Journal of Inequalities in Pure and
Applied Mathematics
ON A CERTAIN CLASS OF p−VALENT FUNCTIONS WITH
NEGATIVE COEFFICIENTS
volume 6, issue 4, article 97,
2005.
H.Ö. GÜNEY AND S. SÜMER EKER
University of Dicle, Faculty of Science & Art
Department of Mathematics
21280-Diyarbakır- TURKEY
Received 09 March, 2005;
accepted 04 October, 2005.
Communicated by: H.M. Srivastava
EMail: ozlemg@dicle.edu.tr
EMail: sevtaps@dicle.edu.tr
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
071-05
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Abstract
In this paper, we introduce the class A∗o (p, A, B, α ) of p-valent functions in the
unit disc U = { z : | z | < 1 }. We obtain coefficient estimate, distortion and closure theorems, radii of close-to convexity, starlikeness and convexity of order
δ ( 0 6 δ < 1 ) for this class. We also obtain class preserving integral operators for this class. Furthermore, various distortion inequalities for fractional
calculus of functions in this class are also given.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: p-valent, Coefficient, Distortion, Closure, Starlike, Convex, Fractional
calculus, Integral operators.
H.Ö. Güney and S. Sümer Eker
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Coefficient Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Distortion Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Radii of Close-To-Convexity, Starlikeness and Convexity . . .
5
Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Closure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Definitions and Applications of Fractional Calculus . . . . . . . .
References
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3
5
8
12
14
16
20
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1.
Introduction
Let A(n) be the class of functions f , analytic and p−valent in U = {z : |z| < 1}
given by
(1.1)
f (z) = z p +
∞
X
ap+n z p+n ,
ap+n > 0.
n=1
A function f belonging to the class A(n) is said to be in the class A∗m (p, A, B, α)
if and only if
(p)
zf (z)
(p − 1) + Re
> 0 for z ∈ U.
f (p−1) (z)
In the other words, f ∈ A∗m (p, A, B, α) if and only if it satisfies the condition
zf (p) (z)
−
p
(p
−
1)
+
f (p−1) (z)
h
i < 1
(p)
(A − B)(p − α) + pB − B (p − 1) + fzf(p−1)(z)
(z)
where −1 ≤ B < A ≤ 1, −1 ≤ B < 0 and 0 ≤ α < p. Let Am denote the
subclass of A(n) consisting of functions analytic and p−valent which can be
expressed in the form
(1.2)
f (z) = z p −
∞
X
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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ap+n z p+n ;
ap+n ≥ 0.
n=1
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Let us define
A∗o (p, A, B, α) = A∗m (p, A, B, α)
\
Am .
J. Ineq. Pure and Appl. Math. 6(4) Art. 97, 2005
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In this paper, we obtain a coefficient estimate, distortion theorems, integral operators and radii of close-to-convexity, starlikeness and convexity, closure properties and distortion inequalities for fractional calculus. This paper is motivated
by an earlier work of Nunokawa [1].
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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2.
Coefficient Estimates
Theorem 2.1. If the function f is defined by (1.1), then f ∈ A∗o (p, A, B, α) if
and only if
(2.1)
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
ap+n ≤ (A − B)(p − α)p!.
The result is sharp.
Proof. Assume that the inequality (2.1) holds true and let |z| = 1. Then we
obtain
(p)
zf (z) − f (p−1) (z) − (A − B)(p − α)f (p−1) − Bzf (p) + Bf (p−1) ∞
X n(p + n)!
n+1
ap+n z − (A − B)(p − α)p!z
= −
(n + 1)!
"n=1
#
∞
∞
X
X
(p + n)!
n(p
+
n)!
n+1
n+1 − (A − B)(p − α)
ap+n z
−B
ap+n z
(n + 1)!
(n + 1)!
n=1
n=1
≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
ap+n − (A − B)(p − α)p! ≤ 0
by hypothesis. Hence, by the maximum modulus theorem, we have f ∈
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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A∗o (p, A, B, α). To prove the converse, assume that
(p)
(p − 1) + fzf(p−1)(z)
−
p
(z)
i
h
zf (p) (z) (A − B)(p − α) + pB − B (p − 1) + f (p−1) (z) ∞
P
n(p+n)!
a z n+1
−
(n+1)! p+n
n=1
= ∞
∞
(A − B)(p − α) p!z − P (p+n)! a z n+1 + B P
p+n
(n+1)!
n=1
n=1
n(p+n)!
n+1 a
z
p+n
(n+1)!
< 1.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
Since Re(z) ≤ |z| for all z, we have
(2.2)


∞
P
n(p+n)!


n+1


−
a z


(n+1)! p+n
n=1
Re
∞
∞
P
P


n(p+n)!

n+1 +B
(A−B)(p−α) p!z− (p+n)!

a
z
a z n+1
p+n
(n+1)!
(n+1)! p+n
n=1
n=1
< 1.
Choosing values of z on the real axis and letting z → 1− through real values,
we obtain
(2.3)
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
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ap+n ≤ (A − B)(p − α)p!,
which obviously is required assertion (2.1). Finally, sharpness follows if we
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take
(2.4)
f (z) = z p −
(A − B)(p − α)p!(n + 1)!
z p+n .
(p + n)! [n(1 − B) + (A − B)(p − α)]
Corollary 2.2. If f ∈ A∗o (p, A, B, α), then
(2.5)
ap+n ≤
(A − B)(p − α)p!(n + 1)!
.
(p + n)! [n(1 − B) + (A − B)(p − α)]
The equality in (2.5) is attained for the function f given by (2.4).
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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3.
Distortion Properties
Theorem 3.1. If f ∈ A∗o (p, A, B, α), then for |z| = r < 1
(3.1) rp −
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
2(A − B)(p − α)
≤ |f (z)| ≤ rp +
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
and
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
2(A − B)(p − α)
(3.2) prp−1 −
rp
(1 − B) + (A − B)(p − α)
≤ |f 0 (z)| ≤ prp−1 +
2(A − B)(p − α)
rp .
(1 − B) + (A − B)(p − α)
H.Ö. Güney and S. Sümer Eker
Title Page
All the inequalities are sharp.
Contents
Proof. Let
f (z) = z p −
∞
X
JJ
J
ap+n z p+n , ap+n > 0.
n=1
From Theorem 2.1, we have
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∞
(p + 1)! [(1 − B) + (A − B)(p − α)] X
ap+n
2
n=1
≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
≤ (A − B)(p − α)p!
(n + 1)!
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ap+n
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which
(3.3)
∞
X
ap+n ≤
n=1
2(A − B)(p − α)
(p + 1) [(1 − B) + (A − B)(p − α)]
and
(3.4)
∞
X
(p + n)ap+n ≤
n=1
2(A − B)(p − α)
.
(1 − B) + (A − B)(p − α)
Consequently, for |z| = r < 1, we obtain
p
|f (z)| ≤ r + r
p+1
∞
X
H.Ö. Güney and S. Sümer Eker
ap+n
n=1
≤ rp +
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
|f (z)| ≥ r − r
p+1
∞
X
ap+n
n=1
≥ rp −
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On A Certain Class Of p−Valent
Functions With Negative
Coefficients
2(A − B)(p − α)
rp+1
(p + 1) [(1 − B) + (A − B)(p − α)]
which prove that the assertion (3.1) of Theorem 3.1 holds.
The inequalities in (3.2) can be proved in a similar manner and we omit the
details.
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The bounds in (3.1) and (3.2) are attained for the function f given by
(3.5)
f (z) = z p −
2(A − B)(p − α)
z p+1 .
(p + 1) [(1 − B) + (A − B)(p − α)]
Letting r → 1− in the left hand side of (3.1), we have the following:
Corollary 3.2. If f ∈ A∗o (p, A, B, α), then the disc |z| < 1 is mapped by f onto
a domain that contains the disc
|w| <
(p + 1)(1 − B) + (A − B)(p − α)(p − 1)
.
(p + 1) [(1 − B) + (A − B)(p − α)]
The result is sharp with the extremal function f being given by (3.5).
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
Putting α = 0 in Theorem 3.1 and Corollary 3.2, we get
Corollary 3.3. If f ∈ A∗o (p, A, B, 0), then for |z| = r
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2p(A − B)
rp −
rp+1
(p + 1) [(1 − B) + p(A − B)]
≤ |f (z)| ≤ rp +
2p(A − B)
rp+1
(p + 1) [(1 − B) + p(A − B)]
and
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prp−1 −
2p(A − B)
rp ≤ |f 0 (z)|
(1 − B) + p(A − B)
2p(A − B)
≤ prp−1 +
rp .
(1 − B) + p(A − B)
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The result is sharp with the extremal function
(3.6)
f (z) = z p −
2p(A − B)
z p+1 ;
(p + 1) [(1 − B) + p(A − B)]
z = ∓r.
Corollary 3.4. If f ∈ A∗o (p, A, B, 0), then the disc |z| < 1 is mapped by f onto
a domain that contains the disc
|w| <
(p + 1)(1 − B) + p(p − 1)(A − B)
.
(p + 1) [(1 − B) + p(A − B)]
The result is sharp with the extremal function f being given by (3.6).
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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4.
Radii of Close-To-Convexity, Starlikeness and
Convexity
Theorem 4.1. Let f ∈ A∗o (p, A, B, α). Then f is p−valent close-to-convex of
order δ (0 ≤ δ < p) in |z| < R1 , where
(
n1 )
(p + n)![n(1 − B) + (A − B)(p − α)] p − δ
(4.1) R1 = inf
.
n
(A − B)(p − α)(n + 1)p!
p+n
Theorem 4.2. If f ∈ A∗o (p, A, B, α), then f is p−valent starlike of order
δ (0 ≤ δ < p) in |z| < R2 , where
(
n1 )
p−δ
(p+n)![n(1−B)+(A−B)(p−α)]
(4.2) R2 = inf
.
n
(A − B)(p − α)(n + 1)!p!
p+n−δ
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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Theorem 4.3. If f ∈ A∗o (p, A, B, α), then f is a p−valent convex function of
order δ (0 ≤ δ < p) in |z| < R3 , where
(
n1 )
[n(1−B)+(A−B)(p−α)](p + n − 1)! p − δ
(4.3) R3 = inf
.
n
(A − B)(p − α)(n + 1)!(p − 1)!
p+n−δ
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In order to establish the required results in Theorems 4.1, 4.2 and 4.3, it is
sufficient to show that
0
f (z)
z p−1 − p ≤ p − δ for |z| < R1 ,
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0
zf (z)
≤p−δ
−
p
f (z)
00
1 + zf (z) − p ≤ p − δ
0
f (z)
for
|z| < R2
for
|z| < R3 ,
and
respectively.
Remark 1. The results in Theorems 4.1, 4.2 and 4.3 are sharp with the extremal function f given by (2.4). Furthermore, taking δ = 0 in Theorems 4.1,
4.2 and 4.3, we obtain radius of close-to-convexity, starlikeness and convexity,
respectively.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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5.
Integral Operators
Theorem 5.1. Let c be a real number such that c > −p. If f ∈ A∗o (p, A, B, α),
then the function F defined by
Z
c + p z c−1
(5.1)
F (z) =
t f (t)dt
zc 0
also belongs to A∗o (p, A, B, α).
Proof. Let
f (z) = z p −
∞
X
ap+n z p+n .
n=1
Then from the representation of F , it follows that
F (z) = z p −
∞
X
bp+n z p+n ,
n=1
c+p
c+p+n
where bp+n =
ap+n . Therefore using Theorem 2.1 for the coefficients
of F , we have
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
bp+n
(n + 1)!
n=1
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
c+p
=
ap+n
(n
+
1)!
c
+
p
+
n
n=1
≤ (A − B)(p − α)p!
since
c+p
c+p+n
< 1 and f ∈ A∗o (p, A, B, α). Hence F ∈ A∗o (p, A, B, α).
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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Theorem 5.2. Let c be a real number such that c > −p. If F ∈ A∗o (p, A, B, α),
then the function f defined by (5.1) is p−valent in |z| < R∗ , where
(5.2) R∗
(
n1 )
c+p
(p + n)! [n(1 − B) + (A − B)(p − α)]
p
= inf
.
n
c+p+n
(n + 1)!(A − B)(p − α)p!
p+n
The result is sharp. Sharpness follows if we take
c+p+n
(n + 1)!(A − B)(p − α)p!
p
f (z) = z −
z p+n .
c+p
(p + n)! [n(1 − B) + (A − B)(p − α)]
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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6.
Closure Properties
In this section we show that the class A∗o (p, A, B, α) is closed under “arithmetic
mean” and “convex linear combinations”.
Theorem 6.1. Let
fj (z) = z p −
∞
X
ap+n,j z p+n ,
j = 1, 2, ...
n=1
and
p
h(z) = z −
∞
X
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
bp+n z p+n ,
H.Ö. Güney and S. Sümer Eker
n=1
where
bp+n =
∞
X
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λj ap+n,j ,
λj > 0
Contents
j=1
P
∗
and ∞
j=1 λj = 1. If fj ∈ Ao (p, A, B, α) for each j = 1, 2, ..., then h ∈
∗
Ao (p, A, B, α).
Proof. If fj ∈ A∗o (p, A, B, α), then we have from Theorem 2.1 that
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
n=1
(n + 1)!
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ap+n,j
≤ (A − B)(p − α)p!,
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j = 1, 2, ....
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Therefore
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
bp+n
"
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
=
(n + 1)!
n=1
∞
X
!#
λj ap+n,j
j=1
≤ (A − B)(p − α)p!.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
Hence, by Theorem 2.1, h ∈ A∗o (p, A, B, α).
Theorem 6.2. The class A∗o (p, A, B, α) is closed under convex linear combinations.
Theorem 6.3. Let fp (z) = z p and
fp+n = z p −
Title Page
(A − B)(p − α)(n + 1)!p!
z p+n
(p + n)! [n(1 − B) + (A − B)(p − α)]
Contents
(n ≥ 1).
Then f ∈ A∗o (p, A, B, α) if and only if it can be expressed in the form
f (z) = λp fp (z) +
H.Ö. Güney and S. Sümer Eker
∞
X
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λn fp+n (z),
z ∈ U,
Close
n=1
Quit
where λn ≥ 0 and λp = 1 −
P∞
n=1 λn .
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Proof. Let us assume that
f (z) = λp fp (z) +
∞
X
λn fp+n (z)
n=1
= zp −
∞
X
n=1
(A − B)(p − α)(n + 1)!p!
λn z p+n .
(p + n)! [n(1 − B) + (A − B)(p − α)]
Then from Theorem 2.1 we have
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
×
H.Ö. Güney and S. Sümer Eker
(A − B)(p − α)(n + 1)!p!
λn
(p + n)! [n(1 − B) + (A − B)(p − α)]
≤ (A − B)(p − α)p!.
Hence f ∈ A∗o (p, A, B, α). Conversely, let f ∈ A∗o (p, A, B, α). It follows from
Corollary 2.2 that
ap+n ≤
(A − B)(p − α)(n + 1)!p!
.
(p + n)! [n(1 − B) + (A − B)(p − α)]
Setting
λn =
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(p + n)! [n(1 − B) + (A − B)(p − α)]
ap+n ,
(A − B)(p − α)(n + 1)!p!
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and λp = 1 −
P∞
n=1
λn , we have
p
∞
X
p
n=1
∞
X
f (z) = z −
=z −
ap+n z p+n
p
λn z +
n=1
−
∞
X
λn z p
n=1
∞
X
λn
n=1
∞
X
= λp fp (z) +
(A − B)(p − α)(n + 1)!p!
z p+n
(p + n)! [n(1 − B) + (A − B)(p − α)]
λn fp+n (z).
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
n=1
This completes the proof of Theorem 6.3.
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7.
Definitions and Applications of Fractional
Calculus
In this section, we shall prove several distortion theorems for functions to general class A∗o (p, A, B, α). Each of these theorems would involve certain operators of fractional calculus we find it to be convenient to recall here the following
definition which were used recently by Owa [2] (and more recently, by Owa and
Srivastava [3], and Srivastava and Owa [4] ; see also Srivastava et al. [5]).
Definition 7.1. The fractional integral of order λ is defined, for a function f ,
by
Z z
f (ζ)
1
−λ
(7.1)
Dz f (z) =
dζ (λ > 0),
Γ(λ) 0 (z − ζ)1−λ
where f is an analytic function in a simply – connected region of the z -plane
containing the origin, and the multiplicity of (z − ζ)λ−1 is removed by requiring
log(z − ζ) to be real when z − ζ > 0.
Definition 7.2. The fractional derivative of order λ is defined, for a function f ,
by
Z z
1
d
f (ζ)
λ
(7.2)
Dz f (z) =
dζ (0 ≤ λ < 1),
Γ(1 − λ) dz 0 (z − ζ)λ
where f is constrained, and the multiplicity of (z − ζ)−λ is removed, as in
Definition 7.1.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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Definition 7.3. Under the hypotheses of Definition 7.2, the fractional derivative
of order (n + λ) is defined by
dn λ
D f (z) (0 ≤ λ < 1),
dz n z
S
where 0 ≤ λ < 1 and n ∈ N0 = N {0}. From Definition 7.2, we have
Dzn+λ f (z) =
(7.3)
Dz0 f (z) = f (z)
(7.4)
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
which, in view of Definition 7.3 yields,
Dzn+0 f (z) =
(7.5)
dn 0
D f (z) = f n (z).
dz n z
H.Ö. Güney and S. Sümer Eker
Thus, it follows from (7.4) and (7.5) that
lim Dz−λ f (z) = f (z) and
λ→0
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lim Dz1−λ f (z) = f 0 (z).
λ→0
Theorem 7.1. Let the function f defined by (1.2) be in the class
Then for z ∈ U and λ > 0,
A∗o (p, A, B, α).
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Γ(p + 1)
Γ(λ + p + 1)
2(A − B)(p − α)Γ(p + 1)
−
|z|
(λ + p + 1)Γ(λ + p + 1) [(1 − B) + (A − B)(p − α)]
−λ
Dz f (z) ≥ |z|p+λ
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and
Γ(p + 1)
Γ(λ + p + 1)
2(A − B)(p − α)Γ(p + 1)
+
|z| .
(λ + p + 1)Γ(λ + p + 1) [(1 − B) + (A − B)(p − α)]
−λ
Dz f (z) ≤ |z|p+λ
The result is sharp.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
Proof. Let
Γ(p + 1 + λ) −λ −λ
z Dz f (z)
Γ(p + 1)
∞
X
Γ(p + n + 1)Γ(p + λ + 1)
ap+n z p+n
= zp −
Γ(p
+
1)Γ(p
+
n
+
λ
+
1)
n=1
F (z) =
p
=z −
∞
X
ϕ(n)ap+n z
p+n
,
ϕ(n) =
Γ(p + n + 1)Γ(p + λ + 1)
,
Γ(p + 1)Γ(p + n + λ + 1)
Then by using 0 < ϕ(n) ≤ ϕ(1) =
p+1
p+λ+1
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n=1
where
H.Ö. Güney and S. Sümer Eker
(λ > 0, n ∈ N).
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and Theorem 2.1, we observe that
∞
(p + 1)! [(1 − B) + (A − B)(p − α)] X
ap+n
2!
n=1
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≤
∞
X
(p + n)! [n(1 − B) + (A − B)(p − α)]
(n + 1)!
n=1
ap+n
≤ (A − B)(p − α)p!,
which shows that F (z) ∈ A∗o (p, A, B, α). Consequently, with the aid of Theorem 3.1, we have
|F (z)| ≥ |z p | − ϕ(1) |z|p+1
∞
X
ap+n
n=1
≥ |z|p −
2(A − B)(p − α)
|z|p+1
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
and
|F (z)| ≤ |z p | + ϕ(1) |z|p+1
∞
X
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ap+n
Contents
n=1
≤ |z|p +
2(A − B)(p − α)
|z|p+1
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
which completes the proof of Theorem 7.1.By letting λ → 0, Theorem 7.1
reduces at once to Theorem 3.1.
Corollary 7.2. Under the hypotheses of Theorem 7.1, Dz−λ f (z) is included in
a disk with its center at the origin and radius R1−λ given by
Γ(p + 1)
2(A − B)(p − α)
−λ
R1 =
1+
.
Γ(λ + p + 1)
(p + λ + 1)[(1 − B) + (A − B)(p − α)]
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Theorem 7.3. Let the function f defined by (1.2) be in the class A∗o (p, A, B, α).
Then,
Γ(p + 1)
Γ(p − λ + 1)
2(A − B)(p − α)Γ(2 − λ)Γ(p + 1)
−
|z|
Γ(p − λ + 1)Γ(p − λ + 2)[(1 − B) + (A − B)(p − α)]
λ
Dz f (z) ≥ |z|p−λ
and
Γ(p + 1)
Γ(p − λ + 1)
2(A − B)(p − α)Γ(2 − λ)Γ(p + 1)
+
|z|
Γ(p − λ + 1)Γ(p − λ + 2)[(1 − B) + (A − B)(p − α)]
λ
Dz f (z) ≤ |z|p−λ
for 0 ≤ λ < 1.
Proof. Using similar arguments as given by Theorem 7.1, we can get the result.
Corollary 7.4. Under the hypotheses of Theorem 7.3, Dzλ f (z) is included in the
disk with its center at the origin and radius R2λ given by
Γ(p + 1)
2(A − B)(p − α)Γ(2 − λ)
λ
R2 =
1+
.
Γ(λ + p + 1)
Γ(p − λ + 1)[(1 − B) + (A − B)(p − α)]
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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References
[1] M. NUNOKAWA, On the multivalent functions, Indian J. of Pure and
Appl.Math., 20(6) (1989), 577–582.
[2] S. OWA, On the distortion theorems, I., Kyunpook Math.J., 18 (1978 ), 53–
59.
[3] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.
[4] H.M. SRIVASTAVA AND S. OWA, Some characterization and distortion
theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Mathematics J., 106 (1987), 1–28.
[5] H.M. SRIVASTAVA, M. SAIGO AND S. OWA, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl.,
131 (1988), 412–420.
On A Certain Class Of p−Valent
Functions With Negative
Coefficients
H.Ö. Güney and S. Sümer Eker
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