Some properties of multivalent functions associated with a certain operator
We obtain certain subordinations and superordinations results involving a new operator. By means of the new introduced operator Cλp,n(a, c)f(z), for certain multivalent functions in the open unit disc, we establish differential Sandwich Theorem
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irk-123456789-1657082020-02-17T01:25:44Z Some properties of multivalent functions associated with a certain operator He, P. Zhang, D. Короткі повідомлення We obtain certain subordinations and superordinations results involving a new operator. By means of the new introduced operator Cλp,n(a, c)f(z), for certain multivalent functions in the open unit disc, we establish differential Sandwich Theorem Отримано деякi субординацiї i результати для суперординацiй iз використанням нового оператора. З допомогою введеного оператора Cλp,n(a, c)f(z) доведено диференцiальну сендвiч-теорему для багатозначних функцiй у вiдкритому одиничному крузi. 2013 Article Some properties of multivalent functions associated with a certain operator / P. He, D. Zhang // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1580–1584. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165708 517.5 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення He, P. Zhang, D. Some properties of multivalent functions associated with a certain operator Український математичний журнал |
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We obtain certain subordinations and superordinations results involving a new operator. By means of the new introduced operator Cλp,n(a, c)f(z), for certain multivalent functions in the open unit disc, we establish differential Sandwich Theorem |
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Article |
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He, P. Zhang, D. |
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He, P. Zhang, D. |
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He, P. |
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Some properties of multivalent functions associated with a certain operator |
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Some properties of multivalent functions associated with a certain operator |
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Some properties of multivalent functions associated with a certain operator |
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Some properties of multivalent functions associated with a certain operator |
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Some properties of multivalent functions associated with a certain operator |
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some properties of multivalent functions associated with a certain operator |
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Інститут математики НАН України |
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2013 |
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Some properties of multivalent functions associated with a certain operator / P. He, D. Zhang // Український математичний журнал. — 2013. — Т. 65, № 11. — С. 1580–1584. — Бібліогр.: 10 назв. — англ. |
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Український математичний журнал |
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AT hep somepropertiesofmultivalentfunctionsassociatedwithacertainoperator AT zhangd somepropertiesofmultivalentfunctionsassociatedwithacertainoperator |
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2025-07-14T19:35:46Z |
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2025-07-14T19:35:46Z |
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1837652244763246592 |
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UDC 517.5
P. He, D. Zhang (Honghe Univ., China)
SOME PROPERTIES OF MULTIVALENT FUNCTIONS
ASSOCIATED A CERTAIN OPERATOR*
ДЕЯКI ВЛАСТИВОСТI БАГАТОЗНАЧНИХ ФУНКЦIЙ,
АСОЦIЙОВАНИХ З ОПЕРАТОРОМ
We obtain certain subordinations and superordinations results involving a new operator. By means of the new introduced
operator Cλp,n(a, c)f(z), for certain multivalent functions in the open unit disc, we establish differential Sandwich Theorem.
Отримано деякi субординацiї i результати для суперординацiй iз використанням нового оператора. З допомогою вве-
деного оператора Cλp,n(a, c)f(z) доведено диференцiальну сендвiч-теорему для багатозначних функцiй у вiдкритому
одиничному крузi.
1. Introduction. Let Σp denote the class of functions f(z) of the form
f(z) = zp +
∞∑
k=n
ap+kz
p+k, p ∈ N = {1, 2, 3, ...}, (1)
which are analytic in the open unit disk U = {z : z ∈ C, |z| < 1}.
For functions f ∈ Σp given by (1) and g ∈ Σp given by
g(z) = zp +
∞∑
k=n
bp+kz
p+k.
We define the Hadamard product (or convolution) of f and g by
(f ∗ g)(z) = zp +
∞∑
k=n
ap+kbp+kz
p+k. (2)
Let f(z) and g(z) be analytic in U. We say that the function g(z) is subordinate to f(z), if there
exists a function w(z) analytic in U, with w(0) = 0 and |w(z)| < 1, and such that g(z) = f(w(z)).
In such a case, we write g(z) ≺ f(z). If the function f is univalent in U, then g(z) ≺ f(z) if and
only if g(0) = f(0) and g(U) ⊂ f(U).
Let H(U) denote the class of analytic functions in U and let H(a, n) denote the subclass of
functions f ∈ H(U) of the form:
f(z) = a+ anz
n + an+1z
n+1 + . . . .
Denote by Q, the set of all functions f(z) that are analytic and injective on U\E(f), where
E(f) = {ξ ∈ ∂U : limz→ξ f(z) =∞}, and such that f ′(ξ) 6= 0 for ξ ∈ ∂U\E(f).
Let ψ : C3×U→ C, let h(z) be univalent in U and q(z) ∈ Q. Miller and Mocanu [1] considered
the problem of determining conditions on admissible function ψ such that
* This work was supported by Nature Science Foundation of Yunnan (2013FZ116), Scientific Research Foundation
from Yunnan Province Education Committee (2010Y167 and 2011C120), Foundation of Honghe University (ZDKC 1111),
and National Science Foundation of China (11301160).
c© P. HE, D. ZHANG, 2013
1580 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOME PROPERTIES OF MULTIVALENT FUNCTIONS ASSOCIATED A CERTAIN OPERATOR 1581
ψ(p(z), zp′(z), z2p′′(z); z) ≺ h(z) (3)
implies p(z) ≺ q(z), for all functions p(z) ∈ H(a, n) that satisfy the differential subordination (3).
Moreover, they found conditions so that q(z) is the smallest function with this property, called the
best dominant of the subordination (3).
Let ϕ : C3×U→ C, let h(z) ∈ H and q(z) ∈ H(a, n). Recently Miller and Mocanu [2] studied
the dual problem and determined conditions on ϕ such that
h(z) ≺ ϕ(p(z), zp′(z), z2p′′(z); z) (4)
implies q(z) ≺ p(z), for all functions p(z) ∈ Q that satisfy the above superordination. They also
found conditions so that the function q(z) is the largest function with this property, called the best
subordinant of the superordination (4).
In [3], N. E. Cho, O. S. Kwon and H. M. Srivastava extended the multiplier transformation and
defined the operator Iλp,n(a, c)f(z) by the following infinite series:
Iλp,n(a, c)f(z) = zp +
∞∑
k=n
(λ+ p)k(c)k
k!(a)k
ak+pz
k+p. (5)
In recent years, Aghalary [4], Patel [5], Patel et al. [6], Sokl and Trojnar-Spelina [7], Zeng et al.
[8] and Wang et al. [9] obtained many interesting results associated with the Cho – Kwon – Srivastava
operator.
We now introduce the following family of linear operators:
Lλp,n(a, c)f(z) = zp +
∞∑
k=n
k!(a)k
(λ+ p)k(c)k
ak+pz
k+p. (6)
It is readily verified from the definition (6) that
z(Lλp,n(a, c+ 1)f(z))′ = cLλp,n(a, c)f(z)− (c− p)Lλp,n(a, c+ 1)f(z) (7)
and
z(Lλp,n(a, c)f(z))′ = (c− 1)Lλp,n(a, c− 1)f(z)− (c− 1− p)Lλp,n(a, c)f(z). (8)
We also note that L1
p,n(p + 1, 1)f(z) = f(z) and L0
p,n(p, 1)f(z) = f(z). In this paper, we
will derive several subordination results, superordination results and sandwich results involving the
operator Lλp,n(a, c)f(z) and some of its special operators.
2. Some lemmas. In order to prove our main results, we need the following lemmas.
Lemma 1 [10]. Let q(z) be univalent in U, γ ∈ C∗ = C\{0} and suppose that
Re
{
1 +
zq′′(z)
q′(z)
}
> max
{
0,−Re
1
γ
}
.
If p(z) is analytic in U and
p(z) + γzp′(z) ≺ q(z) + γzq′(z),
then p(z) ≺ q(z), and q(z) is the best dominant.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1582 P. HE, D. ZHANG
Lemma 2 [10]. Let q(z) be convex in U, q(0) = a and γ ∈ C, Re γ > 0. If p ∈ H(a, 1) and
p(z) + γzp′(z) is univalent in U, then
q(z) + γzq′(z) ≺ p(z) + γzp′(z),
where q(z) ≺ p(z) and q(z) is the best subordinant.
3. Main results. We shall assume in the reminder of this paper that p, n ∈ N and z ∈ U.
Theorem 1. Let q(z) be univalent in U with q(0) = 1, α ∈ C∗, and suppose that
Re
{
1 +
zq′′(z)
q′(z)
}
> max
{
0,−Re
1
α
}
. (9)
If f(z) ∈ Σp satisfies the subordination
R(α, n, p, λ, a, c) ≺ q(z) + αzq′(z), (10)
where R(α, n, p, λ, a, c) is given by
R(α, n, p, λ, a, c) =
= (1− α)
Lλp,n(a, c+ 1) f(z)
Lλp,n(a, c) f(z)
+ α
{
c− (c− 1)
Lλp,n(a, c+ 1)f(z) Lλp,n(a, c− 1) f(z)
(Lλp,n(a, c) f(z))2
}
, (11)
then
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
≺ q(z)
and q(z) is the best dominant.
Proof. Let
p(z) =
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
, (12)
differentiating (12) with respect to z and using the identity (7) and (8) in the resulting equation, we
have
zp′(z) = c− (c− 1)
Lλp,n(a, c+ 1)f(z) · Lλp,n(a, c− 1)f(z)
(Lλp,n(a, c)f(z))2
−
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
.
Therefore, we have
R(α, n, p, λ, a, c) = p(z) + αzp′(z).
By (10), we obtain
p(z) + αzp′(z) ≺ q(z) + αzq′(z).
By Lemma 1,
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
≺ q(z), and the proof of Theorem 1 is completed.
Taking the convex function q(z) =
1 +Az
1 +Bz
in Theorem 1, we have the following corollary.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOME PROPERTIES OF MULTIVALENT FUNCTIONS ASSOCIATED A CERTAIN OPERATOR 1583
Corollary 1. Let A, B, α ∈ C, A 6= B, |B| < 1, Reα > 0. If f(z) ∈ Σp satisfies the
subordination
R(α, n, p, λ, a, c) ≺ 1 +Az
1 +Bz
+ α
(A−B)z
(1 +Bz)2
,
where R(α, n, p, λ, a, c) is given by (11), then
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
≺ 1 +Az
1 +Bz
,
and the function
1 +Az
1 +Bz
is the best dominant.
Theorem 2. Let q(z) be convex in U, q(0) = 1 and α ∈ C, Reα > 0. If f(z) ∈ Σp such
that
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
∈ H(q(0), 1)
⋂
Q, and R(α, n, p, λ, a, c) is univalent in U and satisfies
the superordination
q(z) + αzq′(z) ≺ R(α, n, p, λ, a, c), (13)
where R(α, n, p, λ, a, c) is given by (11), then
q(z) ≺
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
,
and q(z) is the best subordinant.
Proof. Let p(z) be given by (12) and proceeding as in the proof of Theorem 1, the subordination
(13) becomes
q(z) + αzq′(z) ≺ p(z) + αzp′(z).
The proof follows by an application of Lemma 2.
Corollary 2. Let A, B, α ∈ C, A 6= B, |B| < 1, Reα > 0. If f(z) ∈ Σp such that
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
∈ H(q(0), 1)
⋂
Q, and R(α, n, p, λ, a, c) is univalent in U and satisfies the
superordination
1 +Az
1 +Bz
+ α
(A−B)z
(1 +Bz)2
≺ R(α, n, p, λ, a, c),
then
1 +Az
1 +Bz
≺
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
,
and the function
1 +Az
1 +Bz
is the best subordinant.
Combining Theorems 1 and 2, we have the following sandwich theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1584 P. HE, D. ZHANG
Theorem 3. Let q1(z) and q2(z) be convex in U, q1(0) = q2(0) = 1 and q2(z) satisfies
(9), and α ∈ C, Reα > 0. If f(z) ∈ Σp such that
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
∈ H(q(0), 1)
⋂
Q, and
R(α, n, p, λ, a, c) is univalent in U and satisfies
q1(z) + αzq′1(z) ≺ R(α, n, p, λ, a, c) ≺ q2(z) + αzq′2(z),
where R(α, n, p, λ, a, c) is given by (11), then
q1(z) ≺
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
≺ q2(z)
and q1(z), q2(z) are the best subordinant and the best dominant, respectively.
Remark. Combining Corollaries 1, 2, we obtain the corresponding sandwich results for the
operators
Lλp,n(a, c+ 1)f(z)
Lλp,n(a, c)f(z)
.
1. Miller S. S., Mocanu P. T. Differential subordination: theory and applications // Ser. Monogr. and Textbooks in Pure
and Appl. Math. – New York; Basel: Marcel Dekker Inc., 2000. – 225.
2. Miller S. S., Mocanu P. T. Subordinates of differential superordinations // Complex Var. – 2003. – 48, № 10. –
P. 815 – 826.
3. Cho N. E., Kwon O. S., Srivastava H. M. Inclusion relationships and argument properties for certain subclasses
of multivalent functions associated with a family of linear operators // J. Math. Anal. and Appl. – 2004. – 292. –
P. 470 – 483.
4. Aghalary R. On subclasses of p-valent analytic functions defined by integral operators // Kyungpook Math. J. – 2007. –
47. – P. 393 – 401.
5. Patel J. On certain subclasses of multivalent functions involving Cho – Kwonv – Srivastava operator // Ann. Univ.
Mariae Curie-Skaodowska Sect. A. – 2006. – 60. – P. 75 – 86.
6. Patel J., Cho N. E., Srivastava H. M. Certain subclasses of multivalent functions associated with a family of linear
operators // Math. Comput. Modelling. – 2006. – 43. – P. 320 – 338.
7. Sokl J., Trojnar-Spelina L. Convolution properties for certain classes of multivalent functions // J. Math. Anal. and
Appl. – 2008. – 337. – P. 1190 – 1197.
8. Zeng T., Gao C.-Y., Wang Z.-G., Aghalary R. Certain subclass of multivalent functions involving the Cho – Kwon –
Srivastava operator // J. Math. Appl. – 2008. – 30. – P. 161 – 170.
9. Wang Z. G., Aghalaryc R., Darus M., Ibrahim R. W. Some properties of certain multivalent analytic functions
involving the Cho – Kwon – Srivastava operator // Math. and Comput. Modelling. – 2009. – 49. – P. 1969 – 1984.
10. Shanmugam T. N., Ravichandran V., Sivasubramanian S. Differential sandwich theorems for some subclasses of
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Received 04.02.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
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