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Partial Sums of Starlike and Convex Functions
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
209, 221]227 Ž1997.
AY975361
Partial Sums of Starlike and Convex Functions H. Silverman* Department of Mathematics, Uni¨ ersity of Charleston, Charleston, South Carolina 29424 Submitted by William F. Ames Received May 23, 1996
Let f nŽ z . s z q Ý nks2 a k z k be the sequence of partial sums of a function f Ž z . s z q Ý`ks 2 a k z k that is analytic in < z < - 1 and either starlike of order a or convex of order a , 0 F a - 1. When the coefficients a k 4 are ‘‘small,’’ we deterX mine lower bounds on Re f Ž z .rf nŽ z 4, Re f nŽ z .rf Ž z .4, Re f 9Ž z .rf nŽ z .4, and X Re f nŽ z .rf 9Ž z .4. In all cases, the results are sharp for each n. Q 1997 Academic Press
1. INTRODUCTION Let S denote the class of functions of the form f Ž z. s z q
`
Ý ak z k
Ž 1.
ks2
that are analytic and univalent in the unit disk D s z: < z < - 14 . A function f in S is said to be starlike of order a , 0 F a - 1, denoted S*Ž a ., if Re zf 9rf 4 ) a , z g D, and is said to be con¨ ex of order a , denoted K Ž a ., if Re 1 q zf 0rf 94 ) a , z g D. Let T *Ž a . and C Ž a . be the subfamilies of S*Ž a . and K Ž a ., respectively, whose functions are of the form f Ž z. s z y
`
Ý ak z k ,
a k G 0.
Ž 2.
ks2
* This work was completed while the author was on sabbatical leave from the University of Charleston as a Visiting Scholar at the University of California, San Diego. 221 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
222
H. SILVERMAN
A sufficient condition for a function of the form Ž1. to be in S*Ž a . is that `
Ý Ž k y a . < ak < F 1 y a
Ž 3.
ks2
and to be in K Ž a . is that `
Ý k Ž k y a . < ak < F 1 y a .
Ž 4.
ks2
For functions of the form Ž2., these sufficient conditions are also necessary. See w3x. In this note, we will examine the ratio of a function of the form Ž1. to its sequence of partial sums f nŽ z . s z q Ý nks2 a k z k when the coefficients of f are sufficiently small to satisfy either condition Ž3. or Ž4.. We will determine sharp lower bounds for Re f Ž z .rf n Ž z .4 , Re f n Ž z .rf Ž z .4 , Re f 9Ž z .rf nX Ž z .4 , and Re f nX Ž z .rf 9Ž z .4 . Sheil-Small w2x showed that inf Re f Ž z .rf nŽ z .4 for f g K Ž0. occurs when n s 1. In w1x, a sharp lower bound was found for Re f Ž z .rf 1Ž z .4 when f g K Ž a .. E. M. Silvia w4x investigated lower bounds on Re f Ž z .rf nŽ z .4 for f g T *Ž a . and f g C Ž a .. She showed that Re f Ž z .rf nŽ z .4 G 1rŽ2 y a . for f g T *Ž a . and that Re f Ž z .rf nŽ z .4 G Ž3 y a .r Ž4 y 2 a . for f g C Ž a .. These results are sharp when n s 1. Generally, lower bounds on ratios like Re f Ž z .rf nŽ z .4 or Re f nŽ z .rf Ž z .4 have been found to be sharp only when n s 1. In this paper, we determine sharpness for all values of n. The lower bounds in question are strictly increasing functions of n. In the sequel, we will make frequent use of the well-known result that ReŽ1 q w Ž z ..rŽ1 y w Ž z ..4 ) 0, z g D, if and only if w Ž z . s Ý`ks 1 c k z k satisfies the inequality < w Ž z .< F < z <. Unless otherwise stated, we will assume that f is of the form Ž1. and that its sequence of partial sums is denoted by f nŽ z . s z q Ý nks2 a k z k .
2. MAIN RESULTS THEOREM 1. If f of the form Ž1. satisfies condition Ž3., then ReŽ f Ž z .rf nŽ z .. G nrŽ n q 1 y a ., z g D. The result is sharp for e¨ ery n, with extremal function f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1.
223
STARLIKE AND CONVEX FUNCTIONS
Proof. We may write nq1ya
f Ž z.
1ya
fn Ž z .
n
y
nq1ya `
n
Ý ak z ky 1 q Ž Ž n q 1 y a . r Ž 1 y a . . Ý
1q
ks2
s
a k z ky1
ksnq1 n
1q
Ý ak z ky 1 ks2
[
1 q AŽ z .
.
1 q BŽ z.
Set Ž1 q AŽ z ..rŽ1 q B Ž z .. s Ž1 q w Ž z ..rŽ1 y w Ž z .., so that w Ž z . s Ž AŽ z . y B Ž z ..rŽ2 q AŽ z . q B Ž z ... Then `
Ž Ž n q 1 y a . r Ž 1 y a . . Ý ak z ky 1 ksnq1
wŽ z. s
`
n
Ý ak z ky 1 q Ž Ž n q 1 y a . r Ž 1 y a . . Ý
2q2
ks2
a k z ky1
ksnq1
and `
Ž Ž n q 1 y a . r Ž 1 y a . . Ý < ak < ksnq1
wŽ z. F
n
2y2
< ak < y Ž Ž n q 1 y a . r Ž 1 y a . .
Ý ks2
`
Ý
. < ak <
ksnq1
Now < w Ž z .< F 1 if and only if Ž2Ž n q 1 y a .rŽ1 y a ..Ý`ks nq1 < a k < F 2 y 2Ý nks 2 < a k <, which is equivalent to nq1ya
n
`
ks2
ksnq1
Ý < ak < q Ý
1ya
< a k < F 1.
Ž 5.
It suffices to show that the LHS of Ž5. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, which is equivalent to n
Ý ks2
ž
ky1 1ya
/
< ak < q
`
Ý ksnq1
ž
kyny1 1ya
/
< a k < G 0.
224
H. SILVERMAN
To see that f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1 gives the sharp result, we observe for z s rep i r n that f Ž z.
s1q
fn Ž z .
ž
1ya nq1ya
n
s
/
zn ª 1 y
1ya nq1ya
when r ª 1y.
nq1ya
THEOREM 2. If f of the form Ž1. satisfies condition Ž4., then ReŽ f Ž z .rf nŽ z .. G nŽ n q 2 y a .rŽ n q 1.Ž n q 1 y a ., z g D. The result is sharp for e¨ ery n, with extremal function f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. We write
Ž n q 1. Ž n q 1 y a .
f Ž z.
1ya
fn Ž z .
y
nŽ n q 2 y a .
Ž n q 1. Ž n q 1 y a .
n
1 q Ý a k z ky 1 qŽ Ž n q1 . Ž n q1 ya . r Ž 1 y a . . ks2
s
`
a k z ky1
Ý ksnq1
n
1q
Ý ak z
ky 1
ks2
[
1 q wŽ z. 1 y wŽ z.
,
where `
Ž Ž n q 1 . Ž n q 1 y a . r Ž 1 y a . . Ý ak z ky 1 ksnq1
wŽ z. s 2 q2
.
`
n
Ý ak z ky 1 qŽ Ž n q1. Ž n q1 ya . r Ž 1 ya . . Ý ks2
a k z ky1
ksnq1
Now `
Ž Ž n q 1. Ž n q 1 y a . r Ž 1 y a . . Ý < ak < ksnq1
wŽ z. F
n
2 y2
Ý ks2
< a k < y Ž Ž n q1 . Ž n q1 ya . r Ž 1 ya . .
`
Ý ksnq1
< ak <
F1
225
STARLIKE AND CONVEX FUNCTIONS
if n
Ý < ak < q
Ž n q 1. Ž n q 1 y a . 1ya
ks2
`
Ý
< a k < F 1.
Ž 6.
ksnq1
The LHS of Ž6. is bounded above by Ý`ks 2 Ž k Ž k y a .rŽ1 y a ..< a k < if 1 1ya
n
½Ý
k Ž k y a . y Ž 1 y a . < ak <
ks2
q
`
5
k Ž k y a . y Ž n q 1 . Ž n q 1 y a . < a k < G 0,
Ý ksnq1
and the proof is complete. Remark. Our results in Theorems 1 and 2 agree with those in w4x for the special case n s 1 and are an improvement when n ) 1. We next determine bounds for f nŽ z .rf Ž z .. THEOREM 3. Ža. If f of the form Ž1. satisfies condition Ž3., then ReŽ f nŽ z .rf Ž z .. G Ž n q 1 y a .rŽ n q 2 y 2 a ., z g D. Žb. If f satisfies condition Ž4., then ReŽ f nŽ z .rf Ž z .. G Ž n q 1.Ž n q 1 y a .rŽŽ n q 1.Ž n q 1 y a . q Ž1 y a ... Equality holds in Ža. for f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1 and in Žb. for f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. We prove Ža.. The proof of Žb. is similar and will be omitted. We write n q 2Ž 1 y a .
fn Ž z .
1ya
f Ž z.
y
nq1ya n q 2Ž 1 y a .
n
1q s
Ý
a k z ky 1 y Ž Ž n q 1 y a . r Ž 1 y a . .
ks2
1q
`
Ý ak z ky 1 ks2
[
1 q wŽ z. 1 y wŽ z.
,
`
Ý ksnq1
a k z ky1
226
H. SILVERMAN
where `
Ž Ž n q 2 y 2 a . r Ž 1 y a . . Ý < ak < wŽ z. F 2y2
n
ksnq1 `
ks2
ksnq1
Ý < ak < y Ž nr1 y a . Ý
< ak <
F 1.
This last inequality is equivalent to n
Ý
< ak < q
nq1ya 1ya
ks2
`
< a k < F 1.
Ý
Ž 7.
ksnq1
Since the LHS of Ž7. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, the proof is complete. We next turn to ratios involving derivatives. If f of the form Ž1. satisfies condition Ž3., then for z g D,
THEOREM 4.
a ...
Ža. ReŽ f 9Ž z .rf nX Ž z .. G a nrŽ n q 1 y a ., Žb. ReŽ f nX Ž z .rf 9Ž z .. G Ž n q 1 y a .rŽŽ n q 1 y a . q Ž n q 1.Ž1 y
In both cases, the extremal function is f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1. Proof. We prove only Ža., which is similar in spirit to the proof of Theorem 1. The proof of Žb. follows the pattern of that in Theorem 3Ža.. We write nq1ya
Ž n q 1. Ž 1 y a .
f 9Ž z . f nX
Ž z.
y
an nq1ya
[
1 q wŽ z. 1 y wŽ z.
,
where `
Ž Ž n q 1 y a . r Ž n q 1. Ž 1 y a . . Ý kak z ky 1 ksnq1
wŽ z. s 2 q2
Ý
kak z
ky 1
qŽ Ž n q1 ya . r Ž n q1 . Ž 1 ya . .
ks2
.
`
n
Ý
kak z ky1
ksnq1
Now < w Ž z .< F 1 if n
nq1ya
`
Ý k < ak < q Ž n q 1. Ž 1 y a . Ý k < ak < F 1. ks2 ksnq1
Ž 8.
STARLIKE AND CONVEX FUNCTIONS
227
Since the LHS of Ž8. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, the proof is complete. THEOREM 5.
If f of the form Ž1. satisfies condition Ž4., then for z g D,
Ža. ReŽ f 9Ž z .rf nX Ž z .. G nrŽ n q 1 y a ., Žb. ReŽ f nX Ž z .rf 9Ž z .. G Ž n q 1 y a .rŽ n q 2 y 2 a .. In both cases, the extremal function is f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. It is well known that f g K Ž a . m zf 9 g S9Ž a .. In particular, f satisfies condition Ž4. if and only if zf 9 satisfies condition Ž3.. Thus, Ža. is an immediate consequence of Theorem 1 and Žb. follows directly from Theorem 3Ža..
REFERENCES 1. L. Brickman, D. J. Hallenbeck, T. H. MacGregor, and D. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 Ž1973., 413]428. 2. T. Sheil-Small, A note on partial sums of convex schlicht functions, Bull. London Math. Soc. 2 Ž1970., 165]168. 3. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 Ž1975., 109]116. 4. E. M. Silvia, On partial sums of convex functions of order a , Houston J. Math. 11, No. 3 Ž1985., 397]404.
209, 221]227 Ž1997.
AY975361
Partial Sums of Starlike and Convex Functions H. Silverman* Department of Mathematics, Uni¨ ersity of Charleston, Charleston, South Carolina 29424 Submitted by William F. Ames Received May 23, 1996
Let f nŽ z . s z q Ý nks2 a k z k be the sequence of partial sums of a function f Ž z . s z q Ý`ks 2 a k z k that is analytic in < z < - 1 and either starlike of order a or convex of order a , 0 F a - 1. When the coefficients a k 4 are ‘‘small,’’ we deterX mine lower bounds on Re f Ž z .rf nŽ z 4, Re f nŽ z .rf Ž z .4, Re f 9Ž z .rf nŽ z .4, and X Re f nŽ z .rf 9Ž z .4. In all cases, the results are sharp for each n. Q 1997 Academic Press
1. INTRODUCTION Let S denote the class of functions of the form f Ž z. s z q
`
Ý ak z k
Ž 1.
ks2
that are analytic and univalent in the unit disk D s z: < z < - 14 . A function f in S is said to be starlike of order a , 0 F a - 1, denoted S*Ž a ., if Re zf 9rf 4 ) a , z g D, and is said to be con¨ ex of order a , denoted K Ž a ., if Re 1 q zf 0rf 94 ) a , z g D. Let T *Ž a . and C Ž a . be the subfamilies of S*Ž a . and K Ž a ., respectively, whose functions are of the form f Ž z. s z y
`
Ý ak z k ,
a k G 0.
Ž 2.
ks2
* This work was completed while the author was on sabbatical leave from the University of Charleston as a Visiting Scholar at the University of California, San Diego. 221 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
222
H. SILVERMAN
A sufficient condition for a function of the form Ž1. to be in S*Ž a . is that `
Ý Ž k y a . < ak < F 1 y a
Ž 3.
ks2
and to be in K Ž a . is that `
Ý k Ž k y a . < ak < F 1 y a .
Ž 4.
ks2
For functions of the form Ž2., these sufficient conditions are also necessary. See w3x. In this note, we will examine the ratio of a function of the form Ž1. to its sequence of partial sums f nŽ z . s z q Ý nks2 a k z k when the coefficients of f are sufficiently small to satisfy either condition Ž3. or Ž4.. We will determine sharp lower bounds for Re f Ž z .rf n Ž z .4 , Re f n Ž z .rf Ž z .4 , Re f 9Ž z .rf nX Ž z .4 , and Re f nX Ž z .rf 9Ž z .4 . Sheil-Small w2x showed that inf Re f Ž z .rf nŽ z .4 for f g K Ž0. occurs when n s 1. In w1x, a sharp lower bound was found for Re f Ž z .rf 1Ž z .4 when f g K Ž a .. E. M. Silvia w4x investigated lower bounds on Re f Ž z .rf nŽ z .4 for f g T *Ž a . and f g C Ž a .. She showed that Re f Ž z .rf nŽ z .4 G 1rŽ2 y a . for f g T *Ž a . and that Re f Ž z .rf nŽ z .4 G Ž3 y a .r Ž4 y 2 a . for f g C Ž a .. These results are sharp when n s 1. Generally, lower bounds on ratios like Re f Ž z .rf nŽ z .4 or Re f nŽ z .rf Ž z .4 have been found to be sharp only when n s 1. In this paper, we determine sharpness for all values of n. The lower bounds in question are strictly increasing functions of n. In the sequel, we will make frequent use of the well-known result that ReŽ1 q w Ž z ..rŽ1 y w Ž z ..4 ) 0, z g D, if and only if w Ž z . s Ý`ks 1 c k z k satisfies the inequality < w Ž z .< F < z <. Unless otherwise stated, we will assume that f is of the form Ž1. and that its sequence of partial sums is denoted by f nŽ z . s z q Ý nks2 a k z k .
2. MAIN RESULTS THEOREM 1. If f of the form Ž1. satisfies condition Ž3., then ReŽ f Ž z .rf nŽ z .. G nrŽ n q 1 y a ., z g D. The result is sharp for e¨ ery n, with extremal function f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1.
223
STARLIKE AND CONVEX FUNCTIONS
Proof. We may write nq1ya
f Ž z.
1ya
fn Ž z .
n
y
nq1ya `
n
Ý ak z ky 1 q Ž Ž n q 1 y a . r Ž 1 y a . . Ý
1q
ks2
s
a k z ky1
ksnq1 n
1q
Ý ak z ky 1 ks2
[
1 q AŽ z .
.
1 q BŽ z.
Set Ž1 q AŽ z ..rŽ1 q B Ž z .. s Ž1 q w Ž z ..rŽ1 y w Ž z .., so that w Ž z . s Ž AŽ z . y B Ž z ..rŽ2 q AŽ z . q B Ž z ... Then `
Ž Ž n q 1 y a . r Ž 1 y a . . Ý ak z ky 1 ksnq1
wŽ z. s
`
n
Ý ak z ky 1 q Ž Ž n q 1 y a . r Ž 1 y a . . Ý
2q2
ks2
a k z ky1
ksnq1
and `
Ž Ž n q 1 y a . r Ž 1 y a . . Ý < ak < ksnq1
wŽ z. F
n
2y2
< ak < y Ž Ž n q 1 y a . r Ž 1 y a . .
Ý ks2
`
Ý
. < ak <
ksnq1
Now < w Ž z .< F 1 if and only if Ž2Ž n q 1 y a .rŽ1 y a ..Ý`ks nq1 < a k < F 2 y 2Ý nks 2 < a k <, which is equivalent to nq1ya
n
`
ks2
ksnq1
Ý < ak < q Ý
1ya
< a k < F 1.
Ž 5.
It suffices to show that the LHS of Ž5. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, which is equivalent to n
Ý ks2
ž
ky1 1ya
/
< ak < q
`
Ý ksnq1
ž
kyny1 1ya
/
< a k < G 0.
224
H. SILVERMAN
To see that f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1 gives the sharp result, we observe for z s rep i r n that f Ž z.
s1q
fn Ž z .
ž
1ya nq1ya
n
s
/
zn ª 1 y
1ya nq1ya
when r ª 1y.
nq1ya
THEOREM 2. If f of the form Ž1. satisfies condition Ž4., then ReŽ f Ž z .rf nŽ z .. G nŽ n q 2 y a .rŽ n q 1.Ž n q 1 y a ., z g D. The result is sharp for e¨ ery n, with extremal function f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. We write
Ž n q 1. Ž n q 1 y a .
f Ž z.
1ya
fn Ž z .
y
nŽ n q 2 y a .
Ž n q 1. Ž n q 1 y a .
n
1 q Ý a k z ky 1 qŽ Ž n q1 . Ž n q1 ya . r Ž 1 y a . . ks2
s
`
a k z ky1
Ý ksnq1
n
1q
Ý ak z
ky 1
ks2
[
1 q wŽ z. 1 y wŽ z.
,
where `
Ž Ž n q 1 . Ž n q 1 y a . r Ž 1 y a . . Ý ak z ky 1 ksnq1
wŽ z. s 2 q2
.
`
n
Ý ak z ky 1 qŽ Ž n q1. Ž n q1 ya . r Ž 1 ya . . Ý ks2
a k z ky1
ksnq1
Now `
Ž Ž n q 1. Ž n q 1 y a . r Ž 1 y a . . Ý < ak < ksnq1
wŽ z. F
n
2 y2
Ý ks2
< a k < y Ž Ž n q1 . Ž n q1 ya . r Ž 1 ya . .
`
Ý ksnq1
< ak <
F1
225
STARLIKE AND CONVEX FUNCTIONS
if n
Ý < ak < q
Ž n q 1. Ž n q 1 y a . 1ya
ks2
`
Ý
< a k < F 1.
Ž 6.
ksnq1
The LHS of Ž6. is bounded above by Ý`ks 2 Ž k Ž k y a .rŽ1 y a ..< a k < if 1 1ya
n
½Ý
k Ž k y a . y Ž 1 y a . < ak <
ks2
q
`
5
k Ž k y a . y Ž n q 1 . Ž n q 1 y a . < a k < G 0,
Ý ksnq1
and the proof is complete. Remark. Our results in Theorems 1 and 2 agree with those in w4x for the special case n s 1 and are an improvement when n ) 1. We next determine bounds for f nŽ z .rf Ž z .. THEOREM 3. Ža. If f of the form Ž1. satisfies condition Ž3., then ReŽ f nŽ z .rf Ž z .. G Ž n q 1 y a .rŽ n q 2 y 2 a ., z g D. Žb. If f satisfies condition Ž4., then ReŽ f nŽ z .rf Ž z .. G Ž n q 1.Ž n q 1 y a .rŽŽ n q 1.Ž n q 1 y a . q Ž1 y a ... Equality holds in Ža. for f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1 and in Žb. for f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. We prove Ža.. The proof of Žb. is similar and will be omitted. We write n q 2Ž 1 y a .
fn Ž z .
1ya
f Ž z.
y
nq1ya n q 2Ž 1 y a .
n
1q s
Ý
a k z ky 1 y Ž Ž n q 1 y a . r Ž 1 y a . .
ks2
1q
`
Ý ak z ky 1 ks2
[
1 q wŽ z. 1 y wŽ z.
,
`
Ý ksnq1
a k z ky1
226
H. SILVERMAN
where `
Ž Ž n q 2 y 2 a . r Ž 1 y a . . Ý < ak < wŽ z. F 2y2
n
ksnq1 `
ks2
ksnq1
Ý < ak < y Ž nr1 y a . Ý
< ak <
F 1.
This last inequality is equivalent to n
Ý
< ak < q
nq1ya 1ya
ks2
`
< a k < F 1.
Ý
Ž 7.
ksnq1
Since the LHS of Ž7. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, the proof is complete. We next turn to ratios involving derivatives. If f of the form Ž1. satisfies condition Ž3., then for z g D,
THEOREM 4.
a ...
Ža. ReŽ f 9Ž z .rf nX Ž z .. G a nrŽ n q 1 y a ., Žb. ReŽ f nX Ž z .rf 9Ž z .. G Ž n q 1 y a .rŽŽ n q 1 y a . q Ž n q 1.Ž1 y
In both cases, the extremal function is f Ž z . s z q ŽŽ1 y a .rŽ n q 1 y a .. z nq 1. Proof. We prove only Ža., which is similar in spirit to the proof of Theorem 1. The proof of Žb. follows the pattern of that in Theorem 3Ža.. We write nq1ya
Ž n q 1. Ž 1 y a .
f 9Ž z . f nX
Ž z.
y
an nq1ya
[
1 q wŽ z. 1 y wŽ z.
,
where `
Ž Ž n q 1 y a . r Ž n q 1. Ž 1 y a . . Ý kak z ky 1 ksnq1
wŽ z. s 2 q2
Ý
kak z
ky 1
qŽ Ž n q1 ya . r Ž n q1 . Ž 1 ya . .
ks2
.
`
n
Ý
kak z ky1
ksnq1
Now < w Ž z .< F 1 if n
nq1ya
`
Ý k < ak < q Ž n q 1. Ž 1 y a . Ý k < ak < F 1. ks2 ksnq1
Ž 8.
STARLIKE AND CONVEX FUNCTIONS
227
Since the LHS of Ž8. is bounded above by Ý`ks 2 ŽŽ k y a .rŽ1 y a ..< a k <, the proof is complete. THEOREM 5.
If f of the form Ž1. satisfies condition Ž4., then for z g D,
Ža. ReŽ f 9Ž z .rf nX Ž z .. G nrŽ n q 1 y a ., Žb. ReŽ f nX Ž z .rf 9Ž z .. G Ž n q 1 y a .rŽ n q 2 y 2 a .. In both cases, the extremal function is f Ž z . s z q ŽŽ1 y a .rŽ n q 1.Ž n q 1 y a .. z nq 1. Proof. It is well known that f g K Ž a . m zf 9 g S9Ž a .. In particular, f satisfies condition Ž4. if and only if zf 9 satisfies condition Ž3.. Thus, Ža. is an immediate consequence of Theorem 1 and Žb. follows directly from Theorem 3Ža..
REFERENCES 1. L. Brickman, D. J. Hallenbeck, T. H. MacGregor, and D. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 Ž1973., 413]428. 2. T. Sheil-Small, A note on partial sums of convex schlicht functions, Bull. London Math. Soc. 2 Ž1970., 165]168. 3. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 Ž1975., 109]116. 4. E. M. Silvia, On partial sums of convex functions of order a , Houston J. Math. 11, No. 3 Ž1985., 397]404.