A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator
We introduce a subclass of p-valent meromorphic functions involving the Lui–Srivastava operator and investigate various properties of this subclass. We also indicate the relationships between various results presented in the paper and the results obtained in earlier works.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508164910219264 |
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| author | Seoudy, T. M. Суді, Т. М. |
| author_facet | Seoudy, T. M. Суді, Т. М. |
| author_sort | Seoudy, T. M. |
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| description | We introduce a subclass of p-valent meromorphic functions involving the Lui–Srivastava operator and investigate various properties of this subclass. We also indicate the relationships between various results presented in the paper and the results obtained in earlier works. |
| first_indexed | 2026-03-24T02:20:52Z |
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UDC 517.9
T. M. Seoudy (Fayoum. Univ., Egypt),
M. K. Aouf (Mansoura Univ., Egypt)
A CLASS OF p-VALENT MEROMORPHIC FUNCTIONS
DEFINED BY THE LUI – SRIVASTAVA OPERATOR
ПРО КЛАС p-ВАЛЕНТНИХ МЕРОМОРФНИХ ФУНКЦIЙ,
ЩО ВИЗНАЧЕНI ОПЕРАТОРОМ ЛУI – ШРIВАСТАВИ
We introduce а subclass of p-valent meromorphic functions involving the Lui – Srivastava operator and investigate various
properties of this subclass. We also indicate the relationships between various results presented in the paper with the results
obtained in earlier works.
Введено пiдклас p-валентних мероморфних функцiй, що визначаються оператором Луi – Шрiвастави, та вивчено
рiзноманiтнi властивостi цього пiдкласу. Також вказано спiввiдношення мiж рiзноманiтними результатами, що
отриманi в роботi, та результатами, отриманими ранiше.
1. Introduction. Let Σp denote the class of all meromorphic functions f of the form
f(z) = z−p +
∞∑
k=1
akz
k−p, p ∈ N = {1, 2, . . .}, (1.1)
which are analytic and p-valent in the punctured disc U∗ = {z ∈ C : 0 < |z| < 1} = U\{0}. Let
ΣS∗p (λ) denote the class of all meromorphic p-valent starlike of order λ (0 ≤ λ < p) in U.
For functions f ∈ Σp, given by (1.1), and g ∈ Σp defined by
g(z) = z−p +
∞∑
k=1
bkz
k−p, p ∈ N,
then the Hadamard product (or convolution) of f and g is given by
(f ∗ g) = z−p +
∞∑
k=1
akbkz
k−p = (g ∗ f)(z).
For complex parameters α1, . . . , αq and β1, . . . , βs (βj /∈ Z−0 = {0,−1,−2, . . .}; j = 1, 2, . . . , s),
we now define the generalized hypergeometric function qFs(α1, . . . , αq;β1, . . . , βs; z) by (see, for
example, [9, p.19])
qFs(α1, . . . , αq;β1, . . . , βs; z) =
∞∑
k=0
(α1)k . . . (αq)k
(β1)k . . . (βs)k
zk
k!
,
q ≤ s+ 1, q, s ∈ N0 = N ∪ {0}, z ∈ U,
where (θ)ν is the Pochhammer symbol defined, in terms of the Gamma function Γ, by
(θ)ν =
Γ(θ + ν)
Γ(θ)
=
1, ν = 0, θ ∈ C∗ = C\{0},
θ(θ − 1) . . . (θ + ν − 1), ν ∈ N, θ ∈ C.
c© T. M. SEOUDY, M. K. AOUF, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9 1235
1236 T. M. SEOUDY, M. K. AOUF
Corresponding to the function hp(α1, . . . , αq;β1, . . . , βs; z), defined by
hp(α1, . . . , αq;β1, . . . , βs; z) = z−p qFs(α1, . . . , αq;β1, . . . , βs; z),
we consider a linear operator
Hp(α1, . . . , αq;β1, . . . , βs) : Σp → Σp,
which is defined by the following Hadamard product (or convolution):
Hp(α1, . . . , αq;β1, . . . , βs)f(z) = hp(α1, . . . , αq;β1, . . . , βs; z) ∗ f(z).
We observe that, for a function f(z) of the form (1.1), we have
Hp(α1, . . . , αq;β1, . . . , βs)f(z) = z−p +
∞∑
k=1
Γp,q,s (α1) akz
k−p, (1.2)
where
Γp,q,s (α1) =
(α1)k . . . (αq)k
(β1)k . . . (βs)k k!
. (1.3)
If, for convenience, we write
Hp,q,s[α1] = Hp(α1, . . . , αq;β1, . . . , βs) ,
then one can easily verify from the definition (1.2) that (see [5])
z(Hp,q,s[α1]f(z))′ = α1Hp,q,s[α1 + 1]f(z)− (α1 + p)Hp,q,s [α1] f(z). (1.4)
The linear operator Hp,q,s[α1] was investigated recently by Liu and Srivastava [5] and Aouf [2].
In particular, for q = 2, s = 1, α1 > 0, β1 > 0 and α2 = 1, we obtain the linear operator
Hp(α1, 1;β1) f(z) = `p(α1, β1)f(z),
which was introduced and studied by Liu and Srivastava [4] .
We note that,
Hp,2,1(n+ p, 1; 1)f(z) = Dn+p−1f(z) =
1
zp(1− z)n+p
∗ f(z), n > −p,
where Dn+p−1 is the differential operator studied by Uralegaddi and Somanatha [10] and Aouf [1].
Making use of the operator Hp,q,s[α1], we now introduce a subclass of the function class
Σp as follows:
we say that a function f ∈ Σp is in the class Ωp,q,s(α1;λ), if it satisfies the following inequality:
Re
{
α1
Hp,q,s[α1 + 1]f(z)
Hp,q,s[α1]f(z)
− (α1 + p)
}
< −λ, 0 ≤ λ < p, p ∈ N,
or, in view of (1.4), if it satisfies the following inequality:
Re
{
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[α1]f(z)
}
< −λ, 0 ≤ λ < p, p ∈ N.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A CLASS OF p-VALENT MEROMORPHIC FUNCTIONS . . . 1237
2. Main results. In order to establish our main results, we need the following lemma.
Lemma 2.1 [3]. Let w(z) be a non-constant analytic in U with w(0) = 0. If
∣∣w(z)
∣∣ attains its
maximum value on the circle |z| = r < 1 at a point z0, then we have
z0w
′(z0) = ζw(z0), (2.1)
where ζ ≥ 1 is a real number.
Theorem 2.1. Let α1 ≥ 0 and 0 ≤ λ < p, then
Ωp,q,s(α1 + 1;λ) ⊂ Ωp,q,s(α1;λ).
Proof. Let f ∈ Ωp,q,s(α1 + 1;λ), then
Re
{
z
(
Hp,q,s[α1 + 1]f(z)
)′
Hp,q,s[α1 + 1]f(z)
}
< −λ, z ∈ U. (2.2)
We have to show that implies the inequality
Re
{
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[α1]f(z)
}
< −λ, z ∈ U. (2.3)
Define the function w(z) in U by
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[α1]f(z)
= −p+ (p− 2λ)w(z)
1− w(z)
. (2.4)
Clearly, w(z) is analytic in U and w(0) = 0. Using the identity (1.4), (2.4) may be written as
α1
Hp,q,s[α1 + 1]f(z)
Hp,q,s[α1]f(z)
=
α1 − [α1 + 2(p− λ)]w(z)
1− w(z)
. (2.5)
Differentiating (2.5) logarithmically with respect to z and using (1.4), we obtain
z
(
Hp,q,s[α1 + 1]f(z)
)′
Hp,q,s[α1 + 1]f(z)
+ λ =
= − (p− λ)
{
1 + w(z)
1− w(z)
+
2zw′(z)(
1− w(z)
)(
α1 − [α1 + 2(p− λ)]w(z)
)}. (2.6)
We claim that |w(z)| < 1 in U. For otherwise, there exists a point z0 ∈ U such that
max
|z|≤|z0|
∣∣w(z)
∣∣ =
∣∣w(z0)
∣∣ = 1.
Applying Lemma 2.1 to w(z) at the point z0, we have z0w′(z0) = ζw(z0) where ζ ≥ 1. So, (2.6)
yields
Re
{
z
(
Hp,q,s[α1 + 1]f(z0)
)′
Hp,q,s[α1 + 1]f(z0)
+ λ
}
=
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1238 T. M. SEOUDY, M. K. AOUF
= −(p− λ)Re
{
1 + w(z0)
1− w(z0)
+
2ζw(z0)(
1− w(z0)
)(
α1 − [α1 + 2(p− λ)]w(z0)
)} ≥
≥ p− λ
2(α1 + p− λ)
> 0,
which contradicts the inequality (2.2). Hence,
∣∣w(z)
∣∣ < 1 in Uand it follows that f ∈ Ωp,q,s(α1;λ).
Theorem 2.1 is proved.
Theorem 2.2. Let δ > 0 and f(z) ∈ Σp satisfy the following inequality:
Re
{
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[α1]f(z)
}
< −λ+
p− λ
2(δ + λp− λ)
, z ∈ U. (2.7)
Then the function Fδ,p(f) defined by
Fδ,p(f)(z) =
δ
zδ+p
z∫
0
tδ+p−1f(t)dt, δ > 0, (2.8)
belongs to Ωp,q,s(α1;λ).
Proof. From (2.8), we readily have
z
(
Hp,q,s[α1]Fδ,p(f)(z)
)′
= δHp,q,s[α1]f(z)− (δ + p)Hp,q,s[α1]Fδ,p(f)(z). (2.9)
Using the identity (1.4) and (2.9), condition (2.7) may be written as
Re
α1
Hp,q,s[α1 + 1]Fδ,p(f)(z)
Hp,q,s [α1]Fδ,p(f)(z)
z
(
Hp,q,s[α1 + 1]Fδ,p(f)(z)
)′
Hp,q,s[α1 + 1]Fδ,p(f)(z)
+ δ + p
z
(
Hp,q,s[α1]Fδ,p(f)(z)
)′
Hp,q,s[α1]Fδ,p(f)(z)
+ δ + p
− α1 − p
<
< −λ+
p− λ
2 (δ + λp− λ)
. (2.10)
We have to prove that Fδ,p(f) ∈ Ωp,q,s(α1;λ) implies the inequality
Re
{
z (Hp,q,s[α1]Fδ,p(f)(z))′
Hp,q,s[α1]Fδ,p(f)(z)
}
< −λ, 0 ≤ λ < p, z ∈ U. (2.11)
Consider the function w(z) in U defined by
z
(
Hp,q,s[α1]Fδ,p(f)(z)
)′
Hp,q,s[α1]Fδ,p(f)(z)
= −p+ (p− 2λ)w(z)
1− w(z)
, 0 ≤ λ < p, z ∈ U. (2.12)
Clearly, w(z) is analytic and w(0) = 0. (2.12) may be written as
α1
Hp,q,s[α1 + 1]Fδ,p(f)(z)
Hp,q,s[α1]Fδ,p(f)(z)
=
α1 −
[
α1 + 2(p− λ)
]
w(z)
1− w(z)
. (2.13)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A CLASS OF p-VALENT MEROMORPHIC FUNCTIONS . . . 1239
Differentiating (2.13) logarithmically with respect to z and using (2.12), we obtain
z
(
Hp,q,s[α1 + 1]Fδ,p(f)(z)
)′
Hp,q,s[α1 + 1]Fδ,p(f)(z)
=
= −p+ (p− 2λ)w(z)
1− w(z)
− 2(p− λ)zw′(z)
(1− w(z))
(
α1 − [α1 + 2(p− λ)]w(z)
) . (2.14)
Using (2.12) – (2.14) and (2.10), we get
α1
Hp,q,s[α1 + 1]Fδ,p(f)(z)
Hp,q,s[α1]Fδ,p(f)(z)
z
(
Hp,q,s[α1 + 1]Fδ,p(f)(z)
)′
Hp,q,s[α1 + 1]Fδ,p(f)(z)
+ δ + p
z
(
Hp,q,s[α1]Fδ,p(f)(z)
)′
Hp,q,s [α1]Fδ,p(f)(z)
+ δ + p
− α1 − p+ λ =
= −(p− λ)
{
1 + w(z)
1− w (z)
+
2zw′(z)(
1− w(z)
)(
δ − [δ + 2(p− λ)]w(z)
)}. (2.15)
The remaining part of the proof is similar to that of Theorem 2.1.
Theorem 2.2 is proved.
According to Theorem 2.2, we have the following corollary.
Corollary 2.1. If f ∈ Ωp,q,s(α1;λ), then the function Fδ,p(f) defined (2.8) also belongs to
Ωp,q,s(α1;λ).
Theorem 2.3. If f ∈ Ωp,q,s(α1;λ) if and only if the function g defined by
g(z) =
α1
zα1+p
z∫
0
tα1+p−1f (t) dt, α1 > 0, (2.16)
belongs to Ωp,q,s(α1 + 1;λ).
Proof. From (2.16), we have
z (Fδ,p(f)(z))′ = α1f(z)− (α1 + p)Fδ,p(f)(z). (2.17)
Using identity (1.4) and (2.17), hence
Hp,q,s[α1]f(z) = Hp,q,s [α1]Fδ,p(f)(z) (2.18)
and the result follows.
To prove Theorem 2.4, we need the following lemmas.
Lemma 2.2 [8]. The function (1− z)γ = exp
(
γ log(1− z)
)
, γ ∈ C∗ = C\{0}, is univalent in
U if and only if γ is either in the closed disk |γ − 1| ≤ 1 or in the closed disc |γ + 1| ≤ 1.
Lemma 2.3 [7]. Let q(z) be univalent in U and let Q(w) and φ(w) be analytic in a domain D
containing q(U), with φ(w) 6= 0 when w ∈ q(U). Set Q(z) = zq′(z)φ(q(z)), h(z) = θ(q(z)) +Q(z)
and suppose that
(1) Q(z) is starlike (univalent) in U,
(2) Re
{
zh′(z)
Q (z)
}
> 0, z ∈ U.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1240 T. M. SEOUDY, M. K. AOUF
If g(z) is analytic in U, with p(0) = q(0), p(U) ⊂ D, and
θ
(
g(z)
)
+ zg′(z)φ(g(z)) ≺ θ
(
q(z)
)
+ zq′(z)φ
(
q(z)
)
= h(z), (2.19)
then g(z) ≺ q(z), and q(z) is the best dominant of (2.19).
Theorem 2.4. Let f ∈ Ωp,q,s(α1;λ) and γ ∈ C∗ and satisfy either∣∣2γ(p− λ)− 1
∣∣ ≤ 1 or
∣∣2γ(p− λ) + 1
∣∣ ≤ 1. (2.20)
Then (
zpHp,q,s[α1]f(z)
)γ ≺ (1− z)2γ(p−λ) = q(z), (2.21)
and q(z) is the best dominant.
Proof. Set
g(z) =
(
zpHp,q,s[α1]f(z)
)γ
, z ∈ U, (2.22)
then g(z) is analytic in U with g(0) = 1. Differentiating (2.22) logarithmically with respect to z, we
obtain
zg′(z)
g(z)
= γ
[
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[1]f(z)
+ p
]
. (2.23)
Since f ∈ Ωp,q,s(α1;λ), this is equivalent to
z
(
Hp,q,s[α1]f(z)
)′
Hp,q,s[α1]f(z)
≺ −p+ (p− 2λ)z
1− z
, (2.24)
from (2.23), (2.24) can be rewritten as
−p+
zg′(z)
γg(z)
≺ −p+
z
(
(1− z)2γ(p−λ)
)′
(1− z)2γ(p−λ)
. (2.25)
On the other hand, if we take
q(z) = (1− z)2γ(p−λ), θ(z) = −p, φ(w) =
1
γw
in Lemma 2.3, then q(z) is univalent by the condition (2.20) and Lemma 2.2. It is easy to see that
q(z), θ(w), and φ(w) satisfy the conditions of Lemma 2.3. Since
Q(z) = zq′(z)φ(q(z)) = −2(p− λ)z
1− z
is univalent starlike in U and
h(z) = θ(q(z)) +Q(z) = −p+ (p− 2λ)z
1− z
,
from (2.25) and Lemma 2.3, then
g(z) ≺ (1− z)2γ(p−λ) = q(z)
and the function (1− z)2γ(p−λ) is the best dominant.
Theorem 2.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A CLASS OF p-VALENT MEROMORPHIC FUNCTIONS . . . 1241
Corollary 2.2. Let f ∈ Ωp,q,s(α1;λ). Then
Re
{(
zpHp,q,s[α1]f(z)
)γ}
> 22γ(p−λ), z ∈ U,
where γ is a real number and γ ∈
[
− 1
2 (p− λ)
, 0
)
. The result is sharp.
Proof. From Theorem 2.4, we have
Re
{(
zpHp,q,s[α1]f(z)
)γ}
= Re
{(
1− w(z)
)2γ(p−λ)}
, z ∈ U, (2.26)
where w(z) is analytic in U, w(0) = 0, and |w(z)| < 1 for z ∈ U. In view of
Re (tb) ≥ (Re t)b, Re(t) > 0, 0 < b ≤ 1,
(2.26) yields
Re
{(
zpHp,q,s[α1]f(z)
)γ} ≥ {Re
(
1
1− w(z)
)−2γ(p−λ)}
> 22γ(p−λ), z ∈ U,
for −1 ≤ 2γ(p − λ) < 0. To see that the bound 22γ(p−λ) cannot be increased, we consider the
function f(z) which satisfies
zpHp,q,s[α1]f(z) = (1− z)2(p−λ), 0 ≤ λ < p, z ∈ U.
We easily have f ∈ Ωp,q,s(α1;λ) and
Re
{(
zpHp,q,s[α1]f(z)
)γ}→ 22γ(p−λ) as Re(z)→ −1.
Corollary 2.2 is proved.
3. Convolution conditions. We give some necessary and sufficient condition in terms of
convolution operator for meromorphic functions to be in the classes S∗p (λ) and Ωp,q,s(α1;λ).
Lemma 3.1. The function f(z) ∈ Σp belongs to the class ΣS∗p (λ) (0 ≤ λ < p) if and only if
zp
f(z) ∗
(
1− 1− e−iθ + 2(p− λ)
2(p− λ)
z
)
zp(1− z)2
6= 0, 0 < θ < 2π, z ∈ U.
Proof. A function f(z) ∈ ΣS∗p (λ) if and only if
zf ′(z)
f(z)
6= −p+ (p− 2λ) eiθ
1− eiθ
, 0 < θ < 2π, z ∈ U,
which is equivalent to
zp
[(
1− eiθ
)
zf ′(z) +
[
p+ (p− 2λ) eiθ
]
f(z)
]
6= 0, 0 < θ < 2π, z ∈ U. (3.1)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1242 T. M. SEOUDY, M. K. AOUF
Since
f(z) = f(z) ∗ 1
zp(1− z)
and zf ′(z) = f(z) ∗
(
(p+ 1) z − p
zp(1− z)2
)
.
Therefore, we may write (3.1) as
zp
[(
1− eiθ
)
zf ′(z) +
[
p+ (p− 2λ) eiθ
]
f(z)
]
=
= zp
{
f(z) ∗
[
1− eiθ
zp(1− z)2
+
p+ (p− 2λ)eiθ
zp(1− z)
] }
=
= 2(p− λ)eiθ zp
f(z) ∗
(
1− 1− e−iθ + 2(p− λ)
2 (p− λ)
z
)
zp(1− z)2
6= 0.
Lemma 3.1 is proved.
Theorem 3.1. A necessary and sufficient condition for the function f(z) defined by (1.1) to be
in the class Ωp,q,s(α1;λ) is that
1−
∞∑
k=1
(
1− e−iθ
)
k − 2 (p− λ)
2(p− λ)
Γp,q,s (α1) akz
k 6= 0, 0 < θ < 2π, z ∈ U,
where Γp,q,s (α1) is given by (1.3).
Proof. From Lemma 3.1, we find that f ∈ Ωp,q,s(α1;λ) if and only if
zp
Hp,q,s[α1]f(z) ∗
(
1− 1− e−iθ + 2(p− λ)
2 (p− λ)
z
)
zp(1− z)2
6= 0, 0 < θ < 2π, 0 ≤ λ < p, z ∈ U.
(3.2)
From (1.2), the left-hand side of (3.2) may be written as
zp
Hp,q,s[α1]f(z) ∗
(
1− 1− e−iθ + 2(p− λ)
2 (p− λ)
z
)
zp(1− z)2
=
= 1−
∞∑
k=1
(
1− e−iθ
)
k − 2(p− λ)
2(p− λ)
Γp,q,s (α1) akz
k 6= 0.
Theorem 3.1 is proved.
Remarks. 3.1. Putting q = 2, s = α2 = β1 = 1 and α1 = n + p(n > −p) in Theorem 2.1, we
obtain the results obtained by Aouf [1] (Theorem 1).
3.2. Putting q = 2, s = α2 = β1 = 1, δ = c− p + 1 (c > p− 1) and α1 = n + p (n > −p) in
Theorem 2.2, we obtain the results obtained by Aouf [1] (Theorem 2).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
A CLASS OF p-VALENT MEROMORPHIC FUNCTIONS . . . 1243
3.3. Putting q = 2, s = α2 = β1 = 1 and α1 = n+ p(n > −p) in Theorem 2.3, we improve the
result obtained by Aouf [1] (Theorem 2.3).
3.4. Taking q = 2, s = α2 = 1 and α1, β1 > 0 in Theorems 2.1 – 2.4, Corollaries 2.1 and
2.2, respectively, we obtain the results obtained by Liu and Owa [6] (Theorems 2.2, 2.3, 2.5, 2.9,
Corollaries 2.4 and 2.10, respectively).
3.5. Taking e−iθ = −x in Lemma 3.1, we obtain the results obtained by Liu and Owa [6]
(Lemma 3.1).
3.6. Taking q = 2, s = α2 = 1, α1, β1 > 0 and e−iθ = −x in Theorem 3.1, we obtain the result
obtained by Liu and Owa [6] (Theorem 3.2).
1. Aouf M. K. New criteria for multivalent meromorphic starlike functions of order alpha // Proc. Jap. Acad. A. – 1993. –
69. – P. 66 – 70.
2. Aouf M. K. Certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric
function // Comput. Math. and Appl. – 2008. – 55, № 3. – P. 494 – 509.
3. Jack I. S. Functions starlike and convex of order α // J. London Math. Soc. – 1971. – 2, № 3. – P. 469 – 474.
4. Liu J.-L., Srivastava H. M. A linear operator and associated families of meromorphically multivalent functions // J.
Math. Anal. and Appl. – 2000. – 259. – P. 566 – 581.
5. Liu J.-L., Srivastava H. M. Classes of meromorphically multivalent functions associated with the generalized hyper-
geometric function // Math. Comput. Modelling. – 2004. – 39. – P. 21 – 34.
6. Liu J.-L., Owa S. On a classes of meromorphic p-valent starlike functions involving certain linear operators // Int. J.
Math. and Math. Sci. – 2002. – 32. – P. 271 – 280.
7. Miller S. S., Mocanu P. T. On some classes of first-order differential subordinations // Mich. Math. J. – 1985. – 32,
№ 2. – P. 185 – 195.
8. Robertson M. S. Certain classes of starlike functions // Mich. Math. J. – 1985. – 32, № 2. – P. 135 – 140.
9. Srivastava H. M., Karlsson P. W. Multiple Gaussian hypergeometric series. – New York etc.: Halsted Press, Ellis
Horwood Limited, Chichester, John Wiley and Sons, 1985.
10. Uralegaddi B. A., Somanatha C. Certain classes of meromorphic multivalent functions // Tamkang J. Math. – 1992. –
23. – P. 223 – 231.
Received 10.09.12,
after revision — 23.12.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
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| id | umjimathkievua-article-2214 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:52Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-22142019-12-05T10:26:31Z A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator Про клас p-валентних мероморфних функцій, що визначені оператором Луi-Шрiвастави Seoudy, T. M. Суді, Т. М. We introduce a subclass of p-valent meromorphic functions involving the Lui–Srivastava operator and investigate various properties of this subclass. We also indicate the relationships between various results presented in the paper and the results obtained in earlier works. Введено підклас p-валентних мероморфних Функцій, що визначаються оператором Луі - Шрiвастави, та вивчено різноманітні властивості цього підкласу. Також вказано співвідношення між різноманітними результатами, що отримані в роботі, та результатами, отриманими раніше. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2214 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1235–1243 Український математичний журнал; Том 66 № 9 (2014); 1235–1243 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2214/1429 https://umj.imath.kiev.ua/index.php/umj/article/view/2214/1430 Copyright (c) 2014 Seoudy T. M. |
| spellingShingle | Seoudy, T. M. Суді, Т. М. A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title | A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title_alt | Про клас p-валентних мероморфних функцій, що визначені оператором Луi-Шрiвастави |
| title_full | A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title_fullStr | A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title_full_unstemmed | A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title_short | A Class of p-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator |
| title_sort | class of p-valent meromorphic functions defined by the liu–srivastava operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2214 |
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