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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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