Journal of Inequalities in Pure and
Applied Mathematics
STARLIKENESS AND CONVEXITY CONDITIONS FOR
CLASSES OF FUNCTIONS DEFINED BY SUBORDINATION
volume 5, issue 2, article 31,
2004.
R. AGHALARY, JAY M. JAHANGIRI AND S.R. KULKARNI
University of Urmia, Urmia, Iran.
EMail: raghalary@yahoo.com
Received 21 May, 2003;
accepted 18 April, 2004.
Communicated by: H. Silverman
Kent State University, Ohio, USA.
EMail: jay@geauga.kent.edu
Fergussen College, Pune, India.
EMail: srkulkarni40@hotmail.com
Abstract
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Abstract
We consider the family P(1, b), b > 0, consisting of functions p analytic in the
open unit disc U with the normalization p(0) = 1 which have the disc formulation |p − 1| < b in U. Applying the subordination
properties to certain choices
P∞
k
of p using the functions fn (z) = z + k=1+n ak z , n = 1, 2, ..., we obtain inclusion relations, sufficient starlikeness and convexity conditions, and coefficient
bounds for functions in these classes. In some cases our results improve the
corresponding results appeared in print.
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Subordination, Hadamard product, Starlike, Convex.
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Family F(1, b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
The Family Mvλ (1, b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4
Coefficient Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
1.
Introduction
Let A denote the class of functions that are analytic in the open unit disc U =
{z ∈ C : |z| < 1} and let An be the subclass of A consisting of functions fn of
the form
(1.1)
fn (z) = z +
∞
X
ak z k , n = 1, 2, 3, . . . .
k=1+n
The function p ∈ A and normalized by p(0) = 1 is said to be in P(1, b) if
(1.2)
|p(z) − 1| < b,
b > 0,
z ∈ U.
The class P(1, b) which is defined using the disc formulation (1.2) was studied
by Janowski [6] and has an alternative characterization in terms of subordination
(see [5] or [14]), that is, for z ∈ U , we have
(1.3)
p ∈ P(1, b) ⇐⇒ p(z)≺ 1 + bz.
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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For the functions φ and ψ in A, we say that the φ is subordinate to ψ in U ,
denoted by φ ≺ ψ, if there exists a function w(z) in A with w(0) = 0 and
|w(z)| < 1, such that φ(z) = ψ(w(z)) in U. For further references see Duren
[3].
The family P(1, b) contains many interesting classes of functions which
have close inter-relations with different well-known classes of analytic univalent functions. For example, for fn ∈ An if
0
zfn
∈ P(1, 1 − α),
0 ≤ α ≤ 1,
fn
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
then fn is starlike of order α in U and if
zfn00
1+ 0
∈ P(1, 1 − α),
fn
0 ≤ α ≤ 1,
then fn is convex of order α in U.
For 0 ≤ α ≤ 1 we let S ∗ (α) be the class of functions fn ∈ An which are
starlike of order α in U , that is,
0
zfn
∗
S (α) ≡ fn ∈ An : <
≥ α, |z| < 1 ,
fn
and let K(α) be the class of functions fn ∈ An which are convex of order α in
U , that is,
zfn00
K(α) ≡ fn ∈ An : < 1 + 0
≥ α, |z| < 1 .
fn
Alexander [1] showed that fn is convex in U if and only if zfn0 is starlike in U.
In this paper we investigate inclusion relations, starlikeness, convexity, and
coefficient conditions on fn and its related classes for two choices of p(fn ) in
P(1, b). In some cases, we improve the related known results appeared in the
literature.
Define F(1, b) be the subclass of P(1, b) consisting of functions p(f1 ) so
that
zf10 (z)
zf100 (z)
(1.4)
p(f1 (z)) =
1+ 0
f1 (z)
f1 (z)
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
where f1 ∈ A1 is given by (1.1).
For fixed v > −1, n ≥ 1, and for λ ≥ 0, define Mvλ (1, b) be the subclass of
P(1, b) consisting of functions p(fn ) so that
(1.5)
p(fn (z)) = (1 − λ)
Dv fn (z)
+ λ(Dv fn (z))0
z
where fn ∈ An and Dv f is the v-th order Ruscheweyh derivative [10].
The v-th order Ruscheweyh derivative Dv of a function fn in An is defined
by
(1.6)
Dv fn (z) =
∞
X
z
∗
f
(z)
=
z
+
Bk (v)ak z k ,
n
(1 − z)1+v
k=1+n
where
Bk (v) =
(1 + v)(2 + v) · · · (v + k − 1)
(k − 1)!
and the operator “ ∗ ” stands for the convolution or Hadamard product of two
power series
∞
∞
X
X
f (z) =
ai z i and g(z) =
bi z i
i=1
defined by
(f ∗ g)(z) = f (z) ∗ g(z) =
i=1
∞
X
i=1
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
2.
The Family F(1, b)
The class F(1, b) for certain values of b yields a sufficient starlikeness condition
for the functions f1 ∈ A1 .
Theorem 2.1. If 0 < b ≤
9
4
and p(f1 ) ∈ F(1, b) then
√
3 − 9 − 4b
zf10
∈ P 1,
.
f1
2
We need the following lemma, which is due to Jack [4].
Lemma 2.2. Let w(z) be analytic in U with w(0) = 0. If |w| attains its maximum value on the circle |z| = r at a points z0 , we can write z0 w0 (z0 ) = kw(z0 )
for some real k, k ≥ 1.
√
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
zf 0 (z)
Proof of Theorem 2.1. For b1 = 3− 29−4b write f11(z) = 1 + b1 w(z). Obviously,
w is analytic in U and w(0) = 0. The proof is complete if we can show that
|w| < 1 in U. On the contrary, suppose that there exists z0 ∈ U such that
|w(z0 )| = 1. Then, by Lemma 2.2, we must have z0 w0 (z0 ) = kw(z0 ) for some
real k, k ≥ 1 which yields
z0 f10 (z0 )
z0 f100 (z0 )
1+ 0
− 1 = (1 + b1 w(z0 ))2 + b1 z0 w0 (z0 ) − 1
f1 (z0 )
f1 (z0 )
= bk + 2b1 + b21 w(z0 )
≥3b1 − b21 = b.
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This contradicts the hypothesis, and so the proof is complete.
J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
Corollary 2.3. For 0 < b ≤ 2 let p(f1 ) ∈ F(1, b). Then f1 ∈ S ∗
√
−1+ 9−4b
.
2
Corollary 2.4. If p(f1 ) ∈ F(1, b) and 0 < b ≤ 2, then
√
3 − 9 − 4b
zf10 (z)
arg
< arcsin
.
f1 (z)
2
It is not known if the above corollaries can be extended to the case when
b > 2.
√ f1 (z)
1
∗ −1+ 5
Corollary 2.5. If < zf 0 (z)+z
>
.
then
f
∈
S
1
2 f 00 (z)
2
2
1
1
Remark 2.1. For 0 < b < 2, Theorem 2.1 is an improvement to Theorem 1
obtained by Obradović, Joshi, and Jovanović [8].
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
Corollary 2.6. If p(f1 ) ∈ F(1, b) then f1 is convex in U for 0 < b ≤ 0.935449.
Title Page
Proof. For p(f1 ) ∈ F(1, b) we can write | arg p(f1 )| < arcsin b. Therefore,
√
zf100 (z)
3 − 9 − 4b
arg 1 + 0
< arcsin b + arcsin
.
f1 (z)
2
Contents
Now the proof is complete upon noting that the right hand side of the above
inequality is less than π2 for b = 0.935449 .
Remark 2.2. It is not known if the above Corollary 2.6 is sharp but it is an
improvement to Corollary 2 obtained by Obradovic, Joshi, and Jovanovic [8].
Corollary 2.7. If p(f1 ) ∈ F(1, b) then f1 is convex in the disc |z| <
0.935449 ≤ b ≤ 1.
0.935449
b
for
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
Proof. We write p(f1 ) = 1 + bw(z) where w is a Schwarz function. Let |z| ≤ ρ.
Then |w(z)| ≤ ρ and so |p(f1 ) − 1| < bρ for |z| ≤ ρ. Upon choosing bρ =
0.935449 it follows from the above Corollary 2.6 that | arg(1 + zf100 /f10 )| < π/2
for |z| ≤ ρ = 0.935449/b . Therefore the proof is complete.
In the following example we show that there exist functions f which are
not necessarily starlike or univalent in U for p(f1 ) ∈ F(1, b) if b is sufficiently
large.
Example 2.1. For the spirallike function g(z) = z/(1 − z)1+i we have
0
1
1 − |z|2
− π4 i zg (z)
< e
=√
> 0, z ∈ U.
g(z)
2 |1 − z|2
Since
zg 0 (z)
g(z)
=
1+iz
,
1−z
<
zg 0 (z)
g(z)
1 − r( cos θ + sin θ)
=
1 − 2r cos θ + r2
for z = reiθ . Thus g(z) is not starlike for |z| < t,
g(rz)
r
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
we obtain
f (z) =
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
√1
2
< t < 1. This means that
is not starlike in U. Now set
Z z
g(ζ)
h(z) =
dζ = i((1 − z)−i − 1)
ζ
0
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e2π −1
≈0.996
e2π +1
and let z0 =
. Therefore, h(z0 ) = h(−z0 ) and so h is not univalent
in U. Consequently, f (z) = h(zz00 z) is not univalent in U for sufficiently large
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
values of b. On the other hand, p(g) ∈ F(1, b) for sufficiemtly large b, since,
|p(g(z)) − 1| =
z
1 + 3iz
+
−1 <b
(1 − z)2 1 − z
for sufficiemtly large b.
The following theorem is the converse of Theorem 2.1 for a special case.
√ zf10
3− 5
Theorem 2.8. If f1 ∈ P 1, 2
then p(f1 ) ∈ F(1, 1) for |z| < r0 =
0.7851 .
To prove our theorem, we need the following lemma due to Dieudonné [2].
Corollary 2.9. Let z0 and w0 be given points in U, with z0 =
6 0. Then for
all functions f analytic and satisfying |f (z)| < 1 in U, with f (0) = 0 and
f (z0 ) = w0 , the region of values of f 0 (z0 ) is the closed disc
w−
w0
|z0 |2 − |w0 |2
≤
.
z0
|z0 |(1 − |z0 |2 )
Proof of Theorem 2.8. Write
zf 0 (z)
q(z) = 1
=1+
f1 (z)
√ !
3− 5
w(z),
2
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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where w is a Schwarz function. We need to find the largest disc |z| < ρ for
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
which
"
#2
√ !
3− 5
w(z) +
2
1+
=
√ !
3− 5
zw0 (z) − 1
2
√ !2
√
3− 5
w2 (z) + (3 − 5)w(z) +
2
√ !
3− 5
zw0 (z) < 1.
2
For fixed r = |z| and R = |w(z)| we have R ≤ r. Therefore, by Lemma 2.9,
we obtain
R
r 2 − R2
|w0 (z)|≤ +
r
r(1 − r2 )
and so
zf100 (z)
zf10 (z)
|p(f1 ) − 1| =
1+ 0
−1
f1 (z)
f1 (z)
√ !2
√ !
√
3− 5
3
−
5
w2 (z) + (3 − 5)w(z) +
=
zw0 (z)
2
2
≤ t2 R2 + 3tR + t
r 2 − R2
1 − r2
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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t
=
ψ(R),
1 − r2
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where
√
ψ(R) = R2 (t − tr2 − 1) + 3R(1 − r2 ) + r2 and t =
3− 5
.
2
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
2
3(1−r )
We note that ψ(R) attains its maximum at R0 = 2(1+tr
2 −t) . So the theorem
t
follows for r0 ≈0.7851 which is the root of the equation 1−r
2 ψ(R0 ) = 1.
Letting z0 and w0 in Lemma 2.9 be so that |z0 | = r0 and |w0 | =
we conclude that the bound given by Theorem 2.8 is sharp.
3(1−r02 )
2(1+tr02 −t)
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
3.
The Family Mvλ(1, b)
We begin with stating and proving some properties of the family Mvλ (1, b).
Theorem 3.1. If p(fn ) ∈ Mvλ (1, b) then
Dv fn (z)
b
∈ P(1,
).
z
1 + λn
We need the following lemma, which is due to Miller and Mocanu [7].
Lemma 3.2. Let q(z) = 1 + qn z n + · · · (n ≥ 1) be analytic in U and let h(z) be
convex univalent in U with h(0) = 1. If q(z) + 1c zq 0 (z) ≺ h(z) for c > 0, then
Z
c
c −c/n z
q(z) ≺ z
h(t)t n −1 dt.
n
0
v
Proof of Theorem 3.1. For p(fn ) ∈ Mvλ (1, b) set q(z) = D fzn (z) . Then we can
write q(z) + λzq 0 (z) ≺ 1 + bz. Now, applying Lemma 3.2, we obtain
b
q(z) ≺ +1 +
z.
1 + λn
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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Substituting back for q(z) and choosing w(z) to be analytic in U with |w(z)| ≤
|z|n , by the definition of subordination we have
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Dv fn (z)
b
=1+
w(z).
z
(1 + λn)
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Now the theorem follows using the necessary and sufficient condition (1.3). The
estimates in Theorem 3.1 are sharp for p(fn ) where fn is given by
Dv fn (z)
b
=1+
zn.
z
(1 + λn)
Corollary 3.3. If p(fn ) ∈ Mvλ (1, b) then
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
b
Dv fn (z)
≤1+
|z|n .
z
1 + λn
Corollary 3.4. If |fn0 (z) + λzfn00 (z) − 1| < b then
fn0 (z) ≺ 1 +
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
b
z.
1 + λn
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Corollary 3.5. If (1 − λ) fnz(z) + λfn0 (z) − 1 < b then
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fn (z)
b
≺1+
z.
z
1 + λn
In the next two theorems we investigate the inclusion relations for classes of
Mvλ .
Theorem 3.6. For 0 ≤ λ1 < λ and v ≥ 0, let b1 =
1+λ1 n
b.
1+nλ
Mvλ (1, b) ⊂ Mvλ1 (1, b1 ).
Then
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Proof. The case for λ1 = 0 is trivial. For λ1 6= 0 suppose that p(fn ) ∈
Mvλ (1, b). Therefore, we can write
Dv fn (z)
+ λ1 (Dv fn (z))0
z
v
λ1
Dv fn (z)
λ1
D fn (z)
v
0
=
(1 − λ)
+ λ(D fn (z)) + 1 −
.
λ
z
λ
z
(1 − λ1 )
Now, by definition, p(fn ) ∈ Mvλ1 (1, b1 ) and so the proof is complete.
Theorem 3.7. Let v ≥ 0 and b1 =
b(1+v)
.
n+1+v
Then
v
Mv+1
λ (1, b) ⊂ Mλ (1, b1 ).
Proof. For fn ∈ An suppose that p1 (fn ) ∈
p1 (fn (z)) = (1 − λ)
Mv+1
λ (1, b)
where
D1+v fn (z)
+ λ(Dv+1 fn (z))0 .
z
Set
p2 (fn (z)) = (1 − λ)
Dv fn (z)
+ λ(Dv fn (z))0 .
z
An elementary differentiation yields
D1+v fn (z)
p1 (fn (z)) = (1 − λ)
+ λ(Dv+1 fn (z))0
z
1
= p2 (fn (z)) +
zp0 (fn (z)).
1+v 2
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
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R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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From this and Lemma 3.2, we conclude that p1 (fn ) ∈ Mvλ (1, b1 ).
J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
Corollary 3.8.
fn0 (z)
+
λzfn00 (z)
fn (z)
b
0
∈ P(1, b) =⇒ (1 − λ)
+ λfn (z) ∈ P 1,
.
z
1+n
Theorem 3.9. For v ≥ 0 and λ > 0 let b < 1 + λn. If p(fn ) ∈ Mvλ (1, b) then
b(2 + λn)
z(Dv fn (z))0
−1 <
.
v
D fn (z)
λ[(1 + λn) − b]
Proof. First note that, we can write
(1 − λ)
For b1 =
Dv fn (z)
+ λ(Dv fn (z))0 − 1 < b ;
z
b(2+λn)
λ[(1+λn)−b]
Dv fn (z)
b
−1 <
.
z
1 + λn
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
we define w(z) by
1 + b1 w(z) =
[z(Dv fn (z))0 ]
.
[Dv fn (z)]
One can easily verify that w(z) is analytic in U and w(0) = 0. To conclude the
proof, it suffices to show that |w(z)| < 1 in U. If this is not the case, then by
Lemma 2.2, there exists a point z0 ∈ U such that |w(z0 )| = 1 and z0 w0 (z0 ) =
kw(z0 ). Therefore
Dv f (z0 )
+ λ(Dv f (z0 ))0 − 1
z0
Dv fn (z0 )
z0 (Dv fn (z0 ))0
=
(1 − λ) + λ
−1
z0
Dv fn (z0 )
|p(fn (z0 )) − 1| = (1 − λ)
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v
v
z0 (Dv fn (z0 ))0
D fn (z0 )
D fn (z0 )
= λ
−1
+
−1
Dv fn (z0 )
z0
z0
b
b
≥ λb1 1 −
−
= b.
1 + nλ
1 + nλ
This is a contradiction to the hypothesis and so |w(z)| < 1 in U.
Corollary 3.10.
i) If fn0 (z) ∈ P(1, 1+n
) then
3+n
1+n
ii) If fn0 (z) + zfn00 (z) ∈ P(1, 3+n
) then
zfn00 (z)
fn0 (z)
zfn0 (z)
fn (z)
∈ P(1, 1).
∈ P(1, 1).
Theorem 3.11. Let p(fn ) ∈ Mvλ (1, b) for some λ > 0. If

√
(n
−
3)
+
n2 + 2n + 9
λ(1
+
λn)



;
0
<
λ
≤
 2 + λ(n − 1)
2n
b=
√
r

(n − 3) + n2 + 2n + 9
2λ − 1


;
≤λ≤1
 (1 + λn)
λ2 n2 + 2λ(1 + n)
2n
then
<
Dv+1 fn (z)
Dv fn (z)
v
>
.
1+v
We need the following lemma, which is due to Ponnusamy and Singh [9].
Lemma 3.12. Let 0 < λ1 < λ < 1 and let Q be analytic in U satisfying
Q(z) ≺ 1 + λ1 z, and Q(0) = 1. If q(z) is analytic in U, q(0) = 1 and satisfies
Q(z)[c + (1 − c)q(z)] ≺ 1 + λz,
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
where
c=

1−λ


,


 1 + λ1
0 < λ + λ1 ≤ 1


1 − (λ2 + λ21 )


, λ2 + λ21 ≤ 1 ≤ λ + λ1

2(1 − λ21 )
then Re{q(z)} > 0, z ∈ U .
Proof of Theorem 3.11. From Theorem 3.1 and the fact 0 < b < 1 < 1 + λn
we conclude that
Dv fn (z)
b
≺ 1 + b1 z, 0 < b1 =
< b < 1.
z
1 + nλ
On the other hand, we may write
z(Dv fn (z))0
Dv fn (z)
(1 − λ) + λ
≺ 1 + bz.
z
Dv fn (z)
v
v
0
fn (z))
Letting Q(z) = D fzn (z) , q(z) = z(D
, and c = 1 − λ, we see that all
Dv fn (z)
conditions in Lemma 3.12 are satisfied. This implies that Re q(z) > 0 and so
the proof is complete.
Corollary 3.13. Let p(fn ) ∈ Mvλ (1, b) for some λ > 0. Then Dv fn is starlike
in the disc

λ(1 + nλ)



if 0 < λ < λ1 and b1 ≤ b ≤ 1
 (2 + λ(n − 1))b
|z| ≤
r

(1
+
λn)
2λ − 1


if λ1 ≤ λ ≤ 1 and b2 ≤ b ≤ 1,

b
λ2 n2 + 2λ(1 + n)
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
where
λ1 =
(n − 3) +
√
s
b2 = (1 + λn )
i) If
fn0
ii) If
fn0
∈ P 1, √
+
zfn00
n2 + 2n + 9
λ(1 + nλ)
, b1 =
, and
2n
[2 + λ(n − 1)]
2λ − 1
.
λ2 n2 + 2λ(1 + n)
(1+n)
1+(1+n)2
∈ P 1, √
then fn is starlike in U .
(1+n)
1+(1+n)2
then fn is convex in U .
If we let λ = 1 and v = 0, 1 in Corollary 3.13, then we obtain
Corollary 3.14. Let √ (1+n)
1+(1+n)2
≤ b ≤ 1 and fn ∈ An .
i) If fn0 ∈ P(1, b) then f is starlike for |z| < √ (1+n)
b
b
R. Aghalary, Jay M. Jahangiri and
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1+(1+n)
ii) If fn0 + zfn00 ∈ P(1, b) then f is convex for |z| < √
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
.
2
1+n
.
1+(1+n)2
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
4.
Coefficient Bounds
Sufficient coefficient conditions for F(1, b) and Mvλ (1, b) are given next.
Theorem 4.1. Let p(f1 ) be given by (1.4) for f1 as in (1.1). If
(4.1)
∞
X
(k 2 + b − 1)|ak | < b,
k=2
then p(f1 ) ∈ F(1, b).
Proof. We need to show that if (4.1) then |p(f1 (z)) − 1| < b. For p(f1 ) we can
write
zf100
zf10
|p(f1 (z)) − 1| =
1+ 0 −1
f1
f1
P∞
2
− 1)ak z k
k=2 (k
P∞
=
z + k=2 ak z k
P∞
2
− 1)|ak ||z|k−1
k=2 (k
P∞
≤
1 − k=2 |ak ||z|k−1
P∞
2
− 1)|ak |
k=2 (k
P∞
.
<
1 − k=2 |ak |
The above right hand inequality is less than b by (4.1) and so p(f1 ) ∈ F(1, b).
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
Theorem 4.2. Let p(fn ) be given by (1.5) for fn as in (1.1). If
(4.2)
∞
X
(λk − λ + 1)Bk (v)|ak | < b,
k=1+n
then p(fn ) ∈ Mvλ (1, b).
Proof. Apply the Ruscheweyh derivative (1.6) to the function fn (z) and substitute in (1.5) to obtain
v
|p(fn (z)) − 1| = (1 − λ)
=
<
∞
X
D fn (z)
+ λ(Dv fn (z))0 − 1
z
(λk − λ + 1)Bk (v)ak z k−1
k=1+n
∞
X
(λk − λ + 1)Bk (v)|ak |.
k=1+n
Now this latter inequality is less than b by (4.2) and so p(fn ) ∈ Mvλ (1, b).
Next, by judiciously varying the arguments of the coefficients of the functions fn given by (1.1), we shall show that the sufficient coefficient conditions
(4.1) and (4.2) are also necessary for their respective classes with varying arguments.
A function fn given by (1.1) is said to be in V(θk ) if arg(ak ) = θk for all k.
If, further, there exists a real number β such that θk + (k − 1)β ≡ π(mod 2π)
then fn is said to be in V(θk ; β). The union of V(θk ; β) taken over all possible
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
{θk } and all possible real β is denoted by V. For more details see Silverman
[13].
Some examples of functions in V are
i) T ≡ V(π; 0) ⊂ V where T is the class of analytic univalent functions
with negative coefficients studied by Schild [11] and Silverman [12].
P
iθk k
ii) Univalent functions of the form z+ ∞
k=2 |ak |e z are in V(θk ; 2π/k) ⊂ V
for θk = π − 2(k − 1)π/k.
Note that the family V is rotationally invariant since fn ∈ V(θk ; β) implies
that
e−iγ fn (zeiγ ) ∈ V(θk + (k − 1)γ; β − γ).
Finally, we let
VF(1, b) ≡ V ∩ F(1, b) and VMvλ (1, b) ≡ V ∩ Mvλ (1, b).
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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Theorem 4.3.
p(f1 ) ∈ VF(1, b) ⇐⇒
∞
X
(k 2 + b − 1)|ak | < b.
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Proof. In light of Theorem 4.1, we only need to prove the “only if ” part of the
theorem. Suppose p(f1 ) ∈ VF(1, b), then
P∞
2
− 1)ak z k−1
k=2 (k
P∞
|p(f1 ) − 1| =
<b
1 + k=2 ak z k−1
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
or
(4.3)
∞
X
2
(k − 1)ak z
k=2
k−1
<b 1+
∞
X
ak z k−1 .
k=2
The condition (4.3) must hold for all values of z in U. Therefore, for f1 ∈
V(θk ; β) we set z = reiβ in (4.3) and let r −→ 1− . Upon clearing the inequality
(4.3) we obtain the condition
!
∞
∞
X
X
(k 2 − 1)|ak | < b 1 −
|ak |
k=2
k=2
as required.
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
Corollary 4.4. If 0 < b ≤ 1 and p(f1 ) ∈ VF(1, b) then f1 is convex in U.
Corollary 4.5. If 1 < b ≤ 3 and p(f1 ) ∈ VF(1, b) then f1 is starlike in U.
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The above two corollaries can be justifed using Theorem 4.3 and the following lemma due to Silverman [12].
Lemma 4.6. For f1 of the form (1.1) and univalent in U we have
P
2
i) If ∞
k=2 k |ak | ≤ 1, then f1 is convex in U.
P
ii) If ∞
k=2 k|ak | ≤ 1, then f1 is starlike in U.
Next, we show that the above sufficient coefficient condition (4.2) is also
necessary for functions in VMvλ (1, b).
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J. Ineq. Pure and Appl. Math. 5(2) Art. 31, 2004
Theorem 4.7.
p(fn ) ∈
VMvλ (1, b)
⇐⇒
∞
X
(λk − λ + 1)Bk (v)|ak | < b.
k=1+n
Proof. Suppose that p(fn ) ∈ VMvλ (1, b). Then, by (1.5), we have
|p(fn (z)) − 1| = (1 − λ)
Dv fn (z)
+ λ(Dv fn (z))0 − 1 < b.
z
On the other hand, for fn ∈ V(θk ; β) we have
fn (z) = z +
∞
X
|ak |eiθk z k .
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
k=1+n
The condition required for p(fn ) ∈ VMvλ (1, b) must hold for all values of z in
U. Setting z = reiβ yields
∞
X
(λk − λ + 1)Bk (v)|ak |rk−1 < b.
k=1+n
The required coefficient condition follows upon letting z −→ 1− .
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From the above Theorem 4.7 and Lemma 4.6.ii, we obtain
Corollary 4.8. If λ ≥ 2b − 1 and p(fn ) ∈ VMvλ (1, b) then f is starlike in U.
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References
[1] J.W. ALEXANDER, Functions which map the interior of the unit circle
upon simple regions, Ann. of Math. Soc., 17 (1915-1916), 12–22.
[2] J. DIEUDONNÉ, Recherches sur quelques problèms relatifs aux
polynômes et aux functions bornées d’une variable complexe, Ann. Sci.
Ecole Norm. Sup., 48 (1931), 247–358.
[3] P.L. DUREN, Univalent Functions, Grundlehren der Mathematischen
Wissenschaften 259, A series of Comprehensive Studies in Mathematics,
Springer-Verlag New York Inc, 1983.
[4] I.S. JACK, Functions starlike and convex of order α, J. London Math. Soc.,
3 (1971), 469–474.
[5] J.M. JAHANGIRI, H. SILVERMAN AND E.M. SILVIA, Inclusion relations between classes of functions defined by subordination, J. Math. Anal.
Appl., 151 (1990), 318–329.
[6] W. JANOWSKI, Extremal problems for a family of functions with positive
real part and for some related families, Ann. Polon. Math., 23 (1970), 159–
177.
[7] S.S. MILLER AND P.T. MOCANU, Differential subordination and univalent functions, Michigan Math. J., 28 (1981), 157–171.
[8] M. OBRADOVIĆ, S.B. JOSHI AND I. JOVANOVIĆ, On certain sufficient
conditions for starlikeness and convexity, Indian J. Pure Appl. Math., 29
(1998), 271–275.
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
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[9] S. PONNUSAMY AND V. SINGH, Convolution properties of some
classes of analytic functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat.
Inst. Steklov., (POMI) 226 (1996), 138–154.
[10] St. RUSCHEWEYH, New criteria for univalent functions, Proc. Amer.
Math. Soc., 49 (1975), 109–115.
[11] A. SCHILD, On a class of functions schlicht in the unit circle, Proc. Amer.
Math. Soc., 5 (1954), 115–120.
[12] H. SILVERMAN, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51 (1975), 109–116.
Starlikeness and Convexity
Conditions for Classes of
Functions Defined by
Subordination
[13] H. SILVERMAN, Univalent functions with varying arguments, Houston
J. Math., 7 (1981), 283–287.
R. Aghalary, Jay M. Jahangiri and
S.R. Kulkarni
[14] H. SILVERMAN AND E. M. SILVIA, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math., 37 (1985), 48–61.
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