311
Internat. J. Math. & Math. Sci.
VOL. 16 NO. 2 (1993) 311-318
ESTIMATES OF LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNTIONS
STEPHEN M. ZEMYAN
Department of Hathematlcs
The Pennsylvanla State Unlverslty
Hont Alto Campus
Hont Alto, Pennsylvanla 17237-9703
(Received November 7, 1991 and in revised form June 19, 1992)
ABSTRACT. In this paper, we derlve some consequences of Hllln’s Inequallty for the
logarlthmlc coefflclents of a univalent functlon by exploltln a reformulatlon of
It.
KEY WORDS AND PHRASES.
Unlvalent
functlons,
logarlthmlc
coefflclents,
Mllln’s
Inequallty, convex welghts.
1991 MATHEMATICS SUBJECT CLASSIFICATION CODES. 30C50, 30C75.
I.
INTRODUCTION.
Let
S
unlvalent In the unlt dlsk
and
f’ (0)
f(z)
denote the class of all functlons
I.
U
{z:
Izl <
I}
which are analytlc and
and are normalized so that
The logarlthmlc coefflclents
of
k
f(z)
f(O)
0
are deflned by the
re lat lon
2
log
yk
zk.
k=l
In partlcular, the Koebe function
k
I/k (k
1,2
k(z)
z(l
z)
-2
has logarithmic coefficients
}.
In fundamental work published during the 1960’s, I. M. Milln concerned hlmself
with logarlthmlc coefflclents and thelr role in the theory of univalent functions.
Subsequent to a great deal of intense research, Mllln conjectured the inequalities
stated in Theorem
below.
These Inequalltles attracted much attentlon because
their truth would imply the the truth of the Robertson conjecture and the Bleberbach
conjecture, in addltlon to others.
proved these Inequalltles using
Then, in 1984, Louis de Branges [I, p. 146-150]
Loewner’s
parametric method and a special system of
strictly decreasing weight functions. An alternate treatment of these Inequalltles
Since
was subsequently presented by Fitzgerald and Pommerenke [2, pp. 683-690].
then, many authors have explored these Inequalltles.
so, but flrst we restate de Branges’ famous result.
In thls paper we shall also do
S.M. ZEMYAN
312
THEOREM 1.
Let
(de Branges)
[
S, and let
f
k (k
f. Then, for every N z 1,
logarithmic coefficients of
k)lkl2
k{N +
denote the
N+l-k
I
-<
1,2
we have
{1.1)
k=1
k=1
f(z)
Equality holds if and only if
nz)
z/(l
a
Inl
It is our purpose here to derive further consequences of Theorem
by utilizing
the following equivalent reformulation of it.
THEOREM I’.
coefficients of
convex
Let
S, and let
f
Let
f.
P(k)
(k
Assume further that
function.
identically zero; that is, that
k z N + I.
Ik (k
k
non-lncreaslng
be a non-negatlve,
the
weight
N
there exists an
functlon
P
for which
P(k}
Is eventually
0
whenever
N, we have
Then, for thls value of
I
denote the logarlthmlc
1,2
1,2
a
P(k)lkl
P(k)
[
{1.2}
k
k=l
k=l
z/(1
f(z)
Equality holds if and only if
As it Is pointed out in [I],
nz)
a
Inl
by two Abel
Theorem 1’ follows from Theorem
summations (or summations by parts).
Of course, Theorem
follows from Theorem I’
by slmply choosing the welght function
IN
P(k)
Thus,
+
N
k, k
O,
N+I
k
these theorems are equivalent, even though Theorem 1’
appears on the surface
to be more complex.
Several
authors
have
concerned
themselves
with
consequences
inequalities and with other inequalities which are similar in nature.
of
these
Milln and
Grinshpan [3, pp.139-147] derived a necessary condition which must be satisfied by
the weights
P(k)
in Theorem
I’ if the Koebe function is to be extremal, and
Andreev and Duren [3, pp. 721-728] reproved this
explored some of its consequences.
necessary condition in a different manner, and detailed its consequences.
Many choices for
P
for some desired choices, such as
the assumption that
P
-
yield interesting applications of this theorem.
P(k)
k
{ > 0), or
P{k}
r
k
However,
(0 < r < I),
is eventually identically zero could never be met.
simple process of truncation produces a sequence
eventually identically zero,
{P(I}
P(N),O,O
The
which is
but then the truncated sequence need not be convex,
thereby preventing the application of Theorem I’.
In Section 2, we provide a simple method of circumventing this difficulty.
In Section 3, we take advantage of this method in a natural manner to obtain
several interesting consequences which extend and generalize known results.
ESTIMATES OF LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNCTIONS
313
REDEFINING THE CONVEX WEIGHT FUNCTION P(k).
Let N
2 be fixed, and P(k) (k
1,2
2.
be given.
Assume that P Is
Assume further that the weight function
is not eventually identically zero), but
non-negatlve, non-lncreaslng, and convex.
P(k} > 0
that
for all
P{k)
{N
IN
[P(N) -P(N- 1)] x +
y
and the
(R).
I} > P{N}, then there Is a unlque stralght llne which passes through
I}} and (N, P(N}}. The equation of this llne is
I, P{N
P(N
If
the points
P
{i.e., that
k
as k
0
(N- 1} P(N}],
P(N- 1)
x-intercept of this line Is easlly seen to be
xN=N+
P(N)
P(N
P(N)
1)
We conslder two cases.
CASE I:
If
P(N)
O<
P(N
or equlvalently, if
2 P(N}
P(N
1,
1}
P(N)
I}, then we define a new convex weight function
by
{P(k},
A
t
P(k
0
N
k
k
N+I
A
In this case, P
^P
is a simple truncatlon of
P, and it is geometrically evident that
satlsfles all precondltlons necessary to apply Theorem 1’.
CASE II:
If
P(N}
P{N
I}
P{N}
^
then we define a new convex weight function P by including a mld-range llnear
section, the addltlon of which will facilitate the estlmatlons to follow.
Speclflcally, we define
P{k},
P(k}
[P{N) -P{N-I}]k + [NP{N-I)
O,
N
k
{N-I)P{N)], k
k
N+I
Ixu]
[xu]+1
where
denotes the greatest integer function. Since P has been redefined in
a linear fashion for mld-range and end-range values of k, it is again geometrically
clear that
P^
satisfies all pre-condltlons necessary in order to apply Theorem I’.
In so doing, we obtain
S.M. ZEMYAN
314
where we have set
[P(N)/(P(N
Np
P(N))].
I)
For example, if we set
This last inequality allows for convenient estimations.
A
P(k)
ak + b, where
P(N)
a
N+Np
P(N
I)
NP(N
b
and
N+Np
^P(k)
ak + b
Np
+ b In
+ a
Np
+ b in
It
a
<
k:N+l
k=N+l
(N
I)
I)P(N), then
+
Hence, estimates of the form
N
N
k=l
k:l
If ]
+
A
are possible, with no explicit reference to the values of
P.
Also, since we have
^
P(k) -< P(k), we obtain the inequalities
constructed
N+Np
N
P(k}
k
k=l
k=l
k:l
so that
P(k)
k=l
(2.1)
k=l
which is significant whenever the series on the right converges.
The above observations were made under the assumption that
If
P(N
P{N)
I)
for all
P(N
I} > P{N}.
P(N} > O, then the convexity condition would require that
k > N.
But then
P(k}
could never be redefined as above to be
P
In addition, the infinite series on the right side of
Such a choice for P
would diverge by comparison to the harmonic series.
eventually identically zero.
(2.1}
would be inadmissible.
Of course, every admissible choice of
{2.1).
3.
P
yields a new inequality of the form
In the next section, we present some of these choices.
RESULTS.
Andreev and Duren [3, p. 722] asked whether the partial sum inequality
N
N
[
k
k:!
r > O.
{k{2
r
k
<-
[
kr
k
k=l
Although they provide some supportive evidence that this
that for no
inequality should hold for r s I/2, they also establish [3, p. 727]
holds for any
N z 2
can it hold for all
r < I.
315
ESTIMATES OF LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNCTIONS
In Theorem 2(a) below, we shall show that this Inequallty does Indeed hold if
It Is not known whether the constant "1/2" Is best posslble. Wlthln the
proof of Theorem 2(b), a method of estlmatlng the left-hand slde of the above
Is provlded. Although Theorem 2(b) has been obtalned
Inequallty when I/2 < r <
elsewhere (see, for example, Mllln and Grlnshpan [3, p. 143]), our purpose here Is
to show that It may be easlly obtalned from Theorem 1’.
r s 1/2.
Let
THEOREM 2.
0
k (k
1,2
denote the logarlthmlc
f.
coefflcients of
If
S, and let
f
I/2, then for every
r
I, we have
N
N
N
(3.1)
k=l
k--I
f(z)
Equality holds if and only if
(b)
For each
k
.kl
2
r
k
For
0
In
r
I/2, the weight function defined by
|r
P(k}
{b)
If
r
k=1
A
is convex.
I}[
k
k=l
PROOF.
z
r < 1,
0
r,
}z)
z/(1
k,
k
0
k
N
N+I
Hence, Inequallty (3.1} follows Immedlately from Theorem 1’.
< r < 1, then there exists a unlque posltlve Integer J such that
1/2
J/{j + I} < r
For thls value of
(j + 1)/{j + 2}.
J
k
r,
^
P(k)
(r
N
r
-I
)k +
k=l
Nr-I
O,
(N-llr),
N
and any
k
N+I
k
N+J+I
I, we define
N
N+J
Applylng Theorem I’, we now obtain
(3.2)
For
1/2
< r < 1, the left-hand side of (3. I} may now be estlmated by Ignorlng
the second term on the left-hand side of {3.2}, and substltutlng the values of
on the rlght.
that
^
P(k}
P{k}
The calculations Involved in thls estlmatlon may be eased by noting
r
k
for mld-range values of
The result of part
(b)
k.
now follows by letting the index
N
S.M. ZEMYAN
316
Theorem 3 to follow Is a generallzatlon of the result
=!
=!
whlch was prevlously establlshed by Duren and Leung [5, pp.
Of course, the
36-43|.
Koebe functlon Is extremal for thls Inequallty, slnce Its coefflcients are preclsely
In Corollary 3.1, we obtaln an estlmate on the N th partlal
sums of thls serles, whlch Is valld for N
3.
k
I/k (k
1,2
).
THEOREM 3.
Let
f
N
+
coefflclents of
(a)
If
=
> 0
f.
and
-
S, and let
,
2
N
I
(b}
If
N
I
s
(3.3)
k1/
k=l
f(z}
Equality holds if and only if
z/(1
}z)
2
ll
e > O, then
I
kX-lkl-
s
(1+e},
k
k--I
where
1/
denotes the Rlemann Zeta Function.
PROOF.
For
(a)
N
and
in this range, the welght function defined by
N+I
k
Is convex.
If
Hence,
N >
(3.3)
+ I/{2
follows directly from Theorem I’.
I/=
I}, then there exists a unlque Integer
(N
1)
N
P(k}
(N
N, j, and
-
c
,
(N
1}
+ (N(N-I)
O,
c
J+l.
-
N
k=1
(N-1)N-),
k
N+I
k
N+J+I
Applylng Theorem 1’, we now obtaln
N
N+J
k1-lkl a
k--1
+
N
kP{k}
k=N+l
such that
we define
k
(N-1)-}k
J
c
j<
For these values of
N
k=1
^
P(k}
(b}
be the logarlthmlc
1,2
then
kl-lk I
k=l
k (k
lkl
N+J
+
k=t
k/
k
k=N+l
317
ESTIMATES OF LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNCTIONS
A
P(k) sllk
Slnce
=
estlmates of the form
are now posslble, where
is defined as above.
j
The conclusion of part
(b)
N
now follows by lettlng the Index
d
in the
inequality stated above.
In the following corollary, we consider the Duren-Leung conjecture which states
that
N
for all
[3,
N
k=l
k=l
Both Mllln and Grlnshpan [4, p. 144] as well as Andreev and Duren
3.
726]
p.
N
provlde
evidence
supportive
for
thls
although neither
conjecture,
In the event that this conjecture is false,
advantageous to have an accurate estimate of the left-hand slde.
provides
it
proof.
a
Let
COROLLARY 3.1.
coefficients of
k (k
S, and let
f
N
Then, for every
f.
1,2
N
N- 2
1)
2N(N
For
N
-
be the logarithmic
3, we have
N
PROOF.
would be
k=l
k=l
^P(k)
3, we define
as In the proof of Theorem 3(b} which,
after some slmpllflcatlon, becomes
_-k-I
A
P{k)
N
k=
l/k,
k
N+I
k
2N-I
2N-2
1)
(N
O,
After applying Theorem I’, we obtain
N
N
2N-2
k(2N
N(N
k--1
P(k)
I/k
for
(2N
kN(N
1)
N+I
k
2N-2
k
1)
1)
k=N+l
k=l
k--N+1
A
Since
2N-2
1)
k
(this is geometrically clear}, we may
continue our estimation to conclude that
N
N
2N-2
k=l
k=l
k=N+l
..2
Flnally, an integral estimate of the term on the right flnlshes the proof.
S.M. ZEMYAN
318
ACKNOWLEDGEMENT
The author wlshes to recognlze the referee for several helpful comments In his
revlew of thls artlcle.
REFERENCES
I. DE BRANGES, L. A Proof of the Bleberbach Conjecture, A__cta Math.__. 128(1985), 137152.
2. FITZGERD, C. and POMMERENKE, Ch. The de Branges Theorem on Unlvalent Functlons,
Trans. A, Math. Soc. 90(2)(1985), 683-690.
3. ANDREEV, V.V. and DUREN, P.L. Inequalitles for Logarithmic Coefficients of
Univalent Functions and their Derivatives, Indiana Univ. Math. J. 37(4)1988, 721733.
4. MILIN, I.M. and GRINSHPAN, A.Z. Logarlthmlc Coefficient
Functions, Complex Variables Theory Appl. Z(1986), 139-147.
Means of Univalent
5. DUREN, P.L. and LEUNG, Y.J. Logarithmic Coefficients of Univalent Functions, J_
d’ Analyse Math. 3_6 1979), 36-43.