A unified treatment of some special properties of convex conformal functions

J. Korean Math. Soc. 33 (1996), No. 3, pp. 587-599

A UNIFIED TREATMENT OF SOME SPECIAL PROPERTIES OF CONVEX CONFORMAL FUNCTIONS

JONG Su AN AND TAl SUNG SONG

1. Introdution

We assume throughout the paper that fez) is a conformal function in the open unit disk D = {z: Izl <I}. In the rest of paper, conformal means analytic and locally univalent. The Schwarzian derivative of a function f is defined by

Sf(z) ='Pf(z) - (1/2)'P feZ?, 'P j(z) =f"(z)/ f'(z).

A set is convex if it contains the line segment between any two of its points. It can be shown that fez) is convex in D if and only if fez) satisfies either one of the following two inequalities (see[l] p.5):

(1) 1+Rez'Pf(z) > 0, zED

(2) 1(1-lzI2)'PtCz) - 2z\ ~ 2, zED.

We define the following class C(r) in D. Let C(r) be the class of all conformal functions in D which are convex on every hyperbolic disk in D of hyperbolic radiusp,r =tanhp, for some r ~ 1 ([1,p.2]). We have already noted the following four result:

(1) Nehari's result ([7]); Let fez) be a convex function in D. Then

(2) Pommerenke's result (Theorem 2.4 in [10]); For a > 1, if f

satisfies the inequalityI(1-1 Z12)'P f( z) - 2z1 ~ 2a, for allzED, then

Received September 22, 1995.

1991 AMS Subject Classification: 30C99.

Key words: Schwarzian derivative, hyperbolic radius, convex function .

(3) Pommerenke's result (Theorem 2.5 in [10]); For a 2': 1, if f

satisfies the inequality 1(1-lzI2)ep l(z)-2Z1 ::; 2a, for allzED, then

f E G(a- Va²-1).

(4) Pommerenke's result (Corollary 2.3 in [10]); For (3 2': 0, if f

satisfies the inequality (1 - IzI2?ISI(z)1 ::; 2(3, for all zED, then

In this paper we show Theorem 2.2 and Theorem 3.1. Theorem 3.1 is an improvement of Pommerenke's result (2). Both Theorem 2.2 and Theorem 3.1 follow consequences of Theorem 2.1. We find values for a and (3 such that f satisfies

(*) 1(1-lzI2)epl(z) -:-2zl::; 2a, for some a 2': 1,z E D and

(**) (1-lzI2?ISI(z)1 ::; 2(3, for some (32': 0,zED.

Using results of P.Beesack-B.Schwarz[3] and D.Minda[5] we derive an estimation for the uniform radius of univalence of functions in G(r) satisfying the above property(*).

2. The lower bounds of functions having special property Let Q be a simply connected region in the complex plane C (C =J Q) and r E (0,1). Denote

(Jog)"(z) z+a

'Plog(z) = (J ) ( )' whereg(z) = , a ED.

og , z 1+az

IT the condition w = 1+Z'P log E Q, for every z .E Dr = {z : lzl < r}

and for all a E D,holds, then we say that fez) is Q-locally convex with radiusr, and denote byG(Q,r),the class of all such functions. We note 1 EQ. Hence by the Riemann Mapping Theorem there exists a unique analytic function hew) which maps Q onto D, such that h(l) = 0 and h'(l) > O.

Convexco~ormalfunctions LEMMA 2.1. For (z, a) E D x D,

Proof.

'P ⁰ (z) = (J⁰ gy'(z) = f"(g(z))(l -laI^{2) _} 2a(1 +az) . f 9 (J0 g)'(z) f'(g(z))(l +az)Z (1+az)2 This completes the proof. 0

LEMMA 2.2. For zED,

(h(l +z~^{f09(Z)))' +}

I

[=0 ^s :2'

z=o

Proof. By the Schw~z-PickLemma we have

For near z=O, we derive the result. 0

LEMMA 2.3.

Proof.

f"(g(z))(g'(z)) g"(z) 'Pfog{z) = j'(g(z)) + g'(z)'

589

.!!:- _

f"(g(z))g"(z) gll/(z) (gll(Z))²

+ _!'(g(z)) + - - - - -_{g'(z) g'(z)}

Since g(z) = (z +a)j(l+az), we have

d

I -

dz ('Pfog{z)) z=o - (l-Ial) f'(a) - f'(a) - 2a(1 - laI²)'P f( a)+^2(a)2

= (1-la[2)2'Pj(a) - 2a[(1 -laI 2)'Pf(a) - a].

This completes the proof. 0

THEOREM 2.1. Hf E C(Q,r) for a given Q andr E (0,1], then 1(1-lzI2)'Pf(z) - 2z¹~ [h'(l)]-lr-\

(1 -lzI2)2ISf(z)1 +cl(l -lzI2)'P fez) - 2z12 ~ [h'(1)]-lr-² where zED and c= h'(l) - tll+'Ph(l)l.

Proof. By definition of C(Q, r) and h(w),it readily follows that the composition h(l +Z'Pfog(z)) is an analytic function of Zwhich maps Dr into D and it vanishes at z =0. Hence by the Schwarz Lemma we conclude that the function h(l+Z'Pfog(z))jz is also analytic in Zand maps Dr into Dl /r . Therefore we have in particular

Ih(l+Z'Pfog(z))jzl _~ ljr.

For near z = 0, using Lemma 2.1, we have

We get

Convex conformal functions

Hence

On the other hand, using Lemma 2.1, we get

591

Expands h(l+Z'P fog(z ))/ z into a power series in Z : h(l+Z'Pfog(z))jz

= h'(l)'Pfog(O)+ {h'(l)

~'Pf09(Z)\z=0

~h"(l)'P}og(O)}

For near z = 0,

( h(l+Zz'Pf09(Z)))' ₌ h'(l) dz 'Pfog(z) z=od

I

By Lemma 2.3,

[h'(l)]-l (h(l +Z:fog{Z)))' =

~

~'Ph(l)'P}og(O)

z=o

= (1-laI2?'Pj(a) - 2a[(1 -laI²)'P f(a) - a] +12"'P h(l)'P}og(O)

1 1 _

= (1-l aI2)2['Pi(a) - 2"'P}(a)]+2"(1+'Ph(l)) [(1-laI2)'Pf(a) - 2a]2.

By Lemma 2.2, we have the result. 0

COROLLARY 2.1.1. Let n be a simply connected region in C (Ci-

n) and r E(0,1). Hf EC(n,r) then f satisfies (*) and (**), where 2a = _2a~ = [h'(l)]-J. r^{- l} and

2/3=2/3~ =^max{h'(l),(1/2)11+'Ph(1)]}[h'(1)]-2 r -2.

Proof. By Theorem2.1, f satisfies the inequality

Also ifh'(l) ~ !11+'Ph(l)l, then c~

°

for everyf EC(Q, r)and all zED. Finally, ifh'(l) < (1/2)11+'Ph(l)/, so that c< 0, then Theorem2.1 implies

(1-lzI²?ISj(z)1 ::; -cl(1-lzI 2)'Pj(z) - 2:z1²+[h'(_1)]-lr-² S; {(1/2)/1 +'Ph(l)j- h'(1)}1(1-/z1 2)'Pj(z) - 2:z1²+^[h'(_1)]-lr-² S; {(1/2)ll+'Ph(l)l- h'(1)}[h'(1)]-2r -2+[h'(_1)]-lr-²

S; (1/2)11+'Ph(1)l[h'(1)]-2 r -2 = 2j3~

for all f E C(Q,r) and zED. 0

Now we can derive coefficient inequalites for normalized functions in C(Q,r).

COROLLARY 2.1.2. H fez) = z +a2z2 +a3z3 + ... ^E C(Q,r) for given Q and r E (0,1] then

la21 S; ~[h'(l)]-lr-l, la31 ^S; ~max{h'(l), ^11- ~'Ph(1)I}h(1)-2r-2.

By applying Theorem 2.1 to the class C(Q·s,r) where

Q8 = {w: largwl < 1r8/2} for 0::; r::; min(l,8), 8> 0, we obtain generalized form.

THEOREM 2.2. Hf E C(Q8,r), then (1 - IzI²?ISj(z)1 + 21

8 1(1-lzI²)'P j(z) - 2:z1² S; 28r-2, ZED.

Hence we haveinparticular

1(1-lzI2)'PtCz)-2ZI::; (28/r) and (1-lzI2)2IstCz)l::; (28/r^2).

Proof. Let hew) = (w^{1/8 -} 1)(w^I/8 +1). Then h maps Q8 onto D with

h'(l) =1/28, 'Ph(l) =-1 and c=^h'(l)=1/28.

Thus Theorem2.1 readily yields Theorem 2.2 as well. 0

Notice that in the case r =⁸=1, Theorem 2.2 implies an improve- ment .of Nehari's result.

Convex conformal functions 593 COROLLARY 2.2.1. Let fez) be^a convex functionin D. For zED, tben

Minda[5] establishes the following geometric interpretation of The- orem 2.2 in the case 6 = ^1. Let r(z, f) be the hyperbolic radius of the largest disk in D centered at z in which f is univalent. Define r(f) = inf{r(z,f) : ZED}. The function f is called uniformly locally univalent in D provided that r(f) >O. IT

(1-lzI2?ISj(z)1 s: 2(1+^k2),

then r(f) 2:: ^7l"/2k for all k 2:: O.

COROLLARY 2.2.2. H f E C(n,r), tben fez) is uniformly locally univalent, tbat is, f( z) is univalent in every byperbolic disk in D, of byperbolic radius

r(f) 2::7l"/2k =^7l"r_/2~=^(7l"/2)sinbp(f)

wbere p(f) = (1/2)log[(1+r)/(1-r)] is tbe unifonn byperbolic radius ofconvexity of fez).

Proof By Theorem 3 in [5], fez) is univalent in every hyperbolic disk in D of the hyperbolic radius rU) 2:: ^7l"/2k, provided that (1 -

IzI2)2ISj(z)1 s: 2(1 +P). On the other hand by Theorem 2.2, f E C(n,r) then (1 - IzI2)ISj(zW s: 2/r2. Hence k = J(1/r 2)-1, and the result follows. 0

REMARK. From Theorem 4 in [5], f satisfies the inequality

By Pommerenke's result(4), we have

and

3. An estimate of the class of special functions

In this section we use Theorem2.1 and find an estimate for f3, such that if f satisfies (*) then f satisfies (**). First we show that every function f which satisfies(*) is Si~-locallyconvex with radius 1',for all

l'E (0,1), where

01 {

I

Sir = w: w - 1 _ 1'2 < 1 _ 1'2 .

LEMMA 3.1. Hf satisfies(*), thenf EC(Si~,^1')for everyl'E (0,1).

Proof. Note that if f satisfies (*) then fog satisfies the inequality 1(1 - Izl2)cpfog(z) - 2z1 :s ^2a for every Mobius automorphism g(a) =

(aa+z)j(l+aZ-a) of D. Since f satisfies (*) it can be written in the form

I

Hence, fog satisfies (*) which is equivalent to

I

1+ Izl²¹ 2alzl

1+zcp fog(z) - 1 -lzl2 :s ¹-lzI2·

This shows that w = 1+^{zcp fog(z) E} n~l. Note n~1 c n~ whenever Izl <^l'for every ^l'

:s

:s

THEOREM 3.1. Hf satisfies (*), then

(1 -lzI 2)2ISf(z)1 +(1/2)1(1 -lzI 2)cpj{z) - 2z12

:s

where 1:s^(3(a)

:s

and

Convex cQnformal functions 595

for 1 ::;a ::;

JI

Proof. By Lemma 3.1 we can apply Theorem 2.1 to the case that

n= n~ for a fixed a 2:: 1 and a variable r E (0,1). Thus hew) = a(l- r²)(w-1) .n^{a ----+}D

r[(I-r²)w+(2a2 - r^{2 -} 1)]' ^{r .} So we have

, a(I - r²⁾ 1 - r²

h (1) = 2_{r a - r}(2 2)' iph(I) = - _{a - r}^{2 2}

and 1 I-ar

c= h'(I) - -211+ iph(I)1 = ( ) = c(r).

2r a-r By Theorem 2.1 the following inequality holds:

where

(I/2)[h'(I)]-lr-² = (a² _ r²)/ar(I _ r²).

Now put r = a - .../a^2-1. Then

(1 -lzI2?ISf(Z)1+ (I/2)I(I-lzI2ipf(z) - 2z1² ::; 2a²+2(a +(I/2)a

)J

Next, note that c(r) > 0 only for rE (0,I/a) and therefore

{

2 [a~;1-=-":2)]' ^{for 0 <r::; I/a,}

(1-l^zI²?lstC^z)1 ::; ^{[a2_r2 ] 2}

2 ar(1-r2) -4a c(r), for r 2:: l/a.

Observe now that the minimum value of 2 [a~;1-=-":2)] in (0,1) is ob- tained at

ro = J(1/2)[(3a² - 1) - J(a^{2 -} 1)(9a² -1)],

and let f3(a) = 2 [OIr~~;~!~)]' ^If1~ ^a ~ ^JI+v'2 ^{then ro} _{~ ~} ^{and if}

a ~

J

-

r

r)

-.,...----2a c(r) = - - + - -

ar(I - r2 ) a a - I 1 - r 2

which obviously is an increasing function ofr in [I/a,1),and therefore we get in the interval

a² _r² 2 a 2 _(I/a)2 2

ar(I-r2) - 2a c(r) ~ ^[1-(I/a)2] - 2a c(I/a)

a 2 - (I/a? 2

= [1-(I/a)2] ~ ¹+a . This completes the proof 0

Using the argument of the proof of Corollary 2.2.2 we obtain the following result.

COROLLARY 3.1.1. Let f satisfies (*). Then fez) is univalent in every hyperbolic diskin D with the hyperbolic radius

rU) ~ 7r/2Jf3(a) ^-1~ ^7r/2a.

Next we derive some coefficient inequalites for normalized functions.

COROLLARY 3.1.2. Let fez) = z +a2z2 + ... and f satisnes (*).

Then

la3 - a~1+^I/3Ia2/2 _~ ^(I/3)[a2+^(a+^I/2a)Ja2-1],

la3 - a~1 ~ f3(a)/3 _~(1+ (^2)/3,

1 ^{2 2}

I Ia3 ~ 3"max 1 - r[( ²)a,⁽1+2a - 3r )r a2r(I _^{2 2 ] a - r}r2)2' for every r E (0,1).

Proof. The inequalities are localized versions of Theorem 3.1 at near z = O. If we substitute the values ofh'(I) and <Ph(I), in the proof of Theorem 3.1 into Corollary 2.1.2 we obtain the result. 0

Convex conformal functions 597

4. The equivalent properties of convex functions ITf( z) satisfies the improved convexity condition

1+Rezep(z) >u, zED,

then we say that fez) is a convex function of order u, for some u E (0,1)(see[4]). For^Cl:> 0,the technique ofthe proof of Theorem 2.1 may be applied here to derive sharp bounds for 1(1 - IzI2)eptCz) - 2z1 and for (1-lzI2)2ISJCz)/.

THEOREM 4.1. The following statements areequivalent:

(1) f(z) is a convex function of order u in D.

(2) 1(1-lzI2)epf(Z) - 2(1 - u)zl :s; 2(1 - u).

(3) 1(1-lz/2)epf(z) - 2z1 :s; 2[1-u(l-/zl)].

(4) 1(1-lzI2)epf(Z) - 2z1 :s; 2[1 - u(l -lz/2)P/2.

(5) (1-lzI2)2ISf(Z)I+~ 1(1-lzI2)epf(Z) - 2z12 :s; 2[1-u(1-lzI 2)].

Proof We show that (1) = ? (2). Since fez) is a convex function of order u in D, fez) satisfies the improved convexity condition 1 + Rezepf( z) > u, zED. The above inequality tells us that the analytic function w = 1 + zeptCz) maps the unit disk D into the half plane {w : Rew > u}, andg( w) =^{(w -} 1)j (w+ 1 - 2u) maps the half plane onto D. The composition of two functions

zepf(z) g(l+zepfez»~ ^{= 2(1 _ u)}+zepf(z)

satisfies the requirement of the Schwarz Lemma, and therefore g(l +zep_fez»~ epf(z)

Z = 2(1-u)+z'Pf(z)

map D into itself. Hence we have Ig(l + zepf(z»jzl :s; 1 for all zED.

By simple computation we see:

leptCz)12:s; 4(1-u? +4(1-u)Rez'Pf(z)+ IZepf(zW, (1 - Iz1 2)lepfez)1² :s;4(1 - u)2 + 4(1 - u)Rezepf(z),

2 2 4(1-u) 4(1-u) 2

(l-lzl H/epf(z)1 - 1-lz12 Rezepf(Z)+ (1_lzI 2)2Iz1 } :s; 4(1-u?(1+IzI²/1-lzI²)

which is inequality (2). (2)=}(3) and (3)=}(4) are trivial. We show that (4)=}(5). If we square (4) we have 1(1 - IzI 2)cpAz) - 2z1² _~ 4[1 - 0'(1 - IzI^2)] then simply we obtain

o_~ (1-lzI 2)lcpj(z)12_~4(1-0'+^RezcpAz».

Hence 1+^{Rezcpj{z) > 0'.}The above inequality tells us that the analytic function w = 1 +zcp j( z) maps the unit disk D into the half plane {w : Rew > O'}, and g( w) =(w - 1)j (w +1 - 2IT) maps the half plane onto D. The composition of two functions

zcpAz) g(1+^zCPj(z» = 2(1-IT)+^zCPj(z)

satisfies the requirements of the Schwarz-Pick Lemma. Hence we have

(1-lzI^{2 ) (} 2(I-IT)+zcpf(Z)^{cpf(z) )'} <-

1-1

This inequality yields

(1 -lzI 2){2(1 - IT)lcpf(z) - (lj2)cp}(z)l- ITlcpAz)12}

~4(1- IT)[1-IT+Rezcpf(z)] - (1-lzI 2)lcpAz)12 and simplifying above inequality we get

(1-1zI²⁾ {lcpf(Z) -

~<p}(z)1

~lcpj(z)12

~ 2(1 - IT).

Finally, adding (2IzI²)j(1 - Iz1²⁾ to both side, then the inequality (5) follows at once. We show that (5)=}(1). Suppose the below inequality is holds:

Convex conformal functions 599

Then we have 1(1-lzI 2)<pfez)-2Z1 ::; 2[1-a(1-lzI 2)]l/2.Square above inequality and simplify. Then we obtain

Hence fez) is a convex function of order a in D. 0

From the inequalities (2) and (5), we obtain the following conse- quence.

COROLLARY 4.1.1. Let fez) = z +a2z2+a3z3 + ... be a convex function of order a in D, for some a E (0,1). Then

la31 ::; (1/3)(1-17)(3 - 217).

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3.P. Beesack and B. Schwarz, On the zeros of solutions of second-order linear differential equations, Can.J. Math. 8 (1956), 504-515.

4. A.W. Goodman, Univalent Functions, Mariner Publishing Company, Inc, 1983.

5. D.Minda, The Schwarzian derivative and univalence criteria, Contemp. Math.

38 (1985), 43-52.

6. Z. Nehari, The Schwarzian derivative and schlicht function, Bull. Amer. Math.

Soc. 55 (1949), 545-551.

7. , A property of convex conformal maps, J. Analyse Math. 30 (1976), 390-393.

8. B. Osgood, Univalence criteria in multiply connected domains, Trans. Amer.

Math. Soc. 260 (1980),459-473.

9. , Some properties off"/!' and the Poincare metric, Indiana Univ.

Math. J. 31 (1982), 449-461.

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Ann. 155 (1964), 108-154.

Department of Mathematics Education Pusan National University

Pusan 609-735, Korea