolically translating" F and then normalizing,

s(F): { and let HP denote the usual Hardy space. following way. THEOREM 2.

For ~ L i ~ [ )

(I) f = uz + ue where (2) S(F) U

S(I/F)

Theorem I follows,

Part of Theorem I of [3] can be phrased in the

the following are equivalent. I&~,~2EL~[)

is a bounded

and llu211~ < ~/2.

subset of H I.

since F I/e satisfies

(2) when

~

and 0 < I < I.

Theorem 2 may be regarded as a sharpened form of the theorem of Fefferman and Stein [4], which asserts that f = uz + u2 for some pair of bounded functions if and only if f is of bounded mean oscillation. To obtain Conjecture I in the same fashion as Theorem |, we need a result like Theorem 2 in which the