CERTAIN SUBCLASSES OF p-VALENTLY
CLOSE-TO-CONVEX FUNCTIONS
Subclasses of p-Valently
Close-to-convex Functions
OH SANG KWON
Department of Mathematics
Kyungsung University
Busan 608-736, Korea
EMail: oskwon@ks.ac.kr
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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Contents
Received:
28 May, 2007
Accepted:
14 October, 2007
Communicated by:
JJ
II
G. Kohr
J
I
2000 AMS Sub. Class.:
30C45.
Page 1 of 18
Key words:
p-valently starlike functions of order α, p-valently close-to-convex functions of
order α, subordination, hypergeometric series.
Abstract:
The object of the present paper is to drive some properties of certain class
Kn,p (A, B) of multivalent analytic functions in the open unit disk E.
Acknowledgements:
This research was supported by Kyungsung University Research Grants in 2006.
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Contents
1
Introduction
3
2
Preliminaries and Main Results
6
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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1.
Introduction
Let Ap be the class of functions of the form
p
f (z) = z +
(1.1)
∞
X
ap+k z p+k
k=1
which are analytic in the open unit disk E = {z ∈ C : |z| < 1}. A function f ∈ Ap
is said to be p-valently starlike of order α of it satisfies the condition
0 zf (z)
Re
> α (0 ≤ α < p, z ∈ E).
f (z)
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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Sp∗ (α).
We denote by
On the other hand, a function f ∈ Ap is said to be p-valently close-to-convex
functions of order α if it satisfies the condition
0 zf (z)
Re
> α (0 ≤ α < p, z ∈ E),
g(z)
for some starlike function g(z). We denote by Cp (α).
For f ∈ Ap given by (1.1), the generalized Bernardi integral operator Fc is defined
by
Z
c+p z
Fc (z) =
f (t)tc−1 dt
zc 0
∞
X
c+p
p
(1.2)
ap+k z p+k (c + p > 0, z ∈ E).
=z +
c+p+k
k=1
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For an analytic function g, defined in E by
p
g(z) = z +
∞
X
bp+k z p+k ,
k=1
Flett [3] defined the multiplier transform I η for a real number η by
I η g(z) =
∞
X
(p + k + 1)−η bp+k z p+k
(z ∈ E).
k=0
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
η
Clearly, the function I g is analytic in E and
vol. 10, iss. 3, art. 83, 2009
I η (I µ g(z)) = I η+µ g(z)
for all real numbers η and µ.
For any integer n, J. Patel and P. Sahoo [5] also defined the operator Dn , for an
analytic function f given by (1.1), by
−n
∞ X
p+k+1
n
p
D f (z) = z +
ap+k z p+k
1
+
p
k=1
"
#
−n
∞ X
k
+
1
+
p
(1.3)
= f (z) ∗ z p−1 z +
z k+1
(z ∈ E),
1
+
p
k=1
where ∗ stands for the Hadamard product or convolution.
It follows from (1.3) that
(1.4)
z(Dn f (z))0n−1 f (z) − Dn f (z).
We also have
D0 f (z) = f (z) and
D−1 f (z) =
zf 0 (z) + f (z)
.
p+1
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If f and g are analytic functions in E, then we say that f is subordinate to g,
written f < g or f (z) < g(z), if there is a function w analytic in E, with w(0) = 0,
|w(z)| < 1 for z ∈ E, such that f (z) = g(w(z)), for z ∈ U . If g is univalent then
f < g if and only if f (0) = g(0) and f (E) ⊂ g(E).
Making use of the operator notation Dn , we introduce a subclass of Ap as follows:
Definition 1.1. For any integer n and −1 ≤ B < A ≤ 1, a function f ∈ Ap is said
to be in the class Kn,p (A, B) if
(1.5)
p(1 + Az)
z(Dn f (z))0
<
,
zp
1 + Bz
where < denotes subordination.
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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For convenience, we write
Contents
Kn,p
2α
1−
, −1 = Kn,p (α),
p
where Kn,p (α) denote the class of functions f ∈ Ap satisfying the inequality
z(Dn f (z))0
Re
> α (0 ≤ α < p, z ∈ E).
zp
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We also note that K0,p (α) ≡ Cp (α) is the class of p-valently close-to-convex
functions of order α.
In this present paper, we derive some properties of a certain class Kn,p (A, B) by
using differential subordination.
Close
2.
Preliminaries and Main Results
In our present investigation of the general class Kn,p (A, B), we shall require the
following lemmas.
Lemma 2.1 ([4]). If the function p(z) = 1 + c1 z + c2 z 2 + · · · is analytic in E, h(z)
is convex in E with h(0) = 1, and γ is complex number such that Re γ > 0. Then
the Briot-Bouquet differential subordination
Oh Sang Kwon
zp0 (z)
p(z) +
< h(z)
γ
implies
γ
p(z) < q(z) = γ
z
vol. 10, iss. 3, art. 83, 2009
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z
Z
tγ−1 h(t)dt < h(z)
(z ∈ E)
For complex numbers a, b and c 6= 0, −1, −2, . . . , the hypergeometric series
2 F1 (a, b; c; z)
=1+
Contents
0
and q(z) is the best dominant.
(2.1)
Subclasses of p-Valently
Close-to-convex Functions
a(a + 1)b(b + 1) 2
ab
z+
z + ···
c
2!c(c + 1)
represents an analytic function in E. It is well known by [1] that
Lemma 2.2. Let a, b and c be real c 6= 0, −1, −2, . . . and c > b > 0. Then
Z 1
Γ(b)Γ(c − b)
tb−1 (1 − t)c−b−1 (1 − tz)−a dt =
2 F1 (a, b; c; z),
Γ(c)
0
z
−a
(2.2)
2 F1 (a, b; c; z) = (1 − z) 2 F1 a, c − b; c;
z−1
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and
(2.3)
2 F1 (a, b; c; z)
= 2 F1 (b, a; c; z).
Lemma 2.3 ([6]). Let φ(z) be convex and g(z) is starlike in E. Then for F analytic
g
(E) is contained in the convex hull of F (E).
in E with F (0) = 1, φ∗F
φ∗g
Lemma 2.4 ([2]). Let φ(z) = 1 +
∞
P
ck z k and φ(z) <
k=1
1+Az
.
1+Bz
Then
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
|ck | ≤ (A − B).
vol. 10, iss. 3, art. 83, 2009
Theorem 2.5. Let n be any integer and −1 ≤ B < A ≤ 1. If f ∈ Kn,p (A, B), then
(2.4)
p(1 + Az)
z(Dn+1 f (z))0
< q(z) <
p
z
1 + Bz
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(z ∈ E),
where
(2.5)


2 F1 (1, p + 1; p + 2; −Bz)



p+1
+ p+2
Az 2 F1 (1, p + 2; p + 3; −Bz), B 6= 0;
q(z) =



 1 + p+1 Az,
B = 0,
p+2
and q(z) is the best dominant of (2.4). Furthermore, f ∈ Kn+1,p (ρ(p, A, B)), where
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(2.6)

p2 F1 (1, p + 1; p + 2; B)




A2 F1 (1, p + 2; p + 3; B), B 6= 0;
− p(p+1)
p+2
ρ(p, A, B) =




1 − p+1
A,
B = 0.
p+2
Proof. Let
p(z) =
(2.7)
z(Dn+1 f (z))0
,
pz p
where p(z) is analytic function with p(0) = 1.
Using the identity (1.4) in (2.7) and differentiating the resulting equation, we get
(2.8)
zp0 (z)
1 + Az
z(Dn f (z))0
=
p(z)
+
<
(≡ h(z)).
pz p
p+1
1 + Bz
Thus, by using Lemma 2.1 (for γ = p + 1), we deduce that
Z z p
t (1 + At)
−(p+1)
p(z) < (p + 1)z
dt(≡ q(z))
1 + Bt
0
Z 1 p
s (1 + Asz)
= (p + 1)
ds
1 + Bsz
0
Z 1
Z 1
sp
sp+1
= (p + 1)
(2.9)
ds + (p + 1)Az
ds.
0 1 + Bsz
0 1 + Bsz
By using (2.2) in (2.9), we obtain


2 F1 (1, p + 1; p + 2; −Bz)



+ p+1
Az 2 F1 (1, p + 2; p + 3; −Bz), B 6= 0;
p+2
p(z) < q(z) =



 1 + p+1 Az,
B = 0.
p+2
Thus, this proves (2.5).
Now, we show that
(2.10)
Re q(z) ≥ q(−r)
(|z| = r < 1).
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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Since −1 ≤ B < A ≤ 1, the function (1 + Az)/(1 + Bz) is convex(univalent) in E
and
1 + Az
1 − Ar
Re
≥
> 0 (|z| = r < 1).
1 + Bz
1 − Br
Setting
1 + Asz
(0 ≤ s ≤ 1, z ∈ E)
1 + Bsz
and dµ(s) = (p + 1)sp ds, which is a positive measure on [0, 1], we obtain from (2.9)
that
Z 1
q(z) =
g(s, z)dµ(s) (z ∈ E).
g(s.z) =
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
0
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Therefore, we have
Z
1
Z
Re g(s, z)dµ(s) ≥
Re q(z) =
0
0
1
1 − Asr
dµ(s)
1 − Bsr
which proves the inequality (2.10).
Now, using (2.10) in (2.9) and letting r → 1− , we obtain
z(Dn+1 f (z))0
Re
> ρ(p, A, B),
zp
where
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
p2 F1 (1, p + 1; p + 2; B)




− p(p+1)
A2 F1 (1, p + 2; p + 3; B), B 6= 0
p+2
ρ(p, A, B) =




p − p(p+1)
A,
B = 0.
p+2
This proves the assertion of Theorem 2.5. The result is best possible because of the
best dominant property of q(z).
Putting A = 1 −
2α
p
and B = −1 in Theorem 2.5, we have the following:
Corollary 2.6. For any integer n and 0 ≤ α < p, we have
Kn,p (α) ⊂ Kn+1,p (ρ(p, α)),
where
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
(2.11)
ρ(p, α) = p · 2 F1 (1, p + 1; p + 2; −1)
p(p + 1)
−
(1 − 2α)2 F1 (1, p + 2; p + 3; −1).
p+2
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The result is best possible.
Taking p = 1 in Corollary 2.6, we have the following:
Corollary 2.7. For any integer n and 0 ≤ α < 1, we have
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Kn (δ) ⊂ Kn+1 (δ(α)),
where
(2.12)
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δ(α) = 1 + 4(1 − 2α)
∞
X
k=1
1
(−1)k .
k+2
Theorem 2.8. For any integer n and 0 ≤ √
α < p, if f (z) ∈ Kn+1,p (α), then f ∈
−1+ 1+(p+1)2
Kn,p (α) for |z| < R(p), where R(p) =
. The result is best possible.
p+1
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Proof. Since f (z) ∈ Kn+1,p (α), we have
z(Dn+1 f (z))0
= α + (p − α)w(z), (0 ≤ α < p),
zp
where w(z) = 1 + w1 z + w2 z + · · · is analytic and has a positive real part in E.
Making use of logarithmic differentiation and using identity (1.4) in (2.13), we get
z(Dn f (z))0
zw0 (z)
(2.14)
− α = (p − α) w(z) +
.
zp
p+1
(2.13)
and
Re w(z) ≥
1−r
1+r
(|z| = r < 1),
in (2.14), we get
z(Dn f (z))0
1 Re zw0 (z)
Re
− α = (p − α) Re w(z) 1 +
zp
p + 1 Re w(z)
1 |zw0 (z)|
≥ (p − α) Re w(z) 1 −
p + 1 Re w(z)
1−r
1
2r
≥ (p − α)
1−
.
1+r
p + 1 1 − r2
It is easily seen
√ that the right-hand side of the above expression is positive if |z| <
−1+
1+(p+1)2
.
p+1
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
Now, using the well-known (by [5])
|zw0 (z)|
2r
≤
Re w(z)
1 − r2
Subclasses of p-Valently
Close-to-convex Functions
R(p) =
Hence f ∈ Kn,p (α) for |z| < R(p).
To show that the bound R(p) is best possible, we consider the function f ∈ Ap
defined by
1−z
z(Dn+1 f (z))0
=
α
+
(p
−
α)
(z ∈ E).
zp
1+z
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Noting that
z(Dn f (z))0
1−z
1
−2z
− α = (p − α) ·
1+
zp
1+z
p + 1 (p + 1)(1 − z 2 )
1 − z (p + 1) − (p + 1)z 2 − 2z
= (p − α) ·
1+z
(p + 1) − (p + 1)z 2
=0
√
−1+ 1+(p+1)2
for z =
, we complete the proof of Theorem 2.8.
p+1
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
Putting n = −1, p = 1 and 0 ≤ α < 1 in Theorem 2.8, we have the following:
Corollary 2.9. If Re f 0 (z) > α, then Re{zf 00 (z) + 2f 0 (z)} > α for |z| <
√
−1+ 5
.
2
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Theorem 2.10.
(a) If f ∈ Kn,p (A, B), then the function Fc defined by (1.2) belongs to Kn,p (A, B).
(b) f ∈ Kn,p (A, B) implies that Fc ∈ Kn,p (η(p, , c, A, B)) where

p2 F1 (1, p + c; p + c + 1; B)




− p(p+c)
A F (1, p + c + 1; p + c + 2; B), B 6= 0
η(p, c, A, B) =
p+c+1 2 1



 p − p(p+c) A,
B = 0.
p+c+1
Proof. Let
(2.15)
z(Dn Fc (z))0
φ(z) =
,
pz p
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where φ(z) is an analytic function with φ(0) = 1. Using the identity
z(Dn Fc (z))0n f (z) − cDn Fc (z)
(2.16)
in (2.15) and differentiating the resulting equation, we get
zφ0 (z)
z(Dn f (z))0
=
φ(z)
+
.
pz p
p+c
Since f ∈ Kn,p (A, B),
Oh Sang Kwon
0
φ(z) +
zφ (z)
1 + Az
<
.
p+c
1 + Bz
By Lemma 2.1, we obtain Fc (z) ∈ Kn,p (A, B). We deduce that
φ(z) < q(z) <
(2.17)
1 + Az
,
1 + Bz
where q(z) is given by (2.5) and is the best dominant of (2.17).
This proves part (a) of the theorem. Proceeding as in Theorem 2.10, part (b)
follows.
Putting A = 1 −
2α
p
Subclasses of p-Valently
Close-to-convex Functions
and B = −1 in Theorem 2.8, we have the following:
Corollary 2.11. If f ∈ Kn,p (A, B) for 0 ≤ α < p, then Fc ∈ Kn,p H(p, c, α), where
H(p, c, α) = p · 2 F1 (1, p + c; p + c + 1; −1)
p+c
−
(p − 2α)2 F1 (1, p + c; p + c + 1; −1).
p+c+1
Setting c = p = 1 in Theorem 2.10, we get the following result.
vol. 10, iss. 3, art. 83, 2009
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Corollary 2.12. If f ∈ Kn,p (α) for 0 ≤ α < 1, then the function
Z
2 z
G(z) =
f (t)dt
z 0
belongs to the class Kn (δ(α)), where δ(α) is given by (2.12).
Theorem 2.13. For any integer n and 0 ≤ α < p and c > −p, if Fc ∈ Kn,p
√(α) then
the function f defined by (1.1) belongs to Kn,p (α) for |z| < R(p, c) =
The result is best possible.
−1+
1+(p+c)2
.
p+c
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
Proof. Since Fc ∈ Kn,p (α), we write
(2.18)
z(Dn Fc )0
= α + (p − α)w(z),
zp
where w(z) is analytic, w(0) = 1 and Re w(z) > 0 in E. Using (2.16) in (2.18) and
differentiating the resulting equation, we obtain
z(Dn f (z))0
zw0 (z)
(2.19)
Re
− α = (p − α) Re w(z) +
.
zp
p+c
Now, by following the line of proof of Theorem 2.8, we get the assertion of Theorem
2.13.
Theorem 2.14. Let f ∈ Kn,p (A, B) and φ(z) ∈ Ap convex in E. Then
(f ∗ φ(z))(z) ∈ Kn,p (A, B).
Proof. Since f (z) ∈ Kn,p (A, B),
z(Dn f (z))0
1 + Az
<
.
p
pz
1 + Bz
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Now
z(Dn (f ∗ φ)(z))0
φ(z) ∗ z(Dn f )0
=
pz p ∗ φ(z)
φ(z) ∗ pz p
=
(2.20)
z(Dn f (z))0
pz p
pz p
.
φ(z) ∗ pz p
φ(z) ∗
Subclasses of p-Valently
Close-to-convex Functions
Then applying Lemma 2.3, we deduce that
n
0
φ(z) ∗ z(Dpzfp(z)) pz p
φ(z) ∗ pz p
Oh Sang Kwon
<
1 + Az
.
1 + Bz
vol. 10, iss. 3, art. 83, 2009
Hence (f ∗ φ(z))(z) ∈ Kn,p (A, B).
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Theorem 2.15. Let a function f (z) defined by (1.1) be in the class Kn,p (A, B). Then
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(2.21)
|ap+k | ≤
p(A − B)(p + k + 1)
(1 + p)n (p + k)
n
for k = 1, 2, . . . .
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The result is sharp.
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Proof. Since f (z) ∈ Kn,p (A, B), we have
n
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0
z(D f (z))
1 + Az
≡ φ(z) and φ(z) <
.
p
pz
1 + Bz
Hence
(2.22)
JJ
n
0p
z(D f (z)) φ(z) and
φ(z) = 1 +
∞
X
k=1
ck z k .
Close
From (2.22), we have
!0
n
∞ X
1+p
z(Dn f (z))0 = z z p +
ap+k z p+k
p+k+1
k=1
n
∞
X
1+p
p
(p + k)ap+k z p+k
= pz +
p
+
k
+
1
k=1
!
∞
X
= pz p 1 +
ck z k .
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
k=1
Therefore
(2.23)
1+p
p+k+1
By using Lemma 2.4 in (2.23),
n
1+p
p+k+1
n
(p + k)ap+k = pck .
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(p + k)|ap+k |
p
= |ck | ≤ A − B.
p(A − B)(p + k + 1)n
.
(1 + p)n (p + k)
The equality sign in (2.21) holds for the function f given by
|ap+k | ≤
pz p−1 + p(A − B − 1)z p
(Dn f (z))0 =
.
1−z
Hence
z(Dn f (z))0
1 + (A − B − 1)z
1 + Az
=
<
p
pz
1−z
1 + Bz
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Hence
(2.24)
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for k = 1, 2, . . . .
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The function f (z) defined in (2.24) has the power series representation in E,
p
f (z) = z +
∞
X
p(A − B)(p + k + 1)n
k=1
(1 + p)n (p + k)
z p+k .
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN, Hand Book of Mathematical Functions, Dover Publ. Inc., New York, (1971).
[2] V. ANH, k-fold symmetric starlike univalent function, Bull. Austrial Math. Soc.,
32 (1985), 419–436.
[3] T.M. FLETT, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl., 38 (1972), 746–765.
Subclasses of p-Valently
Close-to-convex Functions
Oh Sang Kwon
vol. 10, iss. 3, art. 83, 2009
[4] S.S. MILLER AND P.T. MOCANU, Differential subordinations and univalent
functions, Michigan Math. J., 28 (1981), 157–171.
Title Page
[5] J. PATEL AND P. SAHOO, Certain subclasses of multivalent analytic functions,
Indian J. Pure. Appl. Math., 34(3) (2003), 487–500.
[6] St. RUSCHEWEYH AND T. SHEIL-SMALL, Hadamard products of schlicht
functions and the Polya-Schoenberg conjecture, Comment Math. Helv., 48
(1973), 119–135.
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