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PROCEEDINGS of the
AMERICAN MATHEMATICAL SOCIETY
Volume 118, Number 1, May 1993
UNIFORMLY CONVEX FUNCTIONS AND
A CORRESPONDING CLASS OF STARLIKE FUNCTIONS
FRODE R0NNING
(Communicated by Clifford J. Earle, Jr.)
Abstract.
property that zf'(z)/'f(z)
closely related to a class of functions called uniformly convex and recently in-
troduced by Goodman. We give some particular examples of functions having
the required properties, and we give upper bounds on the coefficients and the
modulus \f(z)\ of the functions in the class.
We investigate starlike functions f(z) = z + Yfk=2akzk wim the
lies inside a certain parabola. These functions are
1. Introduction
We shall be concerned with functions of the form
oo
(1.1) f(z) = z + Yjakzk,
k=2
analytic and univalent in the unit disk U. Goodman [4, 5] introduced the con-
cepts of uniform convexity and uniform starlikeness for such functions, thereby
providing proper subclasses of the usual classes of convex and starlike functions,
here denoted by A^n and So > respectively. The corresponding "uniform classes"
are defined in the following way, by their geometrical mapping properties.
Definition 1. A function / is said to be uniformly convex (starlike) in U if /
is in Ko (So) and has the property that for every circular arc y contained in
U, with center £, also in U, the arc f(y) is convex (starlike w.r.t. /(£))■
Following Goodman's notation, we denote these classes by UCV (UST).
That this definition really gives proper subclasses of Kq and So is established
through numerous results in [4, 5]. An analytic description of UCV and UST
can be found in [5]. We state this as
Theorem A. Let f(z) = z + ££i2 akzk . Then
(a) f£UCV if and only if
(1.2) Re{l + (z-O^|}>0, (z,C)£UxU.
Received by the editors May 13, 1991 and, in revised form, September 11, 1991.
1991 Mathematics Subject Classification. Primary 30C45.
©1993 American Mathematical Society
0002-9939/93 $1.00+ $.25 per page
189
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190 FRODE R0NNING
(b) / £ UST if and only if
(L3) Re(z-Ofi>(z)-°> (z,QeUxU.
Note that by taking £ = 0 in (1.2) and (1.3) we are back to the classes Ko
and So- The classic Alexander's theorem stating that /£rT0o
provides a bridge between these two classes. One might hope that there would
be a similar bridge between UCV and UST, but two examples in [4] show
that this is not the case. However, there is still a hope that there might be a
"one way bridge" in the sense that / e UCV implies zfi'(z) £ UST. In this
paper we proceed in another way, by building the bridge and exploring where
it leads, i.e., we define a class of starlike functions Sp in the following way.
zfi'(z) £ So
Definition 2. Sp := {Fe50| F(z) = zf'(z), fi £ UCV} .
In what follows, we shall investigate the class Sp , and the results that we get
can be directly translated to results about UCV . The question of the "one way
bridge", which could be expressed as Sp c UST, is still an open problem. In the
present paper we give a simple example (Theorem 2) supporting the conjecture
SPC UST.
2. A ONE VARIABLE CHARACTERIZATION OF UCV
We start by stating the theorem that is the basis for our further investigations
of Sp and UCV.
Theorem 1. Let g(z) = z + YJk=i akzk ■ Then g £ UCV if and only if
p.,, Re{1 + ££2£)}> W,
I g'(z) J g'{z)
Proof. Let g 6 UCV. Then from (1.2), we have equivalently
„tt
(2.2) Re/1 + ££3£)UReC|^)
I g'{z) J g'{z)
for every z and £ in U. Choosing £ = e'az in a suitable way we will get
ReCg"(z)/g'(z) = \zg"(z)/g'(z)\. Hence, a necessary condition for g £ UCV
is that
I g'(z) J g'(z)
We show that (2.3) is also a sufficient condition. Let £ £ U be an arbitrary,
but from now on, fixed point in U . Re{ 1 + (z - Qg"(z)/g'(z)}
function, so by the minimum principle it is enough to show (2.2) for \z\ - R >
|£|, R < 1. Assume (2.3) holds. Then for 1 > \z\ = R > |£|, we have
is a harmonic
Re/l + Z*"(z)l> Z8"{Z) > Cg"{z) >ReCg"{z)
which is (2.2). Hence (2.2) and (2.3) are equivalent, and since 1 +zg"(z)/g'(z)
is analytic in U and maps 0 to 1, the Open Mapping Theorem implies that
equality in (2.3) is not possible. □
Taking f(z) = zg'(z), we immediately get
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UNIFORMLY CONVEX FUNCTIONS AND A CLASS OF STARLIKE FUNCTIONS 191
Corollary 1. A function f(z), with f(0) = fi'(0) - 1 - 0, is in Sp if and only if
(24)
(IA)
Zf{z) - 1 < ReZ/,(Z)
/(z)
z£U
< Ke /(z) , Z£U.
Interpreting (2.4) geometrically, we see that Sp is the class of functions for
which the domain of values of zf'(z)/f(z),
(ImiTj)2 < 2 Re w - 1. In particular, we get for f £ Sp that
z e U, is the parabolic region
(2 5)
(2.5)
Re Z/'(Z) > l
Re-^->-
and
/-. ^
(2.6)
zfi'(z) n
arg7(zr<4-
Both conditions (2.5) and (2.6) define well-known subclasses of So. The in-
equality (2.5) defines the class of functions starlike of order \ , here denoted
by 5i/2 • For the functions satisfying (2.6), we refer to the papers [2, 3], which
contain several results about these functions called functions strongly starlike
of order \ . We denote this class by SSX/2.
Both these and other interesting subclasses of So are characterized by the do-
main of values Q of the functional zf'(z)/fi(z).
analytic and univalent in U, which maps U onto Cl, is helpful for investigat-
ing the class of functions in question. In the case Sp , Cl is the parabolic region
\w - 1| < Rew , and the appropriate mapping P : U —> Cl with the normaliza-
tion P(0) = 1 is seen to be
To know a function P(z),
(2.7) ^. + i(.o8Li^)2.
(The branch of yfz is chosen such that Im yfz > 0.)
Using the expansion log( 1 + yfz)/( I - yfz) = 2yfz YJk=o zk/(2k + 1), we get
the power series expansion for P(z),
P(z) = 1 + ^2 2^ I 2^ 4w _ 4m2 + 4mk -i-2k
k=\ \m=l
, 8 ( 2 2 23 3 44 4
= 1 + ttHZ+3Z +45Z +T05Z +-j-
J Z
/
^
3. Some special members of Sp
Theorem 2. f(z) = z + a„z" is in Sp if and only if \a„\ < l/(2n - 1).
Proof. It suffices to study (2.4) for \z\ = 1. Set \an\ = r and a„z"~x = re'9.
Then (2.4) for this / will be
(3.1)
v
(77-1)7^
1 + re'9 ~ 1 + rel<?
<Rel+^
By computing we see that
1 + 777-^'*' _ I + nr2 + (n + l)reoscp
1 + re'f ' |1 + re'i'\2 ~'
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192 FRODE R0NNING
so (3.1) is equivalent to
._ -.
(3.2)
v '
The right-hand side of (3.2) is seen to have a minimum for cp = n, and this
minimal value is 1 - nr. Hence, a necessary and sufficient condition for (3.2)
is (n - l)r < 1 - nr or \a„\ = r < l/(2n - 1). D
, ,, ^. I+ nr2+ (n+l)rcos<p
(77- 1)7* < -7—^-~-
K ' -
I + nr2 + (n + l)r cos cp
= —-x—i--
(I + r2+ 2r cos cp)x I2
,.,, .
ll + re^l
Goodman [5] has proved that \an\ < \f2/2n is sufficient for f(z) = z +
a„z" to be in UST. Note that Theorem 2 is more restrictive than this, so the
inclusion Sp c US T is possible.
Corollary 2. g(z) = z + bnz" is in UCV if and only if \b„\ < l/n(2n - 1).
For the case n — 2 this has also been proved in [4, Lemma 2].
A natural question would be to ask for how large z the Koebe function
A;(z) = z/(l-z)2 will have the properties of Sp . This question has indirectly
been answered by Goodman [4, Theorem 3] by looking at the corresponding
function z/(l - z) = J0z/c(£)/£ti£ in UCV.
completeness.
We state the result here for
Theorem 3. k(z) = z/(l - Az)2 is in Sp if and only if \A\ < j .
It is easily seen, and well known, that the range of values of the functional
zf'(z)/f(z) for f £ So is determined by the Koebe function alone. Hence, we
immediately get what we could call the Sp radius in So .
Corollary 3. If f £S0, then \fi(rz) £ Sp if and only if r<\.
Furthermore, from the properties of the Koebe function, it is clear that if fi is
restricted to So, then F(z) := f(rz)/r is in Sp if and only if Rezf'(z)/f(z)
j . Hence the Sp radius in So is equal to the S1/2 radius. If we turn to the
whole class S of normalized univalent functions then the Sp radius and the
S1/2 radius will not coincide, but as the next theorem shows, the difference
between the two radii is very small.
>
Theorem 4. If fi £ S, then the function \fi(rz)
0.33217... and in Sx/2 if and only if r<\.
Proof. For fi £ S we have the following (sharp) inequality (see, e.g., [6,
p. 168]):
. zfi'(z) ^. l + \z\
108 7(i) -logT^'
is in Sp if and only if r <
zeU-
Using this, the boundary of the range of zf'(z)/ f(z), f £ S ,\z\ = r, can be
parameterized
w) = {r^Y (cos(sineio*TZf)
+/sin ^in01og|^)} , 6 £ [0, 2*}.
We now search for the largest r for which
|J*;(0)-l|<Re»;(0), 0G[O,2jt).
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UNIFORMLY CONVEX FUNCTIONS AND A CLASS OF STARLIKE FUNCTIONS 193
Wr(6) is a smooth, simple, closed curve, symmetric about the real axis. A
straightforward calculation shows that
tan(arg{^^(0)})=-tan(g + s.ngl^(i+r)/(i_r)),
hence
(3.3) arg|^^r(0)} = 0-rsin01og|^ + |
and
^(arg{^(0)}) = l+cos01ogl^>l-logi±^>O
for r < (e - l)/(e + 1) « 0.462... , which shows that Wr(0) is even convex
for those values of r (< I) that are of interest here. If we denote by Clr the
domain enclosed by Wr(0) then 1 £ Clr for all r £ [0, 1}, and if rx < r2 then
Qr, c Qr2. These properties ensure that the value of r that we are looking for
is the smallest (positive) value of r that is a solution of the equation system
(\ COS0 / \
/ \
/ /
-f((-£)
/
\ 2cos0
^(^og|^)
/ \\
+ 1)
(3.5) - tan U + sin 0 log j-±£ J = (|^) sin f sin 0 log j±l) .
Here (3.4) states that Wr(6) intersects the parabola, and (3.5) states that Wr(6)
and the parabola have parallel tangents. We see that one solution of the system
is (r ,6) = (5,7c), which geometrically means that WT(Q) is tangent to the
parabola "from the outside". To find nontrivial solutions we have used the
computer system MAPLE working with 15 digits precision. This provided the
solution r = 0.3321758... and 0 = n ± 0.6240519... . From the geometrical
properties of the two curves we conclude that this value of r is the one that we
want.
From (3.3) we see that Wr(6) has a vertical tangent for 0 = 0, and since
Wr(6) is convex, it is clear that
so the S1/2 radius in S is 5 , i.e., slightly larger than the Sp radius. □
4. General properties of functions in Sp
The question of bounds for the coefficients is a classical one in the theory
of univalent functions. For functions in SX/2 it has long been known [8] that
|a„| < 1 (sharp bound), and since Sp C SX/2, the same bound must hold in
Sp, but not necessarily as a sharp one. As we have mentioned before, S„ C
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194 FRODE R0NNING
S1/2 n SS1/2 (see (2.5) and (2.6)), so the coefficient bounds for SSX/2 must
also hold in Sp . For SSX/2 we find in [3] the bounds |«21 < 1 and |«31 < |
(both sharp). A bound for \a„\, n > 3, in SSX/2 is (as far as we know) not
known. In the class Sp , however, we are able to get bounds on \a„\, but they
are not sharp, except in the case 77 = 2. However, they are lower than the
corresponding bounds in S(/2 and SSX/2, so they represent nontrivial results.
Theorem 5. Let f(z) = z + J^f^a^z11, and let An - maxyeSp \an\. Then
,42 = 4--(« 0.81),
7TZ
For the proof of this theorem we need the following result by Rogosinski [7].
Rogosinski's theorem. Let h(z) = 1 + YJk=\ckzk oe subordinate to H(z) =
1 + JffL\ CkZk in U. If H(z) is univalent in U and H(U) is convex, then
\Cn\<\CX\.
Proof of Theorem 5. Let f(z) — z + YJk=iakzk e Sp > and define cp(z) =
zf'(z)/fi(z) = 1 + Y.h=\ ckzk ■ Then cp -< P ( -< denotes subordination), where
P is the function in (2.7). P(z) is univalent in U and P(U), the parabola
\w - l\ < Rew , is a convex region, so Rogosinski's theorem applies. P(z) =
l + (8/772)z-|— , so we have \c„\ < 8/n2 := c. Now, writing zf'(z) = cp(z)f(z)
and comparing coefficients of z" on both sides, we get
n-\
(n- l)an = ^cn_kak.
k=\
From this we get \a2\ — \cx\ < 8/n2. If we choose / to be that function for
which zfi'(z)/f(z) = P(z), then / is a function in Sp with a2 = $/n2, which
shows that this result is sharp. Further we get
|«3l < 2IC2 + cia2| < £(M + |ci||«2l)< 2-c(l+c)«0.73.
We now proceed by induction. Assume we have
lflfcl^F=T(1 + c)(1 + f)'"(1 + F^2) ' for/c = 3'4, •••'"" L
Then
n-\ n-\
(n- l)\a»\ <^|c„_/fc||flifc|
k=\
<c^]|aA|
k=l
<c(l+c+5(l + c) + |(l+c)(l + |)
+... + _£_(1 + c)(l + £)...(1 + _^_))
= c(l+c>(l + £)...(1 + _^_)
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UNIFORMLY CONVEX FUNCTIONS AND A CLASS OF STARLIKE FUNCTIONS 195
and
w-SiTTnO+i-h)- °
fc=3 v '
For the class SS[/2 there is a fixed number M such that |/(z)/z|
z £ U. In [2] it is proved that M = ^exp(-r'(±)/T(±)
T(z) is the Gamma function and y is Euler's constant. Using the idea from the
proof of this result in [2] we are able to get a corresponding result for f £ Sp
with a value of the upper bound that is considerably smaller.
Theorem 6. If f £ Sp , then
< M,
- y) « 9.62, where
(4.1) |/(z)|<|z|exp^C(3)) = |z|^
for |z| < 1. There is a function f(z)
5.502.
in Sp for which supz€f/ \f(z)\ - K «
(£(7) is the Riemann Zeta function.)
Proof. Let cp(z) = zf'(z)/f(z). Then
«m+*(*£$)'■
Moreover,
z Jo C
and therefore, if z = re'e ,
2 f1 1 / l + v'A2 ^ 2 a
<^\ - lOg-^=
- 7T2 70 t V 1 - -ft)
Substituting u = yft and v = log(l + u)/(l - u) in J*', , we get
(77=^-^.
n2
, f00 «V j 1 f°° v2e~vl2 , 1 _
= 4 / -^-r dv = - --dv
Jo e2v - 1 2 70 1 - e~v
Using a formula for the generalized Zeta function in [9, pp. 265-266] we get
= -J^.
2 2
oo . oc .
J^2 = T(3)y-rj^ = 16^7^-r^
(r(3) = 2). The series £T=o {Kln + 03 is equal to |£(3) [1, p. 807], so we
get Jrx = \jr2 = 7£(3), and (4.1) follows.
The equation
^) = 1 + 4(logi±^)2
fi(z) *2 V 1 - \fz)
defines a function fi £ Sp for which supze(7 |/(z)| = \z\ K. D
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196 FRODE R0NNING
It is clear how both Theorems 5 and 6 give results about the class UC V by
the correspondence g £ UCV ■& zg'(z) £ Sp . Note that Theorem 5 improves
results about bounds for the coefficients in UCV proved in [4].
References
1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York,
1970.
2. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London
Math. Soc.(2) 1 (1969), 431-443.
3. D. A. Brannan, J. Clunie, and W. E. Kirwan, Coefficient estimates for a class of starlike
functions, Canad. J. Math. 22 (1970), 476-485.
4. A. W. Goodman, On uniformly convex functions, preprint.
5. _, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364-370.
6. Chr. Pommerenke, Univalent functions, Vandenhoeck & Ruprect, Gottingen, 1975.
7. W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48
(1943), 48-82.
8. A. Schild, On a class of univalent, star shaped mappings, Proc. Amer. Math. Soc. 9 (1958),
751-757.
9. E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ.
Press, Cambridge, 1927.
Trondheim
E-mail address: frode.ronning@avh.unit.no
College of Education, Rotvoll Alle, N-7050 Charlottenlund, Norway
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