on successive coefficients of functions of bounded

ON SUCCESSIVE COEFFICIENTS OF FUNCTIONS OF BOUNDED BOUNDARY ROTATION J. W. NOONANf AND D. K. THOMAS 1. Introduction Denote by Vk, for k ^ 2, the class of functions of boundary rotation at most kn. Then / given by

f(z)=z + £ anzn

(1)

n=2

belongs to Vk if and only if/is regular in the open unit disc y, f'(z) # 0 for z e y, and, with z = r exp (id), In \

Re

dO ^ kn.

(2)

For fe Vk we shall be concerned with estimates for \an+J — |flj. We shall prove the following two theorems. THEOREM

1. Let fe Vk and be given by (1). Then, for n^ 1, \\an+1\-\an\\^c(k)n*k-2,

where C(k) is a constant depending only on k. We note that the function f0 defined by fc/2

shows that the index \k—2 in Theorem 1 is best possible. In [4] the first author showed that, for/e Vk, co = lim (1 -r)* fc+1 M(r,f)

(3)

exists, isfinite,and that

lim - g n - .

(4)

With this notation we have THEOREM

2. Letfe Vk and be given by (1). Then

ifk>2

Ifk = 2, the result holds only if co > 0. Received 27 April, 1971. tNRC-NRL Post-doctoral Resident Research Associate. [J. LONDON MATH. SOC. (2), 5 (1972), 656-662]

ON SUCCESSIVE COEFFICIENTS OF FUNCTIONS OF BOUNDED BOUNDARY ROTATION

657

We note that Theorems 1 and 2 are analogues for the class Vk of results of Hayman [3] for circumferentially mean one-valent functions. 2. Proof of Theorem 1 Fom (2), write (z/'(z))' = f'(z)h(z),

with h(0) = 1. Put F(z) = (z(zf (z))\.

F(z)=f'(z)[h2(z)+zh'(z)}.

Then (5)

Since fe Vk) we can write [1]

where st and s2 are normalised starlike functions. Thus, if / is given by (I) and

In Z

~^'|Sl(z)l

|Si(z)|ifc

~* {\h\z)+zH{z)\\dO

(7)

where we have used (5), (6) and the fact that for z ey, |s2(z)|~1 < 4/r for any starlike function. Let 0 < r < 1 be fixed. Then, by a result of Golusin [2], there exists zx such that \zx\ = r and such that Iz-zJIs^z)! < 2r 2 /(l-r 2 ) for all z such that \z\ = r. Thus, in (7) with £, = zu we have

r where we have used the distortion theorem for the function It follows from (2) that In

, w, .

f l+zexp(-i7)

oJ l - z e x p ( - i f ) o where ^ is a function defined on [0,2n] with 271

J \dK0\ < kn. 0

Thus, if

A(z) = l+ f cz-, n= l

(9) gives

2n

1 /• cn = — n J JOUR: 20

...

1 f l+zexp(-i7) 2n , .( dfi(t),

exp(-int)dix(t)

(9) vy

658

J. W. NOONAN AND D. K. THOMAS

and so \cn\ ^ k for n ^ 1. Using this estimate, one easily sees from (9) that 2n

Thus from (8)

Choosing

one obtains for n ^ 1, Ik

+

il-IflJ | < k(k+l)

exp(3)2**(4/3)* fc+1 H**"

We also note that with a suitable choice of |£| = r, we can obtain

for « ^ 5 with C(fc) -> 0 as k -> oo. 3. Proof of

Theorem!

For to > 0, it was shown in [4] that there exists 0O such that

Let ton = (l-pn)ik+1f'{pn exp (i0o)) where p n = 1 - 1/n. We shall need the following lemmas, the first of which was essentially given in [4]. LEMMA

1. Let a> > 0. Then with the above notation,

(i) lim \con\ = co and lim Xn exists, where kn = argo n .

(u)exP(-i0o)fln = — (iii) arg exp ( — i90)an = argco exp(i(An—«0o)) + o(l) as n -> oo. o(l) is uniform in any Stolz angle about the point exp (i0o). LEMMA 2.

Let e > 0 be given. If k > 2 and co > 0, f/iere exists C(e) SMC/J

J(l-rexp(i(0-0o)))/'(r where E = {6 : C(e)(l -r) ^ | 0 - 0 O | < TT}. Proo/. Without loss of generality, we may assume 0O = 0. We denote by A(k) a constant depending only on k. It was shown in [4] that if co > 0, then

ON SUCCESSIVE COEFFICIENTS OF FUNCTIONS OF BOUNDED BOUNDARY ROTATION

659

^z) = z/(l - z ) 2 , where sx is defined in (6). Thus, from (6),

Now with z = r exp (i0), |1 -z\ ^ |0| for r ^ £, and so (11) gives

Let C > 0 be given. Then, from (12) since k > 2,

J

r J \(i-z)f(z)\de^A{k)

r j \e\-*kdd= E E

E

and Lemma 2 now follows on suitably choosing C. LEMMA 3. Ifk>2

and co > 0, then as n -> oo

„„ = ( 1 - 1 W e x p ( - f l o ) a , . 1 + Proo/. Let = co exp (iAn) 2L Cm exp ( — Let e > 0 be given, and choose C(e) and E as in Lemma 2. Then, with z = r exp (id),

1

2n

I* ( 1 -

~2n Jo

-T-f+T-f In J In J ^Ii+I2, say, where £ ' = [0, 27r]\£. It is easily seen that Lemma 2 is valid w i t h / ' replaced by (f)'. Thus, using Lemma 2, we have

Now on E', |1 —z exp (—t0o)| < C(e)(l — r) and so \=r- j \f(z)-cf>'(z)\de.

660

J. W. NOONAN AND D. K. THOMAS

It now follows on using Lemma 1 (iv) that

and so

Choosing r = 1 — 1/n we have from (13), (14), and (15)

Lemma 3 now follows on noting that

C._ 1 -C ll - 2 = YQQ C1 +°0))

as « -> oo.

We now prove Theorem 2 when co > 0. We may assume that A: > 2, since if k = 2 then co > 0 implies that /(2) =

(l-zexP(-i0o))'

in which case |aj = 1 for all n ^ 1 and the result is then trivial. Now, from Lemma 3

n

exp ( -

which on using Lemma 1 (ii) gives

Thus

Also (i) and (iii) of Lemma 1 give argexp(-i0 o )fl n _ 1 = arg exp[i(A n -(«-l)0 o )] + o(l). Combining (16) and (17), we have nik-2

as n -*• oo, which gives Theorem 2 for co > 0

(17)

ON SUCCESSIVE COEFFICIENTS OF FUNCTIONS OF BOUNDED BOUNDARY ROTATION

661

Thus it remains to prove Theorem 2 when co = 0 and k > 2. We first suppose that st defined by (6) is not of the form z/(l - z exp (U))2, A real. Then

where /? < 2. As in the proof of Theorem 1, we obtain from (7) ....

..3 . , .

A{k)

1

Choosing

III = r = we have which gives, since /? < 2 and fc > 2, 0 ri* as « -• oo. We now prove Theorem 2 in the case co = 0, k > 2, and Sl (z)

=

(l-zexp(U))2 *

Without loss of generality, we assume A = 0. Then as in the proof of Lemma 2 we see that, given e > 0, there exists C(e) such that J (1 - r exp (id))f'(r exp (i0))«/0

(18)

where E = {0: C(e)(l - r ) < |0| < n). Let £' = [0,2n]\E. Then for 0 e £' we have |1 - r exp (i0)| < C(e)(l - r ) , and so J |1 -r exp (i0)| \f'(r exp (i0))| d9 ^

2C 2 (E)(1

- r ) 2 M(r,/').

E'

But co = 0 and so, from (3),

M(r,f) = as r -> 1. Thus, I |1—rexp(i0)|

exp

(r "> 1).

Since 1 («+1) a n + ! -»flB = — -

2n

r

(1 - r exp (i0))/'(r exp (i0)) exp (-ind)dd,

(19)

662

ON SUCCESSIVE COEFFICIENTS OF FUNCTIONS OF BOUNDED BOUNDARY ROTATION

(18) and (19) give \(n + \)an + i-nan\

= o(\)n*k-x

as n -> oo. Hence

as n -> oo, and the theorem now follows on using (4). If k = 2 and co = 0, the theorem is false, as is shown by the function fe V2 defined by/'(z) = 1/(1 - z 2 ) . It is clear that co = 0, but (2«-2) | | a 2 n _ 1 | - K , - 2 l | = ~

^

~ "> 1

as n -> oo.

1. D. A. Brannan, "On functions of bounded boundary rotation I ",Proc. Edinburgh Math. Soc, 16 (1968-69), 339-347. 2. G. M. Golusin, " On distortion theorems and coefficients of univalent functions", Mat. Sb., 19 (1946), 183-202. 3. W. K. Hayman, "On successive coefficients of univalent functions ", / . London Math. Soc, 38 (1963), 228-243. 4. J. W. Noonan, " Asymptotic behaviour of functions with bounded boundary rotation ", Trans. Amer. Math. Soc, 164 (1972), 397-410.

U.S. Naval Research Laboratory, Washington, D.C. 20390 University of Maryland, College Park, Maryland, 20742 Present address: Department of Pure Mathematics, University College, Swansea.