ON CERTAIN SUBCLASSES OF
MEROMORPHICALLY MULTIVALENT FUNCTIONS
ASSOCIATED WITH THE GENERALIZED
HYPERGEOMETRIC FUNCTION
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
JAGANNATH PATEL
ASHIS KU. PALIT
Department of Mathematics
Utkal University, Vani Vihar
Bhubaneswar-751004, India
EMail: jpatelmath@yahoo.co.in
Department of Mathematics
Bhadrak Institute of Engineering and Technology
Bhadrak-756 113, India
EMail: ashis_biet@rediffmail.com
vol. 10, iss. 1, art. 13, 2009
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20 February, 2009
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Communicated by:
N.E. Cho
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2000 AMS Sub. Class.:
30C45
Key words:
Meromorphic function, p-valent, Subordination, Hypergeometric function,
Hadamard product.
Abstract:
In the present paper, we investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator involving the generalized hypergeometric function. Some interesting applications on Hadamard
product concerning this and other classes of integral operators are also considered.
Received:
12 September, 2008
Accepted:
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Contents
1
Introduction
3
2
Preliminaries
8
3
Main Results
11
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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1.
Introduction
For any integer m > 1 − p, let
f (z) = z
(1.1)
−p
P
+
p,m
∞
X
be the class of functions of the form:
ak z k
(p ∈ N = {1, 2, . . . }),
k=m
∗
which are analytic and p-valent in the
Ppunctured
Punit disk U = {z ∈ C : 0 <
|z| < 1} = U \ {0}. We also denote p,1−p = p . For 0 5 α < p, we denote by
P
P
P
P
S (p; α),
K (p; α) and
C (p; α), the subclasses of
p consisting of all meromorphic functions which are, respectively, p-valently starlike of order α, p-valently
convex of order α and p-valently close-to-convex of order α.
If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or
(more precisely) f (z) ≺ g(z) z ∈ U, if there exists a function ω, analytic in U with
ω(0) = 0 and |ω(z)| < 1 such that f (z) = g(ω(z)), z ∈ U. In particular, if g is
univalent in U, then we have the following equivalence:
f (z) ≺ g(z) (z ∈ U) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).
P
P
For a function f ∈
, given by (1.1) and g ∈
p,m
p,m defined by g(z) =
P∞
−p
k
z + k=m bk z , we define the Hadamard product (or convolution) of f and g by
f (z) ∗ g(z) = (f ∗ g)(z) = z −p +
∞
X
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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ak b k z k
(p ∈ N).
k=m
For real or complex numbers
α1 , α2 , . . . , αq
Meromorphically Multivalent
Functions
and β1 , β2 , . . . , βs βj ∈
/ Z−
0 = {0, −1, −2, . . . }; j = 1, 2, . . . , s ,
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we consider the generalized hypergeometric function q Fs (see, for example, [17])
defined as follows:
(1.2)
q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z)
=
∞
X
(α1 )k · · · (αq )k z k
k=0
(β1 )k · · · (βs )k k!
(q 5 s + 1; q, s ∈ N0 = N ∪ {0}; z ∈ U) ,
where (x)k denotes the Pochhammer symbol (or the shifted factorial) defined, in
terms of the Gamma function Γ, by
(
Γ(x + k)
x(x + 1)(x + 2) · · · (x + k − 1) (k ∈ N);
(x)k =
=
Γ(x)
1
(k = 0).
Meromorphically Multivalent
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Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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Corresponding to the function φp (α1 , . . . , αq ; β1 , . . . , βs ; z) given by
(1.3)
φp (α1 , . . . , αq ; β1 , . . . , βs ; z) = z −p q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z),
we introduce a function φp,µ (α1 , . . . , αq ; β1 , . . . , βs ; z) defined by
φp (α1 , . . . , αq ; β1 , . . . , βs ; z) ∗ φp,µ (α1 , . . . , αq ; β1 , . . . , βs ; z)
1
= p
(µ > −p; z ∈ U∗ ).
z (1 − z)µ+p
P
P
m,µ
We now define a linear operator Hp,q,s
(α1 , . . . , αq ; β1 , . . . , βs ) : p,m −→ p,m
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(1.4)
by
m,µ
(1.5) Hp,q,s
(α1 , . . . , αq ; β1 , . . . , βs )f (z) = φp,µ (α1 , . . . , αq ; β1 , . . . , βs ; z) ∗ f (z)
X
−
∗
; z∈U .
αi , βj ∈ C \ Z0 ; i = 1, 2 . . . , q; j = 1, 2, . . . , s; µ > −p; f ∈
p,m
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For convenience, we write
m,µ
m,µ
Hp,q,s
(α1 , . . . , αq ; β1 , . . . , βs ) = Hp,q,s
(α1 ) and
1−p,µ
µ
Hp,q,s
(α1 ) = Hp,q,s
(α1 ) (µ > −p).
If f is given by (1.1), then from (1.5), we deduce that
(1.6)
m,µ
Hp,q,s
(α1 )f (z)
=z
−p
∞
X
(µ + p)p+k (β1 )p+k · · · (βs )p+k
ak z k
+
(α
)
·
·
·
(α
)
1 p+k
q p+k
k=m
∗
(µ > −p; z ∈ U ).
and it is easily verified from (1.6) that
0
m,µ
m,µ+1
m,µ
(1.7) z Hp,q,s
(α1 )f (z) = (µ + p) Hp,q,s
(α1 )f (z) − (µ + 2p) Hp,q,s
(α1 )f (z)
and
(1.8)
0
m,µ
m,µ
m,µ
z Hp,q,s
(α1 + 1)f (z) = α1 Hp,q,s
(α1 )f (z) − (p + α1 )Hp,q,s
(α1 )f (z).
m,µ
We note that the linear operator Hp,q,s
(α1 ) is closely related to the Choi-SaigoSrivastava operator [5] for analytic functions and is essentially motivated by the
0,µ
operators defined and studied in [3]. The linear operator H1,q,s
(α1 ) was investigated
1−p
recently by Cho and Kim [2], whereas Hp,2,1 (c, 1; a; z) = Lp (a, c) (c ∈ R, a ∈
/ Z−
0)
is the operator studied in [7]. In particular, we have the following observations:
Z z
p
m,0
(i) Hp,s+1,s (p + 1, β1 , . . . , βs ; β1 , . . . , βs )f (z) = 2p
t2p−1 f (t) dt;
z
0
m,0
m,1
(ii) Hp,s+1,s
(p, β1 , ..., βs ; β1 , ..., βs )f (z) = Hp,s+1,s
(p + 1, β1 , ..., βs ; β1 , ..., βs )f (z)
= f (z);
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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m,1
(iii) Hp,s+1,s
(p, β1 , . . . , βs ; β1 , . . . , βs )f (z) =
zf 0 (z) + 2pf (z)
;
p
m,2
(iv) Hp,s+1,s
(p + 1, β1 , . . . , βs ; β1 , . . . , βs )f (z) =
zf 0 (z) + (2p + 1)f (z)
;
p+1
1
= Dn+p−1 f (z)
− z)n+p
(n is an integer > −p), the operator studied in [6], and
Z z
δ
m,1−p
(vi) Hp,s+1,s (δ + 1, β2 , . . . , βs , 1; δ, β2 , . . . , βs )f (z) = δ+p
tδ+p−1 f (t) dt
z
0
(δ > 0; z ∈ U∗ ), the integral operator defined by (3.6).
1−p,n
(v) Hp,s+1,s
(β1 , β2 , . . . , βs , 1; β1 , . . . , βs )f (z) =
z p (1
Let Ω be the class of all functions φ which are analytic, univalent in U and for
which φ(U) is convex with φ(0) = 1 and < {φ(z)} > 0 in U.
m,µ
Next, by making
use of the linear operator Hp,q,s
(α1 ), we introduce the following
P
subclasses of p,m .
P
Definition 1.1. A function f ∈ p,m is said to be in the class MS µ,m
p,α1 (q, s; η; φ), if
it satisfies the following subordination condition:
(
)
0
m,µ
z Hp,q,s
(α1 )f (z)
1
(1.9)
−
+ η ≺ φ(z)
m,µ
p−η
Hp,q,s
(α1 )f (z)
(φ ∈ Ω, 0 5 η < p, µ > −p; z ∈ U).
In particular, for fixed parameters A and B (−1 5 B < A 5 1), we set
1 + Az
µ,m
MS p,α1 q, s; η;
= MS µ,m
p,α1 (q, s; η; A, B) .
1 + Bz
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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It is easy to see that
MS µ,0
1,α1 (q, s; η; φ) = MS µ+1,α1 (q, s; η; φ) and
MS µ,0
1,α1 (q, s; η; A, B) = MS µ+1,α1 (q, s; η; A, B)
are the function classes studied by Cho and Kim [2].
P
Definition 1.2. For fixed parameters A and B, a function f ∈ p,m is said to be in
the class MC µ,m
p,α1 (q, s; λ; A, B), if it satisfies the following subordination condition:
m,µ
m,µ+1
z p+1 (1 − λ)(Hp,q,s
(α1 )f )0 (z) + λ(Hp,q,s
(α1 )f )0 (z)
1 + Az
(1.10)
−
≺
p
1 + Bz
(−1 5 B < A 5 1, λ = 0, µ > −p; z ∈ U) .
2η
To make the notation simple, we write MC µ,m
q,
s;
0;
1
−
,
−1
=
p,α1
p
P
µ,m
MC p,α1 (q, s; η), the class of functions f ∈ p,m satisfying the condition:
n
0 o
m,µ
−< z p+1 Hp,q,s
(α1 )f (z) > η (0 5 η < p; z ∈ U).
Meromorphically multivalent functions have been extensively studied by (for example) Liu and Srivastava [7], Cho et al. [4], Srivastava and Patel [18], Cho and
Kim [2], Aouf [1], Srivastava et al. [19] and others.
The object of the present paper is to investigate several inclusion relationships
and other interesting properties of certain subclasses of meromorphically multivam,µ
(α1 )
lent functions which are defined here by means of the linear operator Hp,q,s
involving the generalized hypergeometric function. Some interesting applications of
the Hadamard product concerning this and other classes of integral operators are also
considered. Relevant connections of the results presented here with those obtained
by earlier workers are also mentioned.
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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2.
Preliminaries
To prove our results, we need the following lemmas.
Lemma 2.1 ([8], see also [10]). Let the function h be analytic and convex(univalent)
in U with h(0) = 1. Suppose also that the function φ given by
φ(z) = 1 + cn z n + cn+1 z n+1 + · · ·
(2.1)
(n ∈ N)
is analytic in U. If
zφ0 (z)
≺ h(z) (<(κ) = 0, κ 6= 0; z ∈ U),
φ(z) +
κ
then
Z
κ − κ z κ −1
φ(z) ≺ q(z) = z n
t n h(t) dt ≺ h(z) (z ∈ U)
n
0
and q is the best dominant.
The following identities are well-known [21, Chapter 14].
Lemma 2.2. For real or complex numbers a, b, c (c ∈
/ Z−
0 ), we have
Z 1
(2.2)
tb−1 (1 − t)c−b−1 (1 − tz)−a dt
0
=
(2.3)
(2.4)
(2.5)
Γ(b)Γ(c − b)
2 F1 (a, b; c; z) (<(c) > <(b) > 0)
Γ(c)
2 F1 (a, b; c; z)
= 2 F1 (b, a; c; z)
−a
2 F1 (a, b; c; z) = (1 − z) 2 F1 a, c − b; c;
z
z−1
(b + 1)2 F1 (1, b; b + 1; z) = (b + 1) + bz 2 F1 (1, b + 1; b + 2; z)
Meromorphically Multivalent
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vol. 10, iss. 1, art. 13, 2009
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and
(2.6)
2 F1
1
1, 1; 2;
2
= 2 ln 2.
We denote by P(γ), the class of functions ψ of the form
(2.7)
ψ(z) = 1 + c1 z + c2 z 2 + · · · ,
which are analytic in U and satisfy the inequality:
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
<{ψ(z)} > γ
(0 5 γ < 1; z ∈ U).
It is known [20] that if fj ∈ P(γj ) (0 5 γj < 1; j = 1, 2), then
(2.8)
(f1 ∗ f2 )(z) ∈ P(γ3 )
(γ3 = 1 − 2(1 − γ1 )(1 − γ2 )) .
The result is the best possible.
We now state
Lemma 2.3 ([12]). If the function ψ, given by (2.7) belongs to the class P(γ), then
<{ψ(z)} = 2γ − 1 +
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2(1 − γ)
1 + |z|
(0 5 γ < 1; z ∈ U).
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2
Lemma 2.4 ([8, 10]). Let the function Ψ : C × U −→ C satisfy the condition
< {Ψ(ix, y; z)} 5 ε for ε > 0, all real x and y 5 −n(1 + x2 )/2, where n ∈ N. If φ
defined by (2.1) is analytic in U and < {Ψ (φ(z), zφ0 (z); z)} > ε, then <{φ(z)} > 0
in U.
We now recall the following result due to Singh and Singh [16].
Close
Lemma 2.5. Let the function Φ be analytic in U with Φ(0) = 1 and <{Φ(z)} > 1/2
in U. Then for any function F , analytic in U, (Φ ∗ F )(U) is contained in the convex
hull of F (U).
Lemma 2.6 ([13]). The function (1 − z)β = eβ log(1−z) , β 6= 0 is univalent in U, if
β satisfies either |β + 1| 5 1 or |β − 1| 5 1.
Lemma 2.7 ([9]). Let q be univalent in U, θ and Φ be analytic in a domain D
containing q(U) with Φ(w) 6= 0 when w ∈ q(U). Set Q(z) = zq 0 (z)φ(q(z)), h(z) =
θ(q(z)) + Q(z) and suppose that
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
0
(i) Q is starlike(univalent) in U with Q(0) = 0, Q (0) 6= 0 and
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(ii) Q and h satisfy
<
zh(z)
Q(z)
=<
Q0 (q(z)) zQ0 (z)
+
Φ(q(z))
Q(z)
Contents
> 0.
If φ is analytic in U with φ(0) = q(0), φ(U) ⊂ D and
(2.9) θ (φ(z)) + zφ0 (z)Φ (φ(z)) ≺ θ (q(z)) + zq 0 (z)Φ (q(z)) = h(z)
then φ ≺ q and q is the best dominant of (2.9).
(z ∈ U),
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3.
Main Results
Unless otherwise mentioned, we assume throughout the sequel that
α1 > 0, αi , βj ∈ R \ Z−
0 (i = 2, 3, . . . , q; j = 1, 2, . . . , s),
λ > 0, µ > −p and − 1 5 B < A 5 1.
Following the lines of proof of Cho and Kim [2] (see, also [4]), we can prove the
following theorem.
z∈U
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
Theorem 3.1. Let φ ∈ Ω with
max < {φ(z)} < min {(µ + 2p − η)/(p − η), (α1 + p − η)/(p − η)}
Meromorphically Multivalent
Functions
(0 5 η < p).
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Then
Contents
µ,m
MS µ+1,m
(q, s; η; φ) ⊂ MS µ,m
p,α1
p,α1 (q, s; η; φ) ⊂ MS p,α1 +1 (q, s; η; φ) .
By carefully choosing the function φ in the above theorem, we obtain the following interesting consequences.
Example 3.1. The function
α
1 + Az
φ(z) =
(0 < α 5 1; z ∈ U)
1 + Bz
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is analytic and convex univalent in U. Moreover,
α
α
1−A
1+A
05
< <{φ(z)} <
1−B
1+B
(0 < α 5 1, −1 < B < A 5 1; z ∈ U).
Thus, by Theorem 3.1 , we deduce that, if
α
1+A
µ + 2p − η α1 + p − η
< min
,
1+B
p−η
p−η
(0 < α 5 1, −1 < B < A 5 1),
then
MS µ+1,m
p,α1
(q, s; η; φ) ⊂
MS µ,m
p,α1
(q, s; η; φ) ⊂
MS µ,m
p,α1 +1
(q, s; η; φ) .
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
Example 3.2. The function
√ 2
2
1+ αz
√
φ(z) = 1 + 2 log
π
1− αz
vol. 10, iss. 1, art. 13, 2009
(0 < α < 1; z ∈ U)
is in the class Ω (cf. [14]) and satisfies
√ 2
2
1+ α
√
<{φ(z)} < 1 + 2 log
π
1− α
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(z ∈ U).
Thus, by using Theorem 3.1, we obtain that, if
√ 2
1+ α
2
µ + 2p − η α1 + p − η
√
,
1 + 2 log
< min
π
p−η
p−η
1− α
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(0 < α < 1),
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then
µ,m
MS µ+1,m
(q, s; η; φ) ⊂ MS µ,m
p,α1
p,α1 (q, s; η; φ) ⊂ MS p,α1 +1 (q, s; η; φ) .
Example 3.3. The function
∞ X
β+1
φ(z) = 1 +
αk z k
β+k
k=1
(0 < α < 1, β = 0; z ∈ U)
Close
belongs to the class Ω (cf. [15]) and satisfies
∞ X
β+1
<{φ(z)} < 1 +
αk
β
+
k
k=1
(0 < α < 1, β = 0).
Thus, by Theorem 3.1, if
∞ X
µ + 2p − η α1 + p − η
β+1
k
1+
α < min
,
β+k
p−η
p−η
k=1
(0 < α < 1, β = 0),
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
then
vol. 10, iss. 1, art. 13, 2009
MS µ+1,m
p,α1
(q, s; η; φ) ⊂
Theorem 3.2. If f ∈
MS µ,m
p,α1
(q, s; η; φ) ⊂
MS µ,m
p,α1 +1
(q, s; η; φ) .
MC µ,m
p,α1
(q, s; λ; A, B), then
0
m,µ
z p+1 Hp,q,s
(α1 )f (z)
1 + Az
(3.1)
−
≺ ψ(z) ≺
(z ∈ U),
p
1 + Bz
where the function ψ given by

µ+p
A
A
Bz
B
+ 1− B
(1 + Bz)−1 2 F1 1, 1; λ(p+m)
+ 1; 1+Bz
(B 6= 0);
ψ(z) =

(µ+p)A
1 + µ+p+λ(p+m)
z
(B = 0)
f ∈ MC µ,m
p,α1 (q, s; pρ) ,
ρ=

1−
(µ+p)A
µ+p+λ(p+m)
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where

µ+p
A
A
B
B
+ 1− B
(1 − B)−1 2 F1 1, 1; λ(p+m)
+ 1; B−1
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is the best dominant of (3.1). Further,
(3.2)
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(B 6= 0);
(B = 0).
The result is the best possible.
Proof. Setting
0
m,µ
z p+1 Hp,q,s
(α1 )f (z)
ϕ(z) = −
p
(3.3)
(z ∈ U),
we note that ϕ is of the form (2.1) and is analytic in U. Making use of the identity
(1.7) in (3.3) and differentiating the resulting equation, we get
0
(3.4)
ϕ(z)+
zϕ (z)
(µ + p)/λ
n
0
0 o
m,µ
m,µ+1
z p+1 (1 − λ) Hp,q,s
(α1 )f (z) + λ Hp,q,s
(α1 )f (z)
=−
p
1 + Az
≺
(z ∈ U).
1 + Bz
Now, by applying Lemma 2.1 (with κ = (µ + p)/λ) in (3.4), we deduce that
0
m,µ
z p+1 Hp,q,s
(α1 )f (z)
−
p
µ+p Z z
µ+p
µ + p − λ(p+m)
1 + Az
−1
λ(p+m)
≺ ψ(z) =
z
dt
t
λ(p + m)
1 + Bz
0

A
A
µ
+
p
Bz

−1


(B =
6 0)
 B + 1 − B (1 + Bz) 2 F1 1, 1; λ(p + m) + 1; 1 + Bz
=

(µ + p)A


z
(B = 0)
1 +
µ + p + λ(p + m)
Meromorphically Multivalent
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by a change of variables followed by the use of the identities (2.2), (2.3), (2.4) and
(2.5), respectively. This proves the assertion (3.3).
To prove (3.2), we follow the lines of proof of Theorem 1 in [18]. The result is
the best possible as ψ is the best dominant. This completes the proof of Theorem
3.2.
Setting A = 1 − (2η/p) , B = −1, µ = 0, m = 1 − p, α1 = λ =
p and αi+1 = βi (i = 1, 2, . . . , s) in Theorem 3.2 followed by the use of the
identity (2.6), we get
P
Corollary 3.3. If f ∈ p satisfies
−< z p+1 ((p + 2)f 0 (z) + zf 00 (z)) > η (0 5 η < p; z ∈ U),
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
Title Page
then
Contents
−<{z
p+1 0
f (z)} > η + 2(p − η)(ln 2 − 1) (z ∈ U).
The result is the best possible.
Putting A = 1 − (2η/p) , B = −1, µ = 0, m = 2 − p, α1 = λ =
p and αi+1 = βi (i = 1, 2, . . . , s) in Theorem 3.2, we obtain the following result
due to Pap [11].
P
Corollary 3.4. If f ∈ p,2−p satisfies
p(π − 2)
−< z p+1 ((p + 2)f 0 (z) + zf 00 (z)) > −
4−π
then
−<{z p+1 f 0 (z)} > 0
The result is the best possible.
(z ∈ U).
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(z ∈ U),
The proof of the following result is much akin to that of Theorem 2 in [18] and
we choose to omit the details.
Theorem 3.5. If f ∈ MC µ,m
p,α1 (q, s; η) (0 5 η < p), then
h
n
0
0 oi
m,µ
m,µ+1
−< z p+1 (1 − λ) Hp,q,s
(α1 )f (z) + λ Hp,q,s
(α1 )f (z) > η
Meromorphically Multivalent
Functions
(|z| < R(p, µ, λ, m)),
Jagannath Patel and Ashis Ku. Palit
where
1
"p
# p+m
(µ + p)2 + λ2 (p + m)2 − λ(p + m)
R(p, µ, λ, m) =
.
µ+p
vol. 10, iss. 1, art. 13, 2009
Title Page
Contents
The result is the best possible.
m,µ
Upon replacing ϕ(z) by z p Hp,q,s
(α1 )f (z) in (3.3) and using the same techniques
as in the proof of Theorem 3.2, we get the following result.
P
Theorem 3.6. If f ∈ p,m satisfies
1 + Az
m,µ
m,µ+1
z p (1 − λ) Hp,q,s
(α1 )f (z) + λ Hp,q,s
(α1 )f (z) ≺
1 + Bz
then
m,µ
z p Hp,q,s
(α1 )f (z) ≺ ψ(z) ≺
1 + Az
1 + Bz
(z ∈ U),
(z ∈ U)
and
m,µ
< z p Hp,q,s
(α1 )f (z) > ρ
(z ∈ U),
where ψ and ρ are given as in Theorem 3.2. The result is the best possible.
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Letting
A = 2 F1 1, 1;
p
1
+ 1;
λ(p + m)
2
−1
2 − 2 F1 1, 1;
1
p
+ 1;
λ(p + m)
2
−1
B = −1, µ = 0, α1 = p and αi+1 = βi (i = 1, 2, . . . , s) in Theorem 3.6, we
obtain
P
Corollary 3.7. If f ∈ p,m satisfies
p
λ(p+m)
1
2
3 − 2 2 F1 1, 1;
+ 1;
o
> n
p
2 2 −2 F1 1, 1; λ(p+m)
+ 1; 12
<{z p f (z)} >
Jagannath Patel and Ashis Ku. Palit
1
2
Title Page
(z ∈ U),
(z ∈ U).
Fδ,p (z) = Fδ,p (f )(z)
Z z
δ
= δ+p
tδ+p−1 f (t) dt
z
0
!
∞
X
δ
= z −p +
z k ∗ f (z)
δ+p+k
k=m
Contents
JJ
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Page 17 of 33
The result is the best possible.
P
For a function f ∈ p,m , we consider the integral operator Fδ,p defined by
(3.6)
Meromorphically Multivalent
Functions
vol. 10, iss. 1, art. 13, 2009
λ p+1 0
(3.5) < (1 + λ)f (z) + z f (z)
p
then
,
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(δ > 0; z ∈ U∗ ).
P
It follows from (3.6) that Fδ,p (f ) ∈ p,m and
0
m,µ
m,µ
m,µ
(3.7) z Hp,q,s
(α1 )Fδ,p (f ) (z) = δ Hp,q,s
(α1 )f (z)−(δ+p) Hp,q,s
(α1 )Fδ,p (f )(z).
Using (3.7) and the lines of proof of Theorem 1 [2], we obtain the following
inclusion relation.
Theorem 3.8. Let φ ∈ Ω with maxz∈U < {φ(z)} < (δ + p − η)/(p − η) (0 5 η <
µ,m
p; δ > 0). If f ∈ MS µ,m
p,α1 (q, s; η; φ), then Fδ,p (f ) ∈ MS p,α1 (q, s; η; φ).
P
Theorem 3.9. If f ∈ p,m and the function Fδ,p (f ), defined by (3.6) satisfies
n
0
0 o
m,µ
m,µ
z p+1 (1 − λ) Hp,q,s
(α1 )Fδ,p (f ) (z) + λ Hp,q,s
(α1 )f (z)
1 + Az
−
≺
(z ∈ U),
p
1 + Bz
then
(
−<
)
0
m,µ
z p+1 Hp,q,s
(α1 )Fδ,p (f ) (z)
>%
p
(z ∈ U),
where
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
Title Page
Contents
JJ
II
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Page 18 of 33
%=

A
A
δ
B
B
+ 1− B
(1 − B)−1 2 F1 1, 1; λ(p+m)
+ 1; B−1

1−
δA
µ+p+λ(p+m)
(B 6= 0)
(B = 0).
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The result is the best possible.
Proof. If we let
(3.8)
0
m,µ
z p+1 Hp,q,s
(α1 )Fδ,p (f ) (z)
ϕ(z) = −
p
(z ∈ U),
then ϕ is of the form (2.1) and is analytic in U. Using the identity (3.7) in (3.8)
followed by differentiation of the resulting equation, we get
ϕ(z) +
zϕ0 (z)
1 + Az
≺
δ/λ
1 + Bz
(z ∈ U).
The proof of the remaining part follows by employing the techniques that proved
Theorem 3.2.
Upon setting A = 1 − (2η/p) , B = −1, λ = µ = 1, α1 = p + 1 and αi+1 =
βi (i = 1, 2, . . . , s) in Theorem 3.9, we have
P
Corollary 3.10. If f ∈ P
C (p; η) (0 5 η < p), then the function Fδ,p (f ) defined by
(3.6) belongs to the class C (p; κ), where
δ
1
κ = η + (p − η) 2 F1 1, 1;
+ 1;
−1 .
p+m
2
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
Title Page
Contents
JJ
II
The result is the best possible.
J
I
Remark 1. Under the hypothesis of Theorem 3.9 and using the fact that
Z z
0
0
δ
p+1
m,µ
m,µ
z
Hp,q,s (α1 )Fδ,p (f ) (z) = δ
tδ+p Hp,q,s
(α1 )f (t) dt (δ > 0; z ∈ U),
z 0
Page 19 of 33
we obtain
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−<
δ
zδ
Z
z
tδ+p
0
m,µ
Hp,q,s (α1 )f (t) dt > %
(z ∈ U),
0
where % is given as in Theorem 3.9.
Following the same lines of proof as in Theorem 3.9, we obtain
Theorem 3.11. If f ∈
P
p,m
and the function Fδ,p (f ) defined by (3.6) satisfies
1 + Az
m,µ
m,µ
z p (1 − λ) Hp,q,s
(α1 )Fδ,p (f )(z) + λ Hp,q,s
(α1 )f (z) ≺
1 + Bz
(z ∈ U),
then
m,µ
< z p Hp,q,s
(α1 )Fδ,p (f )(z) > %
(z ∈ U),
Meromorphically Multivalent
Functions
where % is given as in Theorem 3.9. The result is the best possible.
In the special case when A = 1 − 2η, B = −1, λ = 1, µ = 1 − p, α1 =
δ + 1, β1 = δ, αi = βi (i = 2, 3, . . . , s) and αs+1 = 1 in Theorem 3.11, we get
P
Corollary 3.12. If f ∈ p,m satisfies
<{z p f (z)} > η
<
δ
zδ
Z
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Contents
z
δ+p−1
t
f (t) dt > η + (1 − η) 2 F1 1, 1;
0
δ
1
+ 1;
p+m
2
−1
II
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I
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The result is the best possible.
Theorem 3.13. Let −1 5 Bj < Aj 5 1 (j = 1, 2). If fj ∈
following subordination condition:
JJ
Page 20 of 33
(δ > 0; z ∈ U).
(3.9)
vol. 10, iss. 1, art. 13, 2009
(0 5 η < 1; z ∈ U),
then
Jagannath Patel and Ashis Ku. Palit
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P
p
1 + Aj z
µ
µ+1
z p (1 − λ)Hp,q,s
(α1 )fj (z) + λHp,q,s
(α1 )fj (z) ≺
1 + Bj z
(j = 1, 2; z ∈ U),
satisfies the
Close
then
(3.10)
µ
µ+1
< z p (1 − λ)Hp,q,s
(α1 )g(z) + λHp,q,s
(α1 )g(z) > τ
(z ∈ U),
where
µ
g(z) = Hp,q,s
(α1 )(f1 ∗ f2 )(z)
(3.11)
(z ∈ U∗ )
and
Meromorphically Multivalent
Functions
τ =1−
4(A1 − B1 )(A2 − B2 )
(1 − B1 )(1 − B2 )
1−
1
µ+p
1
+ 1;
2 F1 1, 1;
2
λ
2
.
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
The result is the best possible when B1 = B2 = −1.
Title Page
Proof. Setting
(3.12)
µ
µ+1
ϕj (z) = z p (1 − λ)Hp,q,s
(α1 )fj (z) + λHp,q,s
(α1 )fj (z)
(j = 1, 2; z ∈ U),
we note that ϕj is of the form (2.7) for each j = 1, 2 and using (3.9), we obtain
1 − Aj
ϕj ∈ P(γj )
γj =
; j = 1, 2
1 − Bj
so that by (2.8),
(3.13)
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ϕ1 ∗ ϕ2 ∈ P(γ3 )
(γ3 = 1 − 2(1 − γ1 )(1 − γ2 )) .
Using the identity (1.7) in (3.12), we conclude that
Z
µ + p −p− µ+p z µ+p −1
µ
λ
z
t λ
ϕj (t)dt
Hp,q,s (α1 )fj (z) =
λ
0
(j = 1, 2; z ∈ U∗ )
Close
which, in view of (3.11) yields
µ
Hp,q,s
(α1 )g(z)
µ + p −p− µ+p
λ
=
z
λ
Z
z
t
µ+p
−1
λ
ϕ0 (t)dt
(z ∈ U∗ ),
0
where, for convenience
(3.14)
µ
µ+1
ϕ0 (z) = z p (1 − λ)Hp,q,s
(α1 )g(z) + λHp,q,s
(α1 )g(z)
Z
µ + p − µ+p z µ+p −1
=
t λ
(ϕ1 ∗ ϕ2 )(t) dt (z ∈ U).
z λ
λ
0
Now, by using (3.13) in (3.14) and by appealing to Lemma 2.3 and Lemma 2.5, we
get
Z
µ + p 1 µ+p −1
<{ϕ0 (z)} =
s λ
<(ϕ1 ∗ ϕ2 )(sz) ds
λ
0
Z
µ + p 1 µ+p −1
2(1 − γ3 )
2γ3 − 1 +
=
s λ
ds
λ
1 + s|z|
0
Z
µ + p 1 µ+p −1
2(1 − γ3 )
s λ
ds
>
2γ3 − 1 +
λ
1+s
0
Z
4(A1 − B1 )(A2 − B2 )
µ + p 1 µ+p −1
−1
=1−
1−
s λ
(1 + s) ds
(1 − B1 )(1 − B2 )
λ
0
1
µ+p
1
4(A1 − B1 )(A2 − B2 )
1 − 2 F1 1, 1;
+ 1;
=1−
(1 − B1 )(1 − B2 )
2
λ
2
= τ (z ∈ U).
This proves the assertion (3.10).
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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Page 22 of 33
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P
When B1 = B2 = −1, we consider the functions fj ∈ p defined by
Z
µ + p −p− µ+p z µ+p −1 1 + Aj t
µ
λ
Hp,q,s (α1 )fj (z) =
z
t λ
dt
λ
1−t
0
(j = 1, 2; z ∈ U∗ ).
Then it follows from (3.14) that and Lemma 2.2 that
Meromorphically Multivalent
Functions
ϕ0 (z)
µ+p
λ
Jagannath Patel and Ashis Ku. Palit
Z
1
(1 + A1 )(1 + A2 )
ds
1 − sz
0
µ+p
z
−1
= 1 − (1 + A1 )(1 + A2 ) + (1 + A1 )(1 + A2 )(1 − z) 2 F1 1, 1;
+ 1;
λ
z−1
1
µ+p
1
−→ 1 − (1 + A1 )(1 + A2 ) + (1 + A1 )(1 + A2 ) 2 F1 1, 1;
+ 1;
as z → 1− ,
2
λ
2
=
s
µ+p
−1
λ
vol. 10, iss. 1, art. 13, 2009
1 − (1 + A1 )(1 + A2 ) +
which evidently completes the proof of Theorem 3.13.
By taking Aj = 1 − 2ηj , Bj = −1 (j = 1, 2), µ = 0, α1 = p and αi+1 =
βi (i = 1, 2, . . . , s) in Theorem 3.13, we get the following result which refines the
corresponding work of Yang [22, Theorem 4].
P
Corollary 3.14. If each of the functions fj ∈ p satisfies
λ 0
p
< z (1 + λ) fj (z) + zfj (z)
> ηj
p
(0 5 ηj < 1, j = 1, 2; z ∈ U),
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then
λ
p
0
< z (1 + λ) (f1 ∗ f2 )(z) + z(f1 ∗ f2 ) (z)
>σ
p
where
(z ∈ U),
1
p
1
σ = 1 − 4(1 − η1 )(1 − η2 ) 1 − 2 F1 1, 1; + 1;
.
2
λ
2
The result is the best possible.
Meromorphically Multivalent
Functions
For Aj = 1 − 2ηj , Bj = −1 (j = 1, 2), µ = 0, λ = 1, α1 = p + 1 and
αi+1 = βi (i = 1, 2, . . . , s) in Theorem 3.13, we obtain
P
Corollary 3.15. If each of the functions fj ∈ p satisfies
p
< {z fj (z)} > ηj
vol. 10, iss. 1, art. 13, 2009
Title Page
(0 5 ηj < 1, j = 1, 2; z ∈ U),
Contents
then
1
1
p
< {z (f1 ∗ f2 )(z)} > 1−4(1−η1 )(1−η2 ) 1 − 2 F1 1, 1; p + 1;
2
2
(z ∈ U).
Theorem
P 3.16. Let −1 5 Bj < Aj 5 1 (j = 1, 2). If each of the functions
fj ∈ p,m satisfies
m,µ+1
z p Hp,q,s
(α1 )fj (z) ≺
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Page 24 of 33
The result is the best possible.
(3.15)
Jagannath Patel and Ashis Ku. Palit
1 + Aj z
1 + Bj z
(j = 1, 2; z ∈ U),
m,µ
then the function h = Hp,q,s
(α1 )(f1 ∗ f2 ) satisfies
m,µ+1
Hp,q,s (α1 )h(z)
<
>0
m,µ
Hp,q,s
(α1 )h(z)
(z ∈ U),
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provided
(3.16)
(A1 − B1 )(A2 − B2 )
(1 − B1 )(1 − B2 )
<
2µ + 3p + m
.
n
o2
µ+p 1
2 (p + m) 2 F1 1, 1; p+m ; 2 − 2 + 2(µ + p)
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
Proof. From (3.15), we have
vol. 10, iss. 1, art. 13, 2009
m,µ+1
z p Hp,q,s
(α1 )fj (z) ∈ P(γj )
Thus, it follows from (2.8) that
(
(3.17)
m,µ+1
< z p Hp,q,s
(α1 )h(z) +
1 − Aj
γj =
; j = 1, 2 .
1 − Bj
Contents
0
m,µ+1
z z p Hp,q,s
(α1 )h (z)
µ+p
)
m,µ+1
m,µ+1
= < z p Hp,q,s
(α1 )f1 (z) ∗ z p Hp,q,s
(α1 )f2 (z)
2(A1 − B1 )(A2 − B2 )
(z ∈ U),
>1−
(1 − B1 )(1 − B2 )
which in view of Lemma 2.1 for
(A1 − B1 )(A2 − B2 )
,
(1 − B1 )(1 − B2 )
B = −1, n = p + m and κ = µ + p
A = −1 + 4
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yields
m,µ+1
(3.18) < z p Hp,q,s
(α1 )h(z)
(A1 − B1 )(A2 − B2 )
µ+p 1
>1+
− 2 (z ∈ U).
;
2 F1 1, 1;
(1 − B1 )(1 − B2 )
p+m 2
From (3.18), by using Theorem 3.6 for
(A1 − B1 )(A2 − B2 )
(1 − B1 )(1 − B2 )
B = −1 and λ = 1,
A = −1 − 4
2 F1
1, 1;
µ+p 1
;
p+m 2
−2 ,
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
we deduce that
Title Page
(3.19) < {z p ϑ(z)}
Contents
(A1 − B1 )(A2 − B2 )
>1−2
(1 − B1 )(1 − B2 )
where ϑ(z) = z
p
m,µ
Hp,q,s
(α1 )h(z).
ϕ(z) =
2 F1
µ+p 1
1, 1;
;
p+m 2
2
−2
(z ∈ U),
If we set
m,µ+1
Hp,q,s
(α1 )h(z)
m,µ
Hp,q,s (α1 )h(z)
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Page 26 of 33
(z ∈ U),
then ϕ is of the form (2.1), analytic in U and a simple calculation gives
0
m,µ+1
z z p Hp,q,s
(α1 )h (z)
p m,µ+1
(3.20) z Hp,q,s (α1 )h(z) +
µ+p
zϕ0 (z)
2
= Ψ (ϕ(z), zϕ0 (z); z) (z ∈ U),
= ϑ(z) (ϕ(z)) +
µ+p
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where Ψ(u, v; z) = ϑ(z) {u2 + (v/(µ + p))}. Thus, by applying (3.17) in (3.20),
we get
< {Ψ (ϕ(z), zϕ0 (z); z)} > 1 − 2
(A1 − B1 )(A2 − B2 )
(1 − B1 )(1 − B2 )
(z ∈ U).
Now, for all real x, y 5 −(p + m)(1 + x2 )/2, we have
y
2
< {Ψ(ix, y; z)} =
− x <{ϑ(z)}
µ+p
2(µ + p) 2
p+m
2
1+x +
x <{ϑ(z)}
5−
2(µ + p)
p+m
p+m
(A1 − B1 )(A2 − B2 )
5−
<{ϑ(z)} 5 1 − 2
2(µ + p)
(1 − B1 )(1 − B2 )
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
Title Page
(z ∈ U),
by (3.16) and (3.19). Thus, by Lemma 2.4, we get <{ϕ(z)} > 0 in U. This completes the proof of Theorem 3.16.
Taking Aj = 1 − 2ηj , Bj = −1 (j = 1, 2), µ = 0, λ = 1, α1 = p + 1 and
αi+1 = βi (i = 1, 2, . . . , s) in Theorem 3.16, we have
P
Corollary 3.17. If each of the functions fj ∈ p,m satisfies
p
<{z fj (z)} > ηj
then
<
(0 5 ηj < 1, j = 1, 2; z ∈ U),
z 2p (f1 ∗ f2 )(z)
Rz
t2p−1 (f1 ∗ f2 )(t) dt
0
>0
(z ∈ U),
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provided
(1 − η1 )(1 − η2 ) <
3p + m
n
.
o2
p
1
2 (p + m) 2 F1 1, 1; p+m ; 2 − 2 + 2p
Theorem 3.18. If f ∈ MC µ,m
p,α1 (q, s; λ; A, B) and g ∈
µ,m
f ∗ g ∈ MC p,α1 (q, s; λ; A, B).
P
p,m
satisfies (3.5), then
Proof. From Corollary 3.7, it follows that <{g(z)} > 1/2 in U. Since
m,µ
m,µ+1
z p+1 (1 − λ)(Hp,q,s
(α1 )(f ∗ g))0 (z) + λ(Hp,q,s
(α1 )(f ∗ g))0 (z)
−
p
p+1
m,µ
0
m,µ+1
z
(1 − λ)Hp,q,s (α1 )f ) (z) + λ(Hp,q,s
(α1 )f )0 (z)
=
∗ g(z) (z ∈ U)
p
and the function (1 + Az)/(1 + Bz) is convex(univalent) in U, the assertion of the
theorem follows from (1.10) and Lemma 2.5.
Theorem 3.19. Let 0 P
6= β ∈ C and 0 < γ 5 p be such that either |1 + 2βγ| 5 1 or
|1 − 2βγ| 5 1. If f ∈ p satisfies
µ+1
Hp,q,s (α1 )f (z)
γ
(3.21)
<
<1+
(z ∈ U),
µ
Hp,q,s (α1 )f (z)
µ+p
then
β
µ
z p Hp,q,s
(α1 )f (z) ≺ q(z) = (1 − z)2βγ
and q is the best dominant.
(z ∈ U)
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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Proof. Letting
(3.22)
β
µ
ϕ(z) = z p Hp,q,s
(α1 )f (z)
(z ∈ U)
and choosing the principal branch in (3.22), we note that ϕ is analytic in U with
ϕ(0) = 1. Differentiating (3.22) logarithmically, we deduce that
(
)
0
µ
z Hp,q,s
(α1 )f (z)
zϕ0 (z)
(z ∈ U),
=β p+
µ
ϕ(z)
Hp,q,s
(α1 )f (z)
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
which in view of the identities (1.7) and (3.21) give
2γ
1− 1−
z
zϕ0 (z)
p
≺ −p
(3.23)
−p +
1−z
β ϕ(z)
Title Page
(z ∈ U).
Contents
If we take q(z) = (1 − z)2βγ , θ(z) = −p, Φ(z) = 1/βz in Lemma 3.11, then by
Lemma 2.6, q is univalent in U. Further, it is easy to see that q, θ and Φ satisfy the
hypothesis of Lemma 2.7. Since
Q(z) = zq 0 (z)Φ(q(z)) = −
2γ z
1−z
−p + (p − 2γ)z
1−z
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is starlike (univalent) in U,
h(z) =
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and <
0
zh (z)
Q(z)
=<
1
1−z
>0
(z ∈ U),
it is readily seen that the conditions (i) and (ii) of Lemma 2.7 are satisfied. Thus, the
assertion of the theorem follows from (3.23) and Lemma 2.7.
Putting µ = 0, γ = p(1 − η), β = −1/2γ, α1 = p and αi+1 = βi (i =
1, 2, . . . , s) in Theorem 3.19, we deduce that
P
Corollary 3.20. If f ∈ p satisfies
0 zf (z)
> p η (0 5 η < 1; z ∈ U),
−<
f (z)
then
Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
1
−
1
< {z p f (z)} 2p(1 − η) >
2
The result is the best possible.
vol. 10, iss. 1, art. 13, 2009
(z ∈ U).
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References
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Meromorphically Multivalent
Functions
Jagannath Patel and Ashis Ku. Palit
vol. 10, iss. 1, art. 13, 2009
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Functions
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vol. 10, iss. 1, art. 13, 2009
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Meromorphically Multivalent
Functions
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vol. 10, iss. 1, art. 13, 2009
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