Journal of Inequalities in Pure and
Applied Mathematics
COEFFICIENTS OF INVERSE FUNCTIONS IN A NESTED CLASS OF
STARLIKE FUNCTIONS OF POSITIVE ORDER
volume 7, issue 3, article 94,
2006.
A.K. MISHRA AND P. GOCHHAYAT
Department of Mathematics
Berhampur University
Ganjam, Orissa, 760007, India.
Received 28 July, 2005;
accepted 10 March, 2006.
Communicated by: H. Silverman
EMail: akshayam2001@yahoo.co.in
EMail: pb_gochhayat@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
227-05
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Abstract
In the present paper we find the estimates on the nth coefficients in the Maclaurin’s series expansion of the inverse of functions in the classP
Sδ (α), (0 ≤ δ <
n
∞, 0 ≤ α < 1), consisting of analytic functions
f(z) = z + ∞
n=2 an z in the
P∞ δ n−α
open unit disc and satisfying n=2 n 1−α |an | ≤ 1. For each n these estimates are sharp when α is close to zero or one and δ is close to zero. Further
for the second, third and fourth coefficients the estimates are sharp for every
admissible values of α and δ.
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
2000 Mathematics Subject Classification: 30C45.
Key words: Univalent functions, Starlike functions of order α, Convex functions of
order α, Inverse functions, Coefficient estimates.
A.K. Mishra and P. Gochhayat
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The present investigation is partially supported by National Board For Higher Mathematics, Department of Atomic Energy, Government of India. Sanction No. 48/2/2003R&D-II/844.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Notations and Preliminary Results . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
7
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1.
Introduction
Let U denote the open unit disc in the complex plane
U := {z ∈ C : |z| < 1}.
Let S be the class of normalized analytic univalent functions in U i.e. f is in S
if f is one to one in U, analytic and
∞
X
(1.1)
f (z) = z +
an z n ;
(z ∈ U).
n=2
The function f ∈ S is said to be in S ∗ (α) (0 ≤ α < 1), the class of univalent
starlike functions of order α, if
0 zf (z)
> α,
(z ∈ U)
Re
f (z)
and f is said to be in the class CV(α) of univalent convex functions of order
α if zf 0 ∈ S ∗ (α). The linear mapping f → zf 0 is popularly known as the
Alexander transformation. A well known sufficient condition, for the function
f of the form (1.1) to be in the class S, is
(1.2)
∞
X
n|an | ≤ 1
(see e.g. [17, p. 212]).
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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n=2
In fact, (1.2) is sufficient for f to be in the smaller class S ∗ (0) ≡ S ∗ (see e.g
[4]). An analogous sufficient condition for S ∗ (α) (0 ≤ α < 1) is
∞ X
n−α
(1.3)
|an | ≤ 1
(see [15]).
1−α
n=2
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The Alexander transformation gives that
∞
X
n−α
(1.4)
n
|an | ≤ 1
1−α
n=2
is a sufficient condition for f to be in CV(α). We recall the following:
Definition 1.1 ([8, 12]). The function f given by the series (1.1) is said to be in
the class Sδ (α) (0 ≤ α < 1, −∞ < δ < ∞) if
∞
X
n−α
δ
(1.5)
n
|an | ≤ 1
1−α
n=2
is satisfied.
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
For each fixed n the function nδ is increasing with respect to δ. Thus it
follows that if δ1 < δ2 , then Sδ2 (α) ⊂ Sδ1 (α). Consequently, by (1.3), the
functions in Sδ (α) are univalent starlike of order α if δ ≥ 0 and further if δ ≥ 1,
then by (1.4), Sδ (α) contains only univalent convex functions of order α. Also
we know (see e.g. [12, p. 224]) that if δ < 0 then the class Sδ (α) contains
non-univalent functions as well. Basic properties of the class Sδ (α) have been
studied in [8, 11, 12, 13]. We also note that if f ∈ Sδ (α) then
(1 − α)
|an | ≤ δ
;
n (n − α)
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(n = 2, 3, . . . )
and equality holds for each n only for functions of the form
(1 − α) iθ n
e z ,
fn (z) = z + δ
n (n − α)
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(θ ∈ R).
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We shall use this estimate in our investigation.
The inverse f −1 of every function f ∈ S, defined by f −1 (f (z)) = z, is
analytic in |w| < r(f ), (r(f ) ≥ 41 ) and has Maclaurin’s series expansion
(1.6)
f
−1
(w) = w +
∞
X
bn w
n
|w| < r(f ) .
n=2
The De-Branges theorem [2], previously known as the Bieberbach conjecture;
states that if the function f in S is given by the power series (1.1) then |an | ≤
n (n = 2, 3, . . . ) with equality for each n only for the rotations of the Koebe
z
function (1−z)
2 . Early in 1923 Löwner [10] invented the famous parametric
method to prove the Bieberbach conjecture for the third coefficient (i.e. |a3 | ≤
3, f ∈ S). Using this method he also found sharp bounds on all the coefficients
for the inverse functions in S (or S ∗ ). Thus, if f ∈ S (or S ∗ ) and f −1 is given
by (1.6) then
1
2n
; (n = 2, 3, . . . ) (cf [10]; also see [5, p. 222])
|bn | ≤
n+1 n
with equality for every n for the inverse of the Koebe function k(z) = z/(1 +
z)2 . Although the coefficient estimate problem for inverse functions in the
whole class S was completely solved in early part of the last century; for certain
subclasses of S only partial results are available in literature. For example, if
f ∈ S ∗ (α), (0 ≤ α < 1) then the sharp estimates
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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|b2 | ≤ 2(1 − α)
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and
|b3 | ≤

 (1 − α)(5 − 6α); 0 ≤ α ≤

1 − α;
2
3
2
3
(cf. [7])
≤α<1
hold. Further, if f ∈ CV then |bn | ≤ 1 (n = 2, 3, . . . , 8) (cf. [1, 9]), while
|b10 | > 1 [6]. However the problem of finding sharp bounds for bn for f ∈
S ∗ (α) (n ≥ 4) and for f ∈ CV (n ≥ 9) still remains open.
The object of the present paper is to study the coefficient estimate problem
for the inverse of functions in the class Sδ (α); (δ ≥ 0, 0 ≤ α < 1). We
find sharp bounds for |b2 |, |b3 | and |b4 | for f ∈ Sδ (α) (0 ≤ α < 1 and δ ≥
0). We further show that for every positive integer n ≥ 2 there exist positive
real numbers εn , δn and tn such that for every f ∈ Sδ (α) the following sharp
estimates hold:

1−α n−1
2n−3
2

; (0 ≤ α ≤ εn , 0 ≤ δ ≤ δn )
 n2(n−1)δ
n−2
2−α
(1.7)
|bn | ≤

 δ1−α ;
(1 − tn ≤ α < 1, δ > 0).
n (n−α)
For each n = 2, 3, . . . , there are two different extremal functions; in contrast
to only one extremal function for every n for the whole class S (or S ∗ (0)).
We also obtain crude estimates for |bn | (n = 2, 3, 4, . . . ; 0 ≤ α < 1, δ > 0;
f ∈ Sδ (α)). Our investigation includes some results of Silverman [16] for the
case δ = 0 and provides new information for δ > 0.
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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2.
Notations and Preliminary Results
Let the function s given by the power series
s(z) = 1 + d1 z + d2 z 2 + · · ·
(2.1)
be analytic in a neighbourhood of the origin. For a real number p define the
function h by
p
2
p
h(z) = (s(z)) = (1 + d1 z + d2 z + · · · ) = 1 +
(2.2)
∞
X
(p)
Ck z k .
k=1
(p)
Thus Ck denotes the k th coefficient in the Maclaurin’s series expansion of the
pth power of the function s(z). We need the following:
(p)
Lemma 2.1 ([14]). Let the coefficients Ck be defined as in (2.2), then
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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(p)
(2.3) Ck+1
k X
(p + 1)j
(p)
p−
=
dk+1−j Cj ;
k+1
j=0
(p)
(k = 0, 1, . . . ; C0 = 1).
Lemma 2.2 ([16]). If k and n are positive integers with k ≤ n − 2, then
n+j−1
n(k + 1 − j) + j
Aj =
j
2j (k + 2 − j)
is a strictly increasing function of j, j = 1, 2, . . . , k.
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Lemma 2.3. Let k and n be positive integers with k ≤ n − 2. Write
j
(n(k + 1 − j) + j)
1−α
(1 − α) n + j − 1
,
Aj (α, δ) =
j
2jδ
(k + 2 − j)δ (k + 2 − j − α) 2 − α
(0 ≤ α < 1, δ > 0).
Then for each n there exist positive real numbers εn and δn such that Aj (α, δ)
is strictly increasing in j for 0 ≤ α < εn , 0 ≤ δ < δn and j = 1, 2, . . . , k.
Proof. Write
hj (α, δ)
= Aj+1 (α, δ) − Aj (α, δ)
(n + j)(n(k − j) + (j + 1))(1 − α)
(1 − α)j+1 n + j − 1
= jδ
j
δ
2 (2 − α)
j
2 (j + 1)(k + 1 − j)δ (k + 1 − j − α)(2 − α)
(n(k + 1 − j) + j)
−
.
(k + 2 − j)δ (k + 2 − j − α)
We observe that for each fixed j (j = 1, 2, . . . , k − 1) hj (α, δ) is a continuous function of (α, δ). Also lim(α,δ)→(0,0) hj (α, δ) = hj (0, 0) = Aj+1 (0, 0) −
Aj (0, 0) > 0 by Lemma 2.2. Thus there exists an open circular disc B(0, rj )
with center at (0, 0) and radius rj > 0 such that hj (α, δ) > 0 for (α, δ) ∈
B(0, rj ) for each j = 1, 2, . . . , k − 1. Consequently, hj (α, δ) > 0 for all
j (j = 1, 2, . . . , k −
1) and (α, δ) ∈ B(0, r), where r = min1≤j≤k−1 rj . If we
√
2
choose εn = δn = 2 r, then Aj (α, δ) is strictly increasing in j for 0 ≤ α < εn ,
0 ≤ δ < δn and j = 1, 2, . . . , k. The proof of Lemma 2.3 is complete.
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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3.
Main Results
We have the following:
Theorem 3.1. Let the function f , given by the series (1.1) be in Sδ (α) (0 ≤
α < 1, 0 ≤ δ < ∞). Write
f −1 (w) = w +
(3.1)
∞
X
bn w n ,
(|w| < r0 (f ))
n=2
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
for some r0 (f ) ≥ 41 . Then
(a)
(3.2)
|b2 | ≤
(1 − α)
;
2δ (2 − α)
A.K. Mishra and P. Gochhayat
(0 ≤ α < 1, 0 ≤ δ < ∞).
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Set
(3.3)
δ0 =
log 3 − log 2
log 4 − log 3
δ1 =
and
log 5
− 1.
log 2
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(b) (i) If 0 ≤ δ ≤ δ0 , then
|b3 | ≤
(3.4)



2(1−α)2
;
22δ (2−α)2
(0 ≤ α ≤ α0 ),


(1−α)
;
3δ (3−α)
(α0 ≤ α < 1),
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where α0 is the only root, in the interval 0 ≤ α < 1, of the equation
(3.5)
δ
2δ
2
δ
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δ
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2δ
(2 · 3 − 2 )α − 4(2 · 3 − 2 )α + (6 · 3 − 4 · 2 ) = 0.
J. Ineq. Pure and Appl. Math. 7(3) Art. 94, 2006
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(ii) Further, if δ > δ0 , then
(3.6)
|b3 | ≤
(1 − α)
;
3δ (3 − α)
(0 ≤ α < 1).
(c) (i) If 0 ≤ δ ≤ δ1 , then
(3.7)
|b4 | ≤



5(1−α)3
;
23δ (2−α)3
(0 ≤ α < α1 ),


(1−α)
;
4δ (4−α)
(α1 ≤ α < 1),
where α1 is the only root in the interval 0 ≤ α < 1, of the equation
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
(3.8) (23δ − 5 · 4δ )α3 − 6(23δ − 5 · 4δ )α2
− 3(15 · 4δ − 4 · 23δ )α + 4(5 · 4δ − 2 · 23δ ) = 0.
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(ii) If δ > δ1 , then
(3.9)
|b4 | ≤
(1 − α)
;
4δ (4 − α)
(0 ≤ α < 1).
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All the estimates are sharp.
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Proof. We know from [7] that
1
bn =
2πin
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Z
|z|=r
1
f (z)
n
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dz.
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For fixed n write
n
∞
X
z
1
(−n)
P∞
h(z) =
=
=
1
+
Ck z k .
n
f (z)
(1 + k=2 ak z k−1 )
k=1
Thus
h(z)
h(n−1) (0)
(−n)
dz
=
= Cn−1 .
n
(n − 1)!
|z|=r z
(−n) Therefore a function,
maximizes Cn−1 will also maximize |bn |. Now
P∞ which
k−1
write w(z) = − k=2 ak z
and h(z) = (1 + w(z) + w2 (z) + · · · )n , (z ∈ U).
It follows that all the coefficients in the expansion of h(z) shall be nonnegative
if f (z) is of the form
1
nbn =
2πi
f (z) = z −
(3.10)
Z
∞
X
ak z k ,
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(−n) Consequently, maxf ∈Sδ (α) Cn−1 must occur for a function in Sδ (α) with the
representation (3.10).
(a) Now
z
f (z)
2
=
1−
∞
X
!−2
ak z
k−1
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= 1 + 2a2 z + · · · .
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k=2
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Therefore
(−2)
C1
A.K. Mishra and P. Gochhayat
(ak ≥ 0 ; k = 2, 3, . . . ).
k=2
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
2(1 − α)
= 2a2 = δ
λ2 ;
2 (2 − α)
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(0 ≤ λ2 ≤ 1, 0 ≤ α < 1, 0 ≤ δ < 1)
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(−2)
and the maximum C1
(−2)
C
|b2 | = 1
2
≤
is obtained by replacing λ2 = 1. Equivalently
1−α
;
− α)
(0 ≤ α < 1, 0 ≤ δ < ∞).
2δ (2
We get (3.2). To show that equality holds in (3.2), consider the function
f2 (z) defined by
(3.11) f2 (z) = z −
(1 − α) 2
z ;
2δ (2 − α)
(z ∈ U, 0 ≤ α < 1, 0 ≤ δ < ∞).
For this function
2
2(1 − α)
z
(−2)
=1+ δ
z + · · · = 1 + C1 z + · · ·
f2 (z)
2 (2 − α)
A.K. Mishra and P. Gochhayat
Title Page
and
|b2 | =
(−2)
C1
2
=
(1 − α)
.
2δ (2 − α)
Contents
The proof of (a) is complete.
To find sharp estimates for |b3 |, we consider
3
∞
X
z
(−3)
= (1 − a2 z − a3 z 2 − · · · )−3 = 1 +
Ck z k .
h(z) =
f (z)
k=1
By direct calculation or by taking p = −3, dk = −ak+1 in Lemma 2.1,we get,
(3.12)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
(−3)
C1
= 3a2
and
(−3)
C2
(−3)
= 3a3 + 2a2 C1
= 3a3 + 6a22 .
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2
Substituting a2 = (1−α)λ
and a3 =
2δ (2−α)
the equation (3.12) we obtain
(−3)
C2
=
(1−α)λ3
,
3δ (3−α)
(0 ≤ λ2 , λ3 ≤ 1, λ2 + λ3 ≤ 1) in
6(1 − α)2 2
3(1 − α)
λ
+
λ.
3
3δ (3 − α)
22δ (2 − α)2 2
Equivalently
(−3)
(3.13)
C2
3
= (1 − α)
λ3
2(1 − α)λ22
+
3δ (3 − α) 22δ (2 − α)2
.
In order to maximize the right hand side of (3.13), write
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
G(λ2 , λ3 ) =
2(1 − α)λ22
λ3
+
;
3δ (3 − α) 22δ (2 − α)2
(0 ≤ λ2 ≤ 1, 0 ≤ λ3 ≤ 1, λ2 +λ3 ≤ 1).
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The function G(λ2 , λ3 ) does not have a maximum in the interior of the square
{(λ2 , λ3 ) : 0 < λ2 < 1, 0 < λ3 < 1}, since Gλ2 6= 0, Gλ3 6= 0. Also if λ3 = 1
then λ2 = 0 and if λ2 = 1 then λ3 = 0. Therefore
max G(λ2 , λ3 ) =
λ3 =1
3δ (3
1
− α)
and
max G(λ2 , λ3 ) =
λ2 =1
2(1 − α)
.
− α)2
22δ (2
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Also
2(1 − α)
max G(λ2 , λ3 ) = 2δ
λ3 =0
2 (2 − α)2
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and
1
max G(λ2 , λ3 ) = δ
.
λ2 =0
3 (3 − α)
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We get
max G(λ2 , λ3 ) = max
0≤λ2 ≤1
0≤λ3 ≤1
Thus
(−2)
C3
3
≤ (1 − α) max
1
2(1 − α)
, 2δ
δ
3 (3 − α) 2 (2 − α)2
1
2(1 − α)
, 2δ
δ
3 (3 − α) 2 (2 − α)2
.
.
We now find the maximum of the above two terms. Note that the sign of the
expression
3δ (3
2(1 − α)
−F (α)
1
− 2δ
= 2δ δ
2
− α) 2 (2 − α)
2 3 (3 − α)(2 − α)2
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
2
depends on the sign of the quadratic polynomial F (α) = a(δ)α −4a(δ)α+c(δ),
where a(δ) = 3δ · 2 − 22δ and c(δ) = 2(3δ+1 − 22δ+1 ). Observe that

 ≥ 0 if δ ≤ δ0∗
log 2
∗
a(δ)
;
δ0 =

log 4 − log 3
∗
< 0 if δ > δ0

 ≥ 0 if δ ≤ δ0
log 3 − log 2
c(δ)
;
δ0 =

log 4 − log 3
< 0 if δ > δ0
and δ0 ≤ δ0∗ .
(b) (i) The case 0 ≤ δ ≤ δ0 : Suppose 0 ≤ δ ≤ δ0 then F (0) = c(δ) ≥ 0,
F (1) = −22δ < 0 and since a(δ) ≥ 0, F (α) is positive for large
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values of α. Therefore F (α) ≥ 0 if 0 ≤ α ≤ α0 and F (α) < 0 if
α0 < α < 1 where α0 is the unique root of equation F (α) = 0 in the
interval 0 ≤ α < 1. Or equivalently −F (α) ≤ 0 for 0 < α ≤ α0 and
−F (α) > 0 for α0 < α < 1. Consequently,

2(1−α)2

(0 ≤ α ≤ α0 );
(−3)
2δ (2−α)2 ;

2
C2
|b3 | =
≤

3
 (1−α) ; (α ≤ α < 1).
0
3δ (3−α)
We get the estimate (3.4).
(ii) The case δ0 < δ: We show below that if δ0 < δ ≤ δ0∗ or δ0∗ < δ then
F (α) < 0. Suppose δ0 < δ ≤ δ0∗ , then a(δ) ≥ 0. Consequently,
F (α) > 0 for large positive and negative values of α. Also F (0) =
c(δ) < 0 and F (1) = −22δ < 0. Therefore F (α) < 0 for every α in
the real interval 0 ≤ α < 1. Similarly, if δ0∗ < δ, then a(δ) < 0. Thus
F 0 (α) = 2a(δ)(α − 2) > 0; (0 ≤ α < 1). Or equivalently F (α) is an
increasing function in 0 ≤ α < 1. Also F (1) = −22δ < 0. Therefore
F (α) < 0 in 0 ≤ α < 1.
Since −F (α) > 0 we have
(−3)
C
|b3 | = 2
3
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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(1 − α)
≤ δ
3 (3 − α)
(0 ≤ α < 1; δ > δ0 ).
Close
Quit
This is precisely the estimate (3.6). We note that for the function
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f2 (z) defined by (3.11)
z
f2 (z)
3
−3
(1 − α)
= 1− δ
z
2 (2 − α)
3(1 − α)
6(1 − α)2 2
=1+ δ
z + ··· .
z + 2δ
2 (2 − α)
2 (2 − α)2
Therefore
(−3)
|b3 | =
C2
3
=
2(1 − α)2
.
22δ (2 − α)2
We get sharpness in (3.4) with 0 ≤ α < α0 . Similarly for the function
f3 (z) defined by
(3.14)
(1 − α) 3
z ;
3δ (3 − α)
(z ∈ U, 0 ≤ α < 1, 0 ≤ δ < ∞),
f3 (z) = z −
we have
3 −3
z
(1 − α) 2
3(1 − α) 2
= 1− δ
z
=1+ δ
z + ···
f3 (z)
3 (3 − α)
3 (3 − α)
(−3)
C
|b3 | = 2
3
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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(1 − α)
= δ
.
3 (3 − α)
This establishes the sharpness of (3.4) with α0 ≤ α < 1 and (3.6).
The proof of (b) is complete.
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In order to find sharp estimates for |b4 |, we consider the function
!−4
4
∞
∞
X
X
z
(−4)
= 1−
ak z k−1
=1+
Ck z k .
h(z) =
f (z)
k=2
k=1
Taking p = −4 and dk = −ak+1 in Lemma 2.1, we get
(−4)
C1
= 4a2 ;
(−4)
C2
= 4a3 + 10a22 ;
(−4)
C3
= 4a4 + 20a2 a3 + 20a32 .
(1−α)
Taking a2 = 2(1−α)
δ (2−α) λ2 , a3 = 3δ (3−α) λ3 and a4 =
λ2 , λ3 , λ4 ≤ 1 and λ2 + λ3 + λ4 ≤ 1 we get
(1−α)
λ,
4δ (4−α) 4
where 0 ≤
(−4)
C
|b4 | = 3
4
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
5(1 − α)λ2 λ3
5(1 − α)2 λ32
λ4
= (1 − α)
+
+
4δ (4 − α) 2δ 3δ (2 − α)(3 − α)
23δ (2 − α)3
= (1 − α)L(λ2 , λ3 , λ4 )
(say).
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Since Lλ2 6= 0, Lλ3 6= 0 and Lλ4 6= 0, the function L cannot have a local
maximum in the interior of cube 0 < λ2 < 1, 0 < λ3 < 1, 0 < λ4 < 1.
Therefore the constraint λ2 + λ3 + λ4 ≤ 1 becomes λ2 + λ3 + λ4 = 1. Hence
putting λ4 = 1 − λ2 − λ3 we get
(−4)
C
|b4 | = 3
4
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1 − λ 2 − λ3
5(1 − α)λ2 λ3
5(1 − α)2 λ32
= (1 − α)
+
+
4δ (4 − α)
2δ 3δ (2 − α)(3 − α)
23δ (2 − α)3
= (1 − α)H(λ2 , λ3 )
(say).
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Thus we need to maximize H(λ2 , λ3 ) in the closed square 0 ≤ λ2 ≤ 1, 0 ≤
λ3 ≤ 1. Since
2
5(1 − α)
2
<0
Hλ2 λ2 · Hλ3 λ3 − (Hλ2 λ3 ) = − δ δ
2 3 (2 − α)(3 − α)
the function H cannot have a local maximum in the interior of the square 0 ≤
λ2 ≤ 1, 0 ≤ λ3 ≤ 1. Further, if λ2 = 1 then λ3 = 0 and if λ3 = 1 then λ2 = 0.
Therefore
5(1 − α)2
max H(λ2 , λ3 ) = H(1, 0) = 3δ
,
λ2 =1
2 (2 − α)3
max H(λ2 , λ3 ) = H(0, 1) = 0,
λ3 =1
A.K. Mishra and P. Gochhayat
1 − λ2
5(1 − α)2 λ32
+ 3δ
max H(λ2 , 0) = max
0<λ2 <1
4δ (4 − α)
2 (2 − α)3
5(1 − α)2
1
,
= max
4δ (4 − α) 23δ (2 − α)3
and
1 − λ3
1
=
.
0≤λ3 ≤1 4δ (4 − α)
4δ (4 − α)
Thus
max H(λ2 , λ3 ) = max
0≤λ2 ≤1
0≤λ3 ≤1
1
5(1 − α)2
,
4δ (4 − α) 23δ (2 − α)3
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max H(0, λ3 ) = max
0≤λ3 ≤1
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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.
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The maximum of the above two terms can be found as in the case for |b3 |. We
see that the sign of the expression
1
5(1 − α)2
− δ
3δ
3
2 (2 − α)
4 (4 − α)
is same as the sign of the cubic polynomial P (α) = a(δ)α3 −6a(δ)α2 −3b(δ)α+
4c(δ), where a(δ) = 23δ − 5 · 4δ , b(δ) = 15 · 4δ − 4 · 23δ and c(δ) = 5 · 4δ − 2 · 23δ .
We observe that

 ≥ 0 if δ ≤ δ1
log 5
c(δ)
;
δ1 =
−1 ,

log 2
< 0 if δ > δ1

 ≥ 0 if δ ≤ δ2
log 3
b(δ)
;
δ2 = δ1 +
−1

log 2
< 0 if δ > δ2
and
a(δ)

 ≤ 0 if δ ≤ δ3

> 0 if δ > δ3
;
log 5
δ3 =
log 2
.
Moreover, δ1 < δ2 < δ3 . Also the quadratic polynomial P (α) = 3 a(δ)α2 −
q
4a(δ)α − b(δ) has roots at 2 ± 4 + ab .
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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0
(c) (i) The case 0 ≤ δ ≤ δ1 : In this case c(δ) ≥ 0, b(δ) ≥ 0 and a(δ) ≤ 0.
Note that both the roots of P 0 (α) are complex numbers and P 0 (0) =
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−3b(δ) ≤ 0. Therefore P 0 (α) < 0 for every real number and consequently, P 0 (α) is a decreasing function. Since P (0) = 4c(δ) ≥ 0
and P (1) = −23δ < 0, the function P (α) has a unique root α1 in the
interval 0 < α < 1. Or equivalently, P (α) ≥ 0 for 0 < α ≤ α1 and
P (α) < 0 if α1 < α < 1. Thus

5(1−α)3

 23δ (2−α)3 ; (0 ≤ α ≤ α1 ),
|b4 | ≤

 (1−α) ; (α ≤ α < 1).
1
4δ (4−α)
We get the estimate (3.7).
(ii) The case δ > δ1 : We shall show below, separately, that if δ1 < δ ≤
δ2 or δ2 < δ ≤ δ3 or δ3 < δ then P (α) < 0 in 0 ≤ α < 1.
First suppose that δ1 < δ ≤ δ2 . Then c(δ) < 0, b(δ) ≥ 0 and
a(δ) < 0. Thus, as in case of (c)(i), P 0 (α) has only complex roots
and P 0 (0) < 0. Therefore P (α) is a monotonic decreasing function
in 0 ≤ α < 1. Since P (0) < 0, we get that P (α) < 0 for 0 ≤ α < 1.
Next if δ2 < δ ≤ δ3 , then c(δ) < 0, b(δ) < 0 and a(δ) < 0. Therefore,
P 0 (α) has two real roots: one is negative and the other is greater than
2. The condition P 0 (0) > 0 gives that P 0 (α) > 0 in 0 ≤ α < 1.
Therefore P (α) is a monotonic increasing function in 0 ≤ α < 1.
Since P (1) = −23δ < 0, we get that P (α) < 0 in 0 ≤ α < 1.
Lastly, if δ > δ3 then c(δ) < 0, b(δ) < 0 and a(δ) > 0. Hence
P 0 (α) has only complex roots and the condition P 0 (0) = −3b(δ) > 0
gives P 0 (α) > 0 for every real α. Consequently P (α) is a monotonic
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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increasing function. Since P (1) < 0, we get that P (α) < 0 in 0 ≤
α < 1.
Since P (α) < 0 for 0 ≤ α < 1, we have
|b4 | ≤
(1 − α)
;
4δ (4 − α)
(0 ≤ α < 1).
This is precisely the estimate (3.9). We note that for the function
f2 (z) defined by (3.11)
4
4(1 − α)
20(1 − α)2 2 20(1 − α)3 3
z
= 1+ δ
z+ 2δ
z + 3δ
z +· · · .
f2 (z)
2 (2 − α) 2.2 (2 − α)2
2 (2 − α)3
Therefore
A.K. Mishra and P. Gochhayat
(−4)
C
5(1 − α)3
|b4 | = 3
= 3δ
.
4
2 (2 − α)3
This shows sharpness of the estimate (3.7) with 0 ≤ α ≤ α1 . Similarly, for the function f4 (z) defined by
(1 − α) 4
(3.15) f4 (z) = z− δ
z ;
4 (4 − α)
we have
(−4)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
(z ∈ U, 0 ≤ α < 1, 0 ≤ δ < ∞)
C
(1 − α)
|b4 | = 3
= δ
4
4 (4 − α)
We get sharpness in (3.7) with α1 ≤ α < 1 and in (3.9). The proof of
Theorem 3.1 is complete.
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Theorem 3.2. Let the function f , given by (1.1), be in Sδ (α) (0 ≤ α < 1, δ >
0) and f −1 (w) be given by (3.1). Then for each n there exist positive numbers
εn , δn and tn such that

1−α n−1
2n−3
2

; (0 ≤ α ≤ εn , 0 ≤ δ ≤ δn )
 n2(n−1)δ
n−2
2−α
(3.16)
|bn | ≤

 δ1−α ;
(1 − tn ≤ α < 1, δ > 0).
n (n−α)
The estimate (3.16) is sharp.
Proof. We follow the lines of the proof of Theorem 3.1. Write
n
z
h(z) =
f (z)
= (1 − a2 z − a3 z 2 − · · · )−n
(an ≥ 0, n = 2, 3, . . . )
∞
X (−n)
=1+
Ck z k
k=1
(−n)
and observe that bn =
2.1, we get
Cn−1
n
(−n)
Ck+1
. Now taking p = −n and dk = −ak+1 in Lemma
k X
(1 − n)j
(−n)
=
n+
ak+2−j Cj .
k+1
j=0
(3.17)
A.K. Mishra and P. Gochhayat
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Since f ∈ Sδ (α), we get
(1 − α)
an = δ
λn ;
n (n − α)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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0 ≤ λn ≤ 1,
∞
X
n=2
!
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λn ≤ 1 .
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Therefore
(−n)
Ck+1
(3.18)
k X
(1 − n)j
(1 − α)λk+2−j
(−n)
=
n+
Cj .
δ
k+1
(k + 2 − j) (k + 2 − j − α)
j=0
In order to establish (3.16), we wish to show that for each n = 2, 3, . . . there
(−n)
exist positive real numbers εn and δn such that Cn−1 is maximized when λ2 = 1
for 0 ≤ α ≤ εn and 0 ≤ δ ≤ δn . Using (3.18) we get
(−n)
C1
=
n(1 − α)
n(1 − α)
(−n)
λ2 C0
= δ
λ2
δ
2 (2 − α)
2 (2 − α)
so that
(−n)
C1
(3.19)
(−n)
Thus C1
≤
n(1 − α)
(−n)
= d1
δ
2 (2 − α)
A.K. Mishra and P. Gochhayat
(say).
is maximized when λ2 = 1. Write
(−n)
dj
(−n)
(−n)
= max Cj
f ∈Sδ (α)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
(1 ≤ j ≤ n − 1).
Assume that Cj
(1 ≤ j ≤ n − 2) is maximized for λ2 = 1 when α > 0 and
δ > 0 are sufficiently small. It follows from (3.17) that λ2 = 1 implies λj = 0
for every j ≥ 3. Therefore using (3.18) and (3.19) we get
n+1
(1 − α) (−n)
(−n)
C2
≤
d
2
2δ (2 − α) 1
2
n+2−1 1 1−α
(−n)
=
= d2
(say).
2
22δ 2 − α
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Assume that
(3.20)
(−n)
dj
=
j
n+j−1 1 1−α
j
2jδ 2 − α
(0 ≤ j ≤ n − 2).
Again, using (3.18), we get
(3.21)
(−n)
(−n)
dn−1 = max Cn−1
f ∈Sδ (α)
!
(1 − α)λn−j
(−n)
= max
(n − j)
C
f ∈Sδ (α)
(n − j)δ (n − j − α) j
j=0
!
X
n−2
(n − j)(1 − α)
(−n)
≤ max
C
λn−j
0≤j≤n−2
(n − j)δ (n − j − α) j
j=0
(n − j)(1 − α)
(−n)
≤ max
dj
.
δ
0≤j≤n−2
(n − j) (n − j − α)
n−2
X
Write
Aj (α, δ) =
(n − j)(1 − α)
(−n)
d
;
(n − j)δ (n − j − α) j
(−n)
Substituting d0
(−n)
= 1 and the value of d1
n(1 − α)
A0 (α, δ) = δ
n (n − α)
and
(j = 0, 1, 2, . . . , (n − 2)).
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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from (3.19), we get
n(n − 1)(1 − α)2
A1 (α, δ) = δ
.
2 (n − 1)δ (n − 1 − α)(2 − α)
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Now A0 (α, δ) < A1 (α, δ) (n ≥ 2 and 0 ≤ δ ≤ 2) if and only if
(3.22)
nδ (n
1
1
< δ
.
δ
− 1)(n − α)(1 − α)
2 (n − 1) (n − 1 − α)(2 − α)
The above inequality (3.22) is true, because (n − 1 − α) < (n − α), (1 − α) <
δ
(2 − α) and the maximum value of n2 (n − 1)1−δ is equal to 1 (n ≥ 2, 0 ≤
δ ≤ 2). Also by Lemma 2.3, there exist positive real numbers εn and δn such
that Aj (α, δ) < Ak (α, δ) (0 ≤ α ≤ εn , 0 ≤ δ ≤ δn , 1 ≤ j < k ≤ n − 2).
(−n)
Therefore it follows from (3.21) that the maximum Cn−1 occurs at j = n − 2.
(−n)
Substituting the value of dn−2 , from (3.20) in (3.21) we get
n−1
2n − 3
1−α
2(1 − α) (−n)
2
(−n)
dn−1 = δ
d
= (n−1)δ
2 (2 − α) n−2
2
n−2
2−α
(0 ≤ α ≤ εn , 0 ≤ δ ≤ δn , n = 2, 3, . . . ).
Therefore
n−1
(−n)
Cn−1
2
2n − 3
1−α
|bn | =
≤ (n−1)δ
;
n
n2
n−2
2−α
(0 ≤ α ≤ εn , 0 ≤ δ ≤ δn , n = 2, 3, . . . ).
The above is precisely the first assertion of (3.16). In order to prove the other
case of (3.16), we first observe that in the degenerate case α = 1 we have
(−n)
Sδ (α) = {z}.Therefore Cj
→ 0 as α → 1− for every j = 1, 2, 3, . . . . Hence
there exists a positive real number tn (0 ≤ tn ≤ 1) such that
n
(n − j)
(−n)
≥
Cj
δ
δ
n (n − α)
(n − j) (n − j − α)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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(j = 1, 2, . . . , 1 − tn ≤ α < 1).
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(−n)
Thus the maximum of (3.21) occurs at j = 0 and we get dn−1 =
equivalently
(−n)
C
(1 − α)
|bn | ≤ n−1 = δ
.
n
n (n − α)
n(1−α)
nδ (n−α)
or
This last estimate is precisely the assertion of (3.16) with (1 − tn ≤ α < 1, δ >
0).
We observe that the (n − 1)th coefficient of the function
f2 (z) is defined by (3.11), is equal to
z
f2 (z)
n
, where
n−1
2n − 3
1−α
.
2(n−1)δ n − 2
2−α
n
Similarly, the (n − 1)th coefficient of the function fnz(z) , where fn (z) is
defined by
2
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
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z−
(1 − α) n
z ,
nδ (n − α)
(z ∈ U, 0 ≤ α < 1, 0 ≤ δ < 1)
is equal to
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.
nδ (n − α)
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Therefore the estimate (3.16) is sharp. The proof of Theorem 3.2 is complete.
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Theorem 3.3. Let the function f given by (1.1), be in Sδ (α) (0 ≤ α < 1, δ > 0)
and f −1 (w) be given by (3.2). For fixed α and δ (0 ≤ α < 1, δ > 0) let
Bn (α, δ) = maxf ∈Sδ (α) |bn |. Then
1
2nδ (2 − α)n
· δ
.
n [2 (2 − α) − (1 − α)]n
P
nδ (n−α)
Proof. Since f ∈ Sδ (α), by Definition 1.1 we have ∞
n=2 (1−α) |an | ≤ 1.
δ (2−α) P∞
Therefore 2(1−α)
n=2 |an | ≤ 1 or equivalently
Bn (α, δ) ≤
(3.23)
∞
X
|an | ≤
n=2
(1 − α)
.
2δ (2 − α)
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
A.K. Mishra and P. Gochhayat
This gives
(3.24)
∞
X
|f (z)| = z +
an z n n=2
≥ |z| − |z 2 |
∞
X
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!
|an |
≥ r − r2
n=2
Now using the above estimate (3.24) we have
Z
1
1
|bn | = dz
2inπ |z|=r (f (z))n Z
1
1
1
≤
|dz| ≤
n
2nπ |z|=r |f (z)|
n
(1 − α)
,
2δ (2 − α)
(|z| = r).
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1
r−
r2 (1−α)
2δ (2−α)
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!n
.
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We observe that the function F (r) where
F (r) =
1
r−
!n
r 2 (1−α)
2δ (2−α)
is an increasing function of r (0 ≤ α < 1, δ > 0) in the interval 0 ≤ r < 1.
Therefore
!n
1
1
|bn | ≤
.
n 1 − (1−α)
δ
2 (2−α)
Consequently,
2nδ (2 − α)n
1
Bn (α, δ) ≤
.
n [2δ (2 − α) − (1 − α)]n
The proof of Theorem 3.3 is complete.
Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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References
[1] J.T.P. CAMPSCHROERER, Coefficients of the inverse of a convex function, Report 8227, Nov. 1982, Department of Mathematics, Catholic University, Nijmegen, The Netherlands, (1982).
[2] L. DE BRANGES, A proof of the Bieberbach conjecture, Acta. Math.,
154(1-2) (1985), 137–152.
[3] P.L. DUREN, Univalent Functions, Volume 259, Grunlehren der Mathematischen Wissenchaften, Bd., Springer-Verlag, New York, (1983).
[4] A.W. GOODMAN, Univalent functions and nonanalytic curves, Proc.
Amer. Math. Soc., 8 (1957), 591–601.
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Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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Coefficients of Inverse
Functions in a Nested Class of
Starlike Functions of Positive
Order
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