On the coefficients of starlike functions of some classes

R O C ZN IK I PO LS K IE G O T O W A R Z YST W A M ATEM ATYCZNEGO Seiia I : P R A C E MATEM AT Y CZNE X V I I I (1974)

Z

W

(Kielce)

On the coefficients of starlike functions of some classes

1. Denote by 8 the family of all regular functions of the form w = г + а 322 + ...

which are univalent in the circle К = {z : \z\ < ¹ } and by 8* its subclass of starlike functions, i. e. of functions which map the circle К onto starlike regions with respect to the point w = 0. Then denote by 8* (a), ⁰ < a < ¹ , the set of all functions of the family 8 such that

m

* n * ) x

№

< a

for every ze K . Obviously, 8* (a) c 8* and /S*(l) 8*.

In this paper we give the estimation of the coefficients in 8* {a).

In the proof we employ the method of Clunie [ 1 ] and make use of the properties of the family p { a ) defined below.

2. Let p be the family of all regular functions of the form

( 1 ) P {z) = 1 + + + ...

which are defined in the circle К and which satisfy the condition геР(г) > ⁰ for every zcK -, while p (a), 0 < « < 1 denote the family of functions of form ( 1 ) and such that

P ( z ) - 1 --- -< a

P (*) + 1

for every ze K . Obviously p ( a ) czp and p ( 1) = p .

I t follows immediately from the definition of the family that the functions P (z) of form ( 1 ) belong to p (a ) for а Ф 1 if and only if

(

2

for every ze K .

P ( z ) ~ ^{1 + a 2}

1 — a 2 < ^2a 1 — a 2

8 — R oczniki PTM — P ra e e M atem atyczn e X V III.

1 1 4 Z. W i e c z o r e k

Denote by Ü the set of all regular functions of the form w(z) —

which are defined in the circle К and such that

(3)

\w(z)\ < 1 for every z e K .

L

. The function P (z) belongs to the fam ily p (a ) i f and only i f 1 + aw (z)

P (z) =

aw ( 0 ) fo r some function w(z) e Q and every z* K .

P r o o f. Let P ( z )e p ( a ) . We consider the function (4)

with

and the function ( 5 )

Q ( z )

P (z) a

___ -, K ,

a ^{1 + a 2}

1 — a 2 ’ b ^2a

1 — a 2

w(z) Q M - Q ( 0 )

l - Q ( 0)Q(z) ’ ⁰ e K .

Thus w(0) = 0. I t follows from (2) that \Q(z)\ < 1 by which \w(z)\ < 1 for ⁰ e K . Hence the function w(z) defined by (5) belongs to the family Ü.

Finding $ ( ⁰ ) from (5) and substituting it into (4) we obtain formula (3).

Conversely, if w(z)<= Q, then the function P(z) defined by formula (3) satisfies condition ( ² ) and thus P { z )e p ( a ) . I t is known [2] th at the function f(z ) of the family S belongs to the subclass 8* if and only if

= -p( 0 ) for some function P ( z )e p ( a ) .

3. We shall prove the following

T

1. I f f(z ) e 8* (a), 0 < a < 1, then ( 6 )

and (7)

la j < n a‘ i n = ² , .. Я ,

a„ < N ( N - l )

n — 1 a

^{- 1} 1 n = W + l , N + 2 , . . . ,

where N € [ ^{1 + a 1 — a ’ 1 — a 2} is a natural number.

( 8 )

(9)

Estim ation ( ⁶ ) is sharp and equality holds in (6) fo r the function / » ₍₁ — saz)2 ’ z

P ro o f. If f( z ) e S * ( a ) , then

zf'(z) 1 + aw{z) e\ = 1 .

f{z ) l — aw{z) for some function w(z)e Q.

I t follows from (9) that

*/ '(« )-/ (« ) = « (»/'(«) +/(»))'w(z), thus

CO oo

( 10 ) ^ ( f c — 1 )аАг* = aw(«) ^ ( f c + l)% z * (a, = 1 ).

A=2 *= 1

Equating the coefficients at г 2 on both sides of equality ( 1 0 ) we find a 2 = 2aw'(0). Since w '( 0 ) ^ 1 then

( ^{1 1} ) |u ² | < ² a.

Thus estimation ( ⁶ ) is true for n = ² .

Let n > ² . We write equality (10) as follows:

n OO t t — 1

( ^{1 2} ) ^ ( f c - l j a ^ d - £ dkzk = aw(z) JT* (fc + l )ak^ ,

k= 2 + l &=l

oo

where the series £ djczk converges in the circle K .

k —n + l

Since |w(æ)| < 1, we obtain from (12)

n OO *1 — 1 •

I - ! ) « * « * + £ dkZk\< a i ^ (* + !)« *« * |-

fc=2 & = tt+ l fc = l

Hence putting z = 0 < r < 1, 0 < i < 271, we have

2rr n

0 k—2 ^{k = n + l}

— f I 'S'{h — ±)akrkeltk + V dt

!ir J I Z-J !

2я и—1

< 7~T j I (* + 1 ) V V

0 *=1

Integrating we get

11 OO n —1

(13) ^ ( f c — 1 ) 2 I«*|V* + £ \dk\2r2k < a 2 J ? (h + 1)2 \ак\2г*к.

f c = l & = w + l

In particular from (13) follows

lc=l

^ (fc - 1)2Kl V* ^{< a 2} JT1 ^{{Je +} 1)2 ^\ak\ V *.

k = l

(14)

1 1 6 Z. W i e c z o r e k

Passing in (14) to the limit as r -> 1 we obtain

n M— 1

^ ( J c - 1 ) 2 l^ l2 < a2 (Tc + 1)21%|2.

A=1 A;=]

Thus

тг—1

fc=i

We observe that a 2n 2 — (n — 2 )2 > 0 if and only if

< W.

Estimations ( 6 ) and (7) are obtained by induction. The function f*(z ) defined by formula ( 8 ) belongs to the family 8* (a) and /*(n)( 0 )/w

= nan~} en_1, thus estimation ( 6 ) is sharp. By which the theorem has been proved.

C

. The values o f the function f(z )e 8* (a) include the circle

(15) \w\ < ¹

2 ( l + a) '

In fact, let f{z )e 8* (a) and f(z ) Ф c for every 0 € K . Then the function

9(*) №

1 ~/(»)/c ² + (a2 + 1 /v)z2 + • • ■

belongs to the family 8. Thus |u ² +l/c| < ² . Hence by (11) we find (15).

4. Denote by E the family of regular and univalent functions

F {z) = — + a } z + a2z2+ ... + anzn + ...

z

defined in the ring Q = {z: 0 < \z\ < 1} and by E* its subclass of starlike functions, i. e. functions which map Q onto regions whose complement to the closed plane is a starlike region with respect to the origin.

I t is known that F e E* if and only if

re [ ^{zF '(z)l F{z) J > 0 .}

One may prove that F e E* if an only if zF'(z)

F (z) P ( z ) e p .

Now we shall define the class E *{a), 0 < a < 1.

Denote by 27*(a), 0 < a < 1 the set of all functions of the family 27 such that

zF' (z) --- — + ¹

F ( z )

zF'{z) F (z )'

< a

for every z*Q .

Obviously, 27* (a) c .27*, 27*(1) = 27*.

In this part of the paper we shall find the estimations of the coefficients of the functions of the family 27*(a).

5. We shall prove the theorem.

T

2 . I f F (z)e 2 7*(a ) , then

(16) \an\ < fo r n = ¹ , ² , ...

n + 1

Estim ation (16) is sharp, equality in (16) holds i f and only i f ,

(17) F*(z) = - ( l

0 eazn+1)n+1

(18)

P ro o f. If F ( z ) e 2 7*(a), then

zF'(z) 1 - f a w(z) F (z) ¹ — aw(z) for some function w(z)e Q.

From (18) we have

F (z) -\-zF’ (z) — a[zF '(z) — F(z)]w (z) hence

o o o o

(19) ^ { J c + l ) a kzk+] = aw{z) ( - 2 + £ ( T c - l) a kzk+l]j.

k= 1 fc= 2

Equating in (19) the coefficients at ⁰ , z2, z ³ because of |cfc| < 1 (Jc — 1 , 2 , ...) , we obtain

( 20 ) Ia j < a, \a2\ < f a .

Let n > 3. Equality (19) may be written in the form

n OO n — 1

J ? ( k + l ) a k2p + l+ £ dkzk = [ - 2 a + {1c — 1 )aakzk+l\w{z).

к — 1 k = n + 2 k= 2

Since |ад>(г)| < 1, then

n oo n — ¹

I ^ ( f c + l) « * 2fc+1 + dkzk I < a I — 2 + ^ Г ( й - 1 )алг*+1|.

/с== 1 k = n+ 2 /c= 2

118

Accepting 2 = relt, 0 < г < 1 , 0 < £ < 2 т и after integrating we obtain

^ ( f c + l ) 2 |eyV(fc+1) + \dk\2r2k^ a 2 |4-f ^ ( f c - 1 )2 |afc| 2 r2(*+])] .

*=i Hence

k —n+2 k —2

■Ц, n — X

(21) J^ , (fc -fl) ² |aA| ² r2(*+1) < a2|4 + (к — l ⁾² |«y ² r2(fc+1)|.

fc=l k = 2]

Passing to the limit in ( 2 1 ) as r -> 1 we get

n n — 1

£

( * + 1 ) 2 M 2 «: ce2 [4 + J T (fc - 1 ) 2 K l 2] .

fc= 1 fc

Hence

/» — i

(» + l ) 2K l 2 < 4a2- {4 |a,|« + £ [(* + l ) 2- ( k - l ) 2a2] Ы 2}

2

Since (fc + ¹⁾² — (fc — l ) ² a ² > ⁰ for к = 2 , 3 , . . . , then (w + l ) 2|a„|2 < 4a2

or

la-1 < 2a

w + 1 for n = 3 , 4 , . . .

Hence, because of (20), we obtain estimation (16). The function (17) belongs to the family 27*(a) and if F *(z) = - — \-a*zn jr . . . , then |a*|

Z

2 a for an arbitrary but fixed n thus estimation (16) is sharp.

n -\-1

C orollary . I f F ( z ) e 2 7*(a), then

( 22 ) [ ( l - a 2)/c2 + 2 ( l + a 2)& + (1 - a 2)] • |%l2 < 4 a 2.

In fact, let F ( z ) e 2 7*(a); then equality (19) may be written in the following equivalent form

00 o o

^ ( h 1 ) akzk+1 = a — ² -+- ^ (A; l ) a fc ⁰ *+ ¹ jw ( ⁰ ).

* = i fc=i

( 2 3 )

By^ \w(z)\ < 1 and for 0 = ret4>, 0 < r < 1 , 0 < 9 ? < 2iz, we get from (23) the integral inequality

2k 00 2 n 00

f | JT\& + l K r * +V<A+1)* | % < a2 f I - 2 + ^ { k - l ) a krk+1ei(k+l)<p^d(p.

0 1 0 1 1 .

Hence after integrating and passing to the limit as r tends to 1 we obtain ( ²² ).

References

[1] J. Clunie, On meromorphic schlicht functions, J . London Math. Soc. 34 (1959) p. 215-216.

[2] Z. N eh ari, Conformal mapping, New York 1952.