Let $q_1$ and $q_2$ belong to a certain class of normalized analytic
univalent functions in the open unit disk of the complex plane. Sufficient
conditions are obtained for normalized analytic functions $p$ to satisfy the
double subordination chain $q_1(z)\prec p(z)\prec q_2(z)$. The differential
sandwich-type result obtained is applied to normalized univalent functions and
to $\Phi$-like functions

Brickman

Bulboacă

Kanas

Lecko

Lewandowski

Miller

Nak Eun Cho

Oh Sang Kwon

Rosihan M. Ali

Ruscheweyh

Singh

V. Ravichandran

English

arXiv

Crossref

A first-order differential double subordination with applications

AbstractLet q1 and q2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions p to satisfy the double subordination chain q1(z)≺p(z)≺q2(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Φ-like functions

Ali, Rosihan M.

Cho, Nak Eun

Kwon, Oh Sang

Ravichandran, V.

Elsevier - Publisher Connector 

A first-order differential double subordination with applications 

Applied Mathematics Letters 25 (2012) 268–274
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
A first-order differential double subordination with applications✩
Rosihan M. Ali a,∗, Nak Eun Cho b, Oh Sang Kwon c, V. Ravichandran a,d
a School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
b Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea
c Department of Mathematics, Kyungsung University, Busan 608-736, South Korea
d Department of Mathematics, University of Delhi, Delhi-110 007, India
a r t i c l e i n f o
Article history:
Received 22 October 2010
Received in revised form 3 May 2011
Accepted 26 August 2011
Keywords:
Differential subordination
Differential superordination
Best subordinant
Best dominant
Φ-like functions
a b s t r a c t
Let q1 and q2 belong to a certain class of normalized analytic univalent functions in the open
unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic
functions p to satisfy the double subordination chain q1(z) ≺ p(z) ≺ q2(z). The differential
sandwich-type result obtained is applied to normalized univalent functions and to Φ-like
functions.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
LetH be the class consisting of analytic functions in the open unit disk D := {z ∈ C : |z| < 1} of the complex plane C.
For a ∈ C, letH [a, n] := {f ∈ H : f (z) = a+ anzn + an+1zn+1 + · · ·}, and A := {f ∈ H : f (0) = 0, f ′(0) = 1}. A function
f ∈ H is said to be subordinate to an analytic function g ∈ H , or g superordinates f , written as f (z) ≺ g(z)(z ∈ D), if there
exists a Schwarz function w, analytic in D with w(0) = 0 and |w(z)| < 1, satisfying f (z) = g(w(z)). If the function g is
univalent in D, then f (z) ≺ g(z) is equivalent to f (0) = g(0) and f (D) ⊆ g(D). An exposition on the widely used theory of
differential subordination, developed in the main byMiller andMocanu, with numerous applications to univalent functions
can be found in their monograph [1]. Miller andMocanu [2] also introduced the dual concept of differential superordination.
Let p, h ∈ H andφ(r, s, t; z) : C3×D→ C. If p andφ(p(z), zp′(z), z2p′′(z); z) are univalent and p satisfies the second-order
superordination
h(z) ≺ φ(p(z), zp′(z), z2p′′(z); z), (1)
then p is a solution of the differential superordination (1). An analytic function q is called a subordinant if q ≺ p for all p
satisfying (1). A univalent subordinantq satisfying q ≺q for all subordinants q of (1) is said to be the best subordinant. Miller
and Mocanu [2] obtained conditions on h, q and φ for which the following differential implication holds:
h(z) ≺ φ(p(z), zp′(z), z2p′′(z); z)⇒ q(z) ≺ p(z).
✩ The works by the first and the fourth authors were supported in part by grants from the Universiti Sains Malaysia and the University of Delhi
respectively. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded
by the Ministry of Education, Science and Technology (No. 2010-0017111). This work was completed during the visit of the first and the fourth authors to
Pukyong National University. The authors are thankful to the referees for their suggestions that helped improve the presentation of this manuscript.∗ Corresponding author.
E-mail addresses: rosihan@cs.usm.my (R.M. Ali), necho@pknu.ac.kr (N.E. Cho), oskwon@ks.ac.kr (O.S. Kwon), vravi68@gmail.com (V. Ravichandran).
0893-9659/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2011.08.025
R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274 269
Using these results, Bulboacă gave a treatment on certain classes of first-order differential superordinations [3,4], as
well as superordination-preserving integral operators [5]. Ali et al. [6] gave several applications of first-order differential
subordination and superordination to obtain sufficient conditions for normalized analytic functions f to satisfy q1(z) ≺
zf ′(z)/f (z) ≺ q2(z), where q1 and q2 are given univalent analytic functions in D. In [7], they have also applied differential
superordination to functions defined by means of linear operators. Recently Ali and Ravichandran [8] investigated first-
order superordination to a class ofmeromorphicα-convex functions. Several differential subordination and superordination
associated with various linear operators were also investigated in [9].
Generalizing the familiar starlike and convex functions, Lewandowski et al. [10] introduced γ -starlike functions
consisting of f ∈ A satisfying the inequality
Re

zf ′(z)
f (z)
1−γ 
1+ zf
′′(z)
f ′(z)
γ
> 0.
These functions are starlike. With p(z) := zf ′(z)/f (z), to show that γ -starlike functions are indeed starlike, is to analytically
make the implication
Re

p(z)

1+ zp
′(z)
p2(z)
γ
> 0⇒ Re p(z) > 0.
Following the work of Lewandowski et al. [10,11], Kanas et al. [12] determined conditions on p and h satisfying
p(z)

1+ zp
′(z)
p(z)
α
≺ h(z)⇒ p(z) ≺ h(z)
for a fixed α ∈ [0, 1]. Lecko [13] (see [12] for a symmetric version) investigated the more general subordination
p(z)

1+ zp
′(z)
p(z)
ϕ(p(z))
α
≺ h(z)⇒ p(z) ≺ h(z).
Singh and Gupta [14] subsequently investigated the following first-order differential subordination that included the
important Briot–Bouquet differential subordination.
(p(z))α

p(z)+ zp
′(z)
βp(z)+ γ
µ
≺ (q(z))α

q(z)+ zq
′(z)
βq(z)+ γ
µ
⇒ p(z) ≺ q(z).
For a closely related class, see [15].
The present paper investigates differential subordination and superordination implications of expressions similar to the
form considered above by Singh and Gupta [14]. Special cases of the results obtained include one involving the expression
αp2(z)+ (1− α)p(z)+ αzp′(z), a result which cannot be deduced from the work of Singh and Gupta [14]. The sandwich-
type results obtained in our present investigation are then applied to normalized analytic univalent functions and toΦ-like
functions.
The following definition and results will be required.
Lemma 1.1 (cf. Miller and Mocanu [1, Theorem 3. 4h, p. 132]). Let q be univalent in the unit disk D, and let ϑ and ϕ be analytic
in a domain D ⊃ q(D) with ϕ(w) ≠ 0, w ∈ q(D). With Q (z) := zq′(z)ϕ(q(z)), let h(z) := ϑ(q(z))+ Q (z). Suppose that Q is
starlike univalent in D and
Re

zh′(z)
Q (z)

> 0 (z ∈ D).
If p is analytic in D with p(0) = q(0), p(D) ⊂ D and
ϑ(p(z))+ zp′(z)ϕ(p(z)) ≺ ϑ(q(z))+ zq′(z)ϕ(q(z)),
then p(z) ≺ q(z), and q is the best dominant.
Definition 1.2 ([2, Definition 2, p. 817]). Denote by Q the set of all functions f that are analytic and injective on D − E(f ),
where
E(f ) =

ζ ∈ ∂D : lim
z→ζ f (z) = ∞

,
and are such that f ′(ζ ) ≠ 0 for ζ ∈ ∂D− E(f ).
Lemma 1.3 ([4]). Let q be univalent in the unit disk D, ϑ and ϕ be analytic in a domain D containing q(D). Suppose that
Re[ϑ ′(q(z))/ϕ(q(z))] > 0 for z ∈ D and zq′(z)ϕ(q(z)) is starlike univalent in D. If p ∈ H [q(0), 1] ∩ Q with p(D) ⊆ D,
and ϑ(p(z))+ zp′(z)ϕ(p(z)) is univalent in D, then
ϑ(q(z))+ zq′(z)ϕ(q(z)) ≺ ϑ(p(z))+ zp′(z)ϕ(p(z))
implies q(z) ≺ p(z), and q is the best subordinant.
270 R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274
2. A sandwich theorem
Our main result involves the following class of functions.
Definition 2.1. Let α and µ be fixed numbers with 0 < µ ≤ 1, α + µ ≥ 0. Also let β , γ and δ be complex numbers with
β ≠ 0. The class R(α, β, γ , δ, µ) consists of analytic functions p with p(0) = 1, p(z) ≠ 0 in D, and are such that the
functions
P(z) := (p(z))α

p(z)+ δ + zp
′(z)
βp(z)+ γ
µ
(z ∈ D)
are well-defined in D. (Here the powers are principal values.)
By making use of Lemma 1.1, the following result is derived.
Theorem 2.2. Let q ∈ R(α, β, γ , δ, µ) be analytic and univalent in D. Set
R(z) := zq
′(z)
βq(z)+ γ (z ∈ D). (2)
Assume that
Re

(βq(z)+ γ )

1+ α
µ
+ αδ
µq(z)

> 0 (z ∈ D), (3)
and
Re

α
µ
zq′(z)
q(z)
+ zR
′(z)
R(z)

> 0 (z ∈ D). (4)
If p ∈ R(α, β, γ , δ, µ) satisfies
(p(z))α

p(z)+ δ + zp
′(z)
βp(z)+ γ
µ
≺ (q(z))α

q(z)+ δ + zq
′(z)
βq(z)+ γ
µ
, (5)
then p(z) ≺ q(z), and q is the best dominant.
Proof. We first write the differential subordination (5) as
(p(z))
α
µ+1 + δ(p(z)) αµ + (p(z)) αµ zp
′(z)
βp(z)+ γ ≺ (q(z))
α
µ+1 + δ(q(z)) αµ + (q(z)) αµ zq
′(z)
βq(z)+ γ .
Define the functions ϑ and ϕ by
ϑ(w) := w αµ+1 + δw αµ and ϕ(w) := w
α
µ
βw + γ .
Since q ∈ R(α, β, γ , δ, µ), then q(z) ≠ 0 and therefore ϕ(w) ≠ 0 when w ∈ q(D). Also ϕ and ϑ are analytic in a domain
containing q(D). Define the function
Q (z) := zq′(z)ϕ(q(z)) = (q(z)) αµ zq
′(z)
βq(z)+ γ = (q(z))
α
µ R(z),
where R is given by (2). It follows from (4) that
Re
zQ ′(z)
Q (z)
= ℜ

α
µ
zq′(z)
q(z)
+ zR
′(z)
R(z)

> 0,
and so Q is a starlike function. Now define h by
h(z) := ϑ(q(z))+ Q (z) = (q(z)) αµ+1 + δ(q(z)) αµ + Q (z).
In view of the assumptions (3) and (4), it follows that
Re
zh′(z)
Q (z)
= Re

(βq(z)+ γ )

1+ α
µ
+ αδ
µq(z)

+ α
µ
zq′(z)
q(z)
+ zR
′(z)
R(z)

> 0 (z ∈ D).
The result is now deduced from Lemma 1.1. 
R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274 271
Example 2.3. Let q : D → C be defined by q(z) = (1 + Az)/(1 + Bz) with −1 < B < A ≤ 1. It is evident that
q ∈ R(α, β, γ , δ, µ)whenever
δ + 1− A
1− B >
A− B
(1− B)| |β + γ | − |βA+ γ B| | .
With additional constraints on the parameters, there exist functions q satisfying the hypothesis of Theorem 2.2. For instance,
in addition to the above condition, assuming that all the parameters α, β, γ , δ, and µ are positive with
1− 2A
1− A >
|βA+ γ B|
|β + γ − |βA+ γ B| ,
then q satisfies the conditions of Theorem 2.2.
By a similar application of Lemma 1.3, the following result can be established, which we state without proof.
Theorem 2.4. Let q ∈ R(α, β, γ , δ, µ) be as in Theorem 2.2. Let p ∈ R(α, β, γ , δ, µ) satisfies p ∈ H ∩ Q and
(p(z))
α
µ+1 + δ(p(z)) αµ + (p(z)) αµ zp′(z)
βp(z)+γ be univalent. If p satisfies
(q(z))α

q(z)+ δ + zq
′(z)
βq(z)+ γ
µ
≺ (p(z))α

p(z)+ δ + zp
′(z)
βp(z)+ γ
µ
,
then q(z) ≺ p(z), and q is the best subordinant.
Combining Theorems 2.2 and 2.4, the following ‘‘sandwich theorem’’ is obtained.
Theorem 2.5. Let qi ∈ R(α, β, γ , δ, µ) (i = 1, 2) be analytic and univalent in D. Set
Ri(z) := zq
′
i(z)
βqi(z)+ γ (i = 1, 2; z ∈ D),
hi(z) := (qi(z))α

qi(z)+ δ + zq
′
i(z)
βqi(z)+ γ
µ
(i = 1, 2).
Assume that
Re

(βqi(z)+ γ )

1+ α
µ
+ αδ
µqi(z)

> 0 (z ∈ D)
and
Re

α
µ
zq′i(z)
qi(z)
+ zR
′
i(z)
Ri(z)

> 0 (i = 1, 2; z ∈ D).
If p ∈ R(α, β, γ , δ, µ) satisfies p ∈ H ∩ Q and (p(z)) αµ+1 + δ(p(z)) αµ + (p(z)) αµ zp′(z)
βp(z)+γ is univalent, then
h1(z) ≺ (p(z))α

p(z)+ δ + zp
′(z)
βp(z)+ γ
µ
≺ h2(z) (6)
implies q1(z) ≺ p(z) ≺ q2(z). Further q1 and q2 are the best subordinant and the best dominant respectively.
3. Applications to univalent functions
By use of Theorem 2.5, the following result is obtained.
Theorem 3.1. Let α, µ be fixed numbers with 0 < µ ≤ 1, α + µ > 0, and λ ∈ C. Let f , g ∈ A , and qi(z) = zg ′i (z)/gi(z)
(i = 1, 2) be univalent in D satisfying
Re

1
λ
qi(z)

> 0
and
Re

α
µ
− 1

zq′i(z)
qi(z)
+ 1+ zq
′′
i (z)
q′i(z)

> 0.
272 R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274
Let
hi(z) :=

zg ′i (z)
gi(z)
α 
(1− λ) zg
′
i (z)
gi(z)
+ λ

1+ zg
′′
i (z)
g ′i (z)
µ
(i = 1, 2).
If f ∈ A satisfies 0 ≠ zf ′(z)f (z) ∈ H [1, 1] ∩ Q and ( zf
′(z)
f (z) )
α((1− λ) zf ′(z)f (z) + λ(1+ zf
′′(z)
f ′(z) ))
µ is univalent in D, then
h1(z) ≺

zf ′(z)
f (z)
α 
(1− λ) zf
′(z)
f (z)
+ λ

1+ zf
′′(z)
f ′(z)
µ
≺ h2(z)
implies
zg ′1(z)
g1(z)
≺ zf
′(z)
f (z)
≺ zg
′
2(z)
g2(z)
.
Proof. The result follows from Theorem 2.5 by taking γ = δ = 0, β = 1/λ, and
p(z) := zf
′(z)
f (z)
and qi(z) := zg
′
i (z)
gi(z)
(i = 1, 2). 
The following two corollaries are immediate consequences of Theorem 2.5 (or Theorem 3.1).
Corollary 3.2 ([6]). Let α ∈ C, and qi(z) ≠ 0 (i = 1, 2) be univalent in D. Assume that Re[αqi(z)] > 0 for i = 1, 2 and
zq′i(z)/qi(z) (i = 1, 2) is starlike univalent in D. If f ∈ A , 0 ≠ zf ′(z)/f (z) ∈ H [1, 1] ∩ Q, (1 − α) zf
′(z)
f (z) + α(1 + zf
′′(z)
f ′(z) ) is
univalent in D, then
q1(z)+ α zq
′
1(z)
q1(z)
≺ (1− α) zf
′(z)
f (z)
+ α

1+ zf
′′(z)
f ′(z)

≺ q2(z)+ α zq
′
2(z)
q2(z)
implies
q1(z) ≺ zf
′(z)
f (z)
≺ q2(z).
Further q1 and q2 are the best subordinant and best dominant respectively.
Corollary 3.3 ([6]). Let qi(z) ≠ 0 be univalent in D with Re qi(z) > 0. Let zq′i(z)/q2i (z) be starlike univalent in D for i = 1, 2.
If f ∈ A , 0 ≠ zf ′(z)/f (z) ∈ H [1, 1] ∩ Q , 1+zf ′′(z)/f ′(z)zf ′(z)/f (z) is univalent in D, then
1+ zq
′
1(z)
q21(z)
≺ 1+ zf
′′(z)/f ′(z)
zf ′(z)/f (z)
≺ 1+ zq
′
2(z)
q22(z)
implies q1(z) ≺ zf ′(z)/f (z) ≺ q2(z). Further q1 and q2 are the best subordinant and best dominant respectively.
Another application of Theorem 2.5 yields the following result.
Corollary 3.4 ([6]). Let q1 and q2 be convex univalent in D. Let 0 ≠ α ∈ C, and assume that Re qi(z) > Re α−12α for i = 1, 2. If
f ∈ A , zf ′(z)/f (z) ∈ H [1, 1] ∩ Q, zf ′(z)f (z) + α z
2f ′′(z)
f (z) is univalent in D, then
(1− α)q1(z)+ αq21(z)+ αzq′1(z) ≺
zf ′(z)
f (z)

1+ α zf
′′(z)
f ′(z)

≺ (1− α)q2(z)+ αq22(z)+ αzq′2(z)
implies
q1(z) ≺ zf
′(z)
f (z)
≺ q2(z).
Further q1 and q2 are the best subordinant and best dominant respectively.
R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274 273
4. Application toΦ-like functions
LetΦ be an analytic function in a domain containing f (D),Φ(0) = 0 andΦ ′(0) > 0. A function f ∈ A is calledΦ-like if
Re
zf ′(z)
Φ(f (z))
> 0 (z ∈ D).
This concept was introduced by Brickman [16] and it was shown that an analytic function f ∈ A is univalent if and only if
f is Φ-like for some Φ . When Φ(w) = w and Φ(w) = λw, the Φ-like function f is respectively starlike and spiral-like of
type arg λ. Ruscheweyh [17] introduced and studied the following general class ofΦ-like functions.
Definition 4.1. LetΦ be analytic in a domain containing f (D),Φ(0) = 0,Φ ′(0) = 1 andΦ(ω) ≠ 0 for ω ∈ f (D)− {0}. Let
q be a fixed analytic function in D, with q(0) = 1. A function f ∈ A is calledΦ-like with respect to q if
zf ′(z)
Φ(f (z))
≺ q(z) (z ∈ D).
Theorem 4.2. Let α ≠ 0 be a complex number and qi (i = 1, 2) be convex univalent in D. Define hi by
hi(z) := αq2i (z)+ (1− α)qi(z)+ αzq′i(z) (i = 1, 2),
and suppose that
Re

1− α
α
+ 2qi(z)

> 0 (i = 1, 2; z ∈ D).
If f ∈ A satisfies f ∈ H [1, 1] ∩ Q and zf ′(z)
Φ(f (z)) (1+ αzf
′′(z)
f ′(z) + αz(f
′(z)−(Φ(f (z)))′)
Φ(f (z)) ) is univalent in D, then
h1(z) ≺ zf
′(z)
Φ(f (z))

1+ αzf
′′(z)
f ′(z)
+ αz(f
′(z)− (Φ(f (z)))′)
Φ(f (z))

≺ h2(z) (7)
implies
q1(z) ≺ zf
′(z)
Φ(f (z))
≺ q2(z).
Further q1 and q2 are the best subordinant and the best dominant respectively.
Proof. Define the function p by
p(z) := zf
′(z)
Φ(f (z))
(z ∈ D). (8)
Then the function p is analytic in Dwith p(0) = 1. From (8), it follows that
zf ′(z)
Φ(f (z))

1+ αzf
′′(z)
f ′(z)
+ αz(f
′(z)− (Φ(f (z)))′)
Φ(f (z))

= p(z)

1+ α

zp′(z)
p(z)
− 1

+ αp(z)

= αp2(z)+ (1− α)p(z)+ αzp′(z). (9)
Substituting (9) in the subordination (7) yields
h1(z) ≺ αp2(z)+ (1− α)p(z)+ αzp′(z) ≺ h2(z).
The result now follows from Theorem 2.5. 
Remark 1. WhenΦ(w) = w, Theorem 4.2 reduces to Corollary 3.4.
References
[1] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, in: Series in Pure and Applied Mathematics, vol. 225, Marcel Dekker,
New York, 2000.
[2] S.S. Miller, P.T. Mocanu, Subordinants of differential superordinations, Complex Variables 48 (10) (2003) 815–826.
[3] T. Bulboacă, Sandwich-type theorems for a class of integral operators, Bull. Belg. Math. Soc. Simon Stevin 13 (3) (2006) 537–550.
[4] T. Bulboacă, Classes of first-order differential superordinations, Demonstratio Math. 35 (2) (2002) 287–292.
[5] T. Bulboacă, A class of superordination-preserving integral operators, Indag. Math. (N.S.) 13 (3) (2002) 301–311.
[6] R.M. Ali, V. Ravichandran, M. Hussain Khan, K.G. Subramanian, Differential sandwich theorems for certain analytic functions, Far East J. Math. Sci. 15
(1) (2004) 87–94.
274 R.M. Ali et al. / Applied Mathematics Letters 25 (2012) 268–274
[7] R.M. Ali, V. Ravichandran, M. Hussain Khan, K.G. Subramanian, Applications of first order differential superordinations to certain linear operators,
Southeast Asian Bull. Math. 30 (5) (2006) 799–810.
[8] R.M. Ali, V. Ravichandran, Classes of meromorphic α-convex functions, Taiwanese J. Math. 14 (4) (2010) 1479–1490.
[9] R.M. Ali, V. Ravichandran, N. Seenivasagan, Differential subordination and superordination of analytic functions defined by theDziok–Srivastava linear
operator, J. Franklin Inst. 347 (2010) 1762–1781.
[10] Z. Lewandowski, S. Miller, E.Zł otkiewicz, Gamma-starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 28 (1974) 53–58. 1976.
[11] Z. Lewandowski, S. Miller, E.Zł otkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56 (1976) 111–117.
[12] S. Kanas, A. Lecko, J. Stankiewicz, Differential subordinations and geometric means, Complex Variables Theory Appl. 28 (3) (1996) 201–209.
[13] A. Lecko, On differential subordinations and inclusion relation between classes of analytic functions, Complex Variables Theory Appl. 40 (4) (2000)
371–385.
[14] S. Singh, S. Gupta, Some applications of a first order differential subordination, J. Inequal. Pure Appl. Math. 5 (3) (2004) Article 78.
[15] S. Kanas, J. Kowalczyk, A note on Briot–Bouquet–Bernoulli differential subordination, Comment. Math. Univ. Carolin. 46 (2) (2005) 339–347.
[16] L. Brickman,Φ-like analytic functions, I, Bull. Amer. Math. Soc. 79 (1973) 555–558.
[17] St. Ruscheweyh, A subordination theorem forΦ-like functions, J. London Math. Soc. 13 (2) (1976) 275–280.


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A first-order differential double subordination with applications

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