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Axioms 2018, 7(2), 27; doi:10.3390/axioms7020027

Article
Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
1
Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt
2
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
Received: 19 January 2018 / Accepted: 17 April 2018 / Published: 24 April 2018

Abstract

:
Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved.
Keywords:
differential subordination; p-valent functions; generalized fractional differintegral operator
JEL Classification:
30C45; 30C50

1. Introduction and Definitions

Denote by A ( p ) the class of analytic and p-valent functions of the form:
f ( z ) = z p + n = 1 a p + n z p + n ( p N = { 1 , 2 , } ; z U = { z C : z < 1 } ) .
For functions f , g analytic in U , f is subordinate to g, written f ( z ) g ( z ) if there exists a function w, analytic in U with w ( 0 ) = 0 and w ( z ) < 1 , such that f ( z ) = g ( w ( z ) ) , z U . If g is univalent in U , then (see [1,2]):
f ( z ) g ( z ) f ( 0 ) = g ( 0 ) and f ( U ) g ( U ) .
If φ ( z ) is analytic in U and satisfies:
H ( φ ( z ) , z φ ( z ) ) h ( z ) ,
then φ is a solution of (2). The univalent function q is called dominant, if φ ( z ) q ( z ) for all φ . A dominant q ˜ is called the best dominant, if q ˜ ( z ) q ( z ) for all dominants q.
Let 2 F 1 ( a , b ; c ; z ) c 0 , 1 , 2 , be the well-known (Gaussian) hypergeometric function defined by:
2 F 1 ( a , b ; c ; z ) : = n = 0 ( a ) n ( b ) n ( c ) n ( 1 ) n z n , z U ,
where:
( λ ) n : = Γ λ + n Γ λ .
We will recall some definitions that will be used in our paper.
Definition 1.
For f ( z ) A ( p ) , the fractional integral and fractional derivative operators of order λ are defined by Owa [3] (see also [4]) as:
D z λ f ( z ) : = 1 Γ ( λ ) 0 z f ( ζ ) ( z ζ ) 1 λ d ζ ( λ > 0 ) ,
D z λ f ( z ) : = 1 Γ ( 1 λ ) d d z 0 z f ( ζ ) ( z ζ ) λ d ζ ( 0 λ < 1 ) ,
where f is an analytic function in a simply-connected region of the complex z-plane containing the origin, and the multiplicity of ( z ζ ) λ 1 ( ( z ζ ) λ ) is removed by requiring log ( z ζ ) to be real when z ζ > 0 .
Definition 2.
For f ( z ) A ( p ) and in terms of 2 F 1 , the generalized fractional integral and generalized fractional derivative operators defined by Srivastava et al. [5] (see also [6]) as:
I 0 , z λ , μ , η f ( z ) : = z λ μ Γ ( λ ) 0 z ( z ζ ) λ 1 f ( ζ ) 2 F 1 μ + λ , η ; λ ; 1 ζ z d ζ λ > 0 , μ , η R ,
J 0 , z λ , μ , η f ( z ) : = d d z z λ μ 0 z ( z ζ ) λ f ( ζ ) 2 F 1 μ λ , 1 η ; 1 λ ; 1 ζ z d ζ Γ ( 1 λ ) ( 0 λ < 1 ) , d n d z n J 0 , z λ n , μ , η f ( z ) ( n λ < n + 1 ; n N ) ,
where f ( z ) is an analytic function in a simply-connected region of the complex z plane containing the origin with the order f ( z ) = O ( z ε ) , z 0 when ε > max { 0 , μ η } 1 , and the multiplicity of ( z ζ ) λ 1 ( ( z ζ ) λ ) is removed by requiring log ( z ζ ) to be real when z ζ > 0 .
We note that:
I 0 , z λ , λ , η f ( z ) = D z λ f ( z ) ( λ > 0 ) and J 0 , z λ , λ , η f ( z ) = D z λ f ( z ) ( 0 λ < 1 ) ,
where D z λ f ( z ) and D z λ f ( z ) are the fractional integral and fractional derivative operators studied by Owa [3].
Goyal and Prajapat [7] (see also [8]) defined the operator:
S 0 , z λ , μ , η , p f ( z ) = Γ ( p + 1 μ ) Γ ( p + 1 λ + η ) Γ ( p + 1 ) Γ ( p + 1 μ + η ) z μ J 0 , z λ , μ , η f ( z ) ( 0 λ < η + p + 1 ; z U ) , Γ ( p + 1 μ ) Γ ( p + 1 λ + η ) Γ ( p + 1 ) Γ ( p + 1 μ + η ) z μ I 0 , z λ , μ , η f ( z ) ( < λ < 0 ; z U ) .
For f ( z ) A ( p ) , we have:
S 0 , z λ , μ , η , p f ( z ) = z 3 p F 2 ( 1 , 1 + p , 1 + p + η μ ; 1 + p μ , 1 + p + η λ ; z ) f ( z ) = z p + n = 1 ( p + 1 ) n ( p + 1 μ + η ) n ( p + 1 μ ) n ( p + 1 λ + η ) n a p + n z p + n ( p N ; μ , η R ; μ < p + 1 ; < λ < η + p + 1 ) ,
where “∗” stands for convolution of two power series, and q F s ( q s + 1 ; q , s N 0 = N { 0 } ) is the well-known generalized hypergeometric function.
Let:
G p , η , μ λ ( z ) = z p + n = 1 ( p + 1 ) n ( p + 1 μ + η ) n ( p + 1 μ ) n ( p + 1 λ + η ) n z p + n ( p N ; μ , η R ; μ < p + 1 ; < λ < η + p + 1 ) .
and:
G p , η , μ λ ( z ) G p , η , μ λ ( z ) 1 = z p ( 1 z ) δ + p ( δ > p ; z U ) .
Tang et al. [9] (see also [10,11,12,13,14,15]) defined the operator H p , η , μ λ , δ : A ( p ) A ( p ) , where:
H p , η , μ λ , δ f ( z ) = z p + n = 1 ( δ + p ) n ( p + 1 μ ) n ( p + 1 λ + η ) n ( 1 ) n ( p + 1 ) n ( p + 1 μ + η ) n a p + n z p + n ( p N , δ > p , μ , η R , μ < p + 1 , < λ < η + p + 1 ) .
It is easy to verify that:
z H p , η , μ λ , δ f ( z ) = ( δ + p ) H p , η , μ λ , δ + 1 f ( z ) δ H p , η , μ λ , δ f ( z ) ,
and:
z H p , η , μ λ + 1 , δ f ( z ) = ( p + η λ ) H p , η , μ λ , δ f ( z ) ( η λ ) H p , η , μ λ + 1 , δ f ( z ) .
By using the operator H p , η , μ λ , δ , we introduce the following class.
Definition 3.
For A , B ( 1 B < A 1 ) , f A ( p ) is in the class T p , η , μ λ , δ ( A , B ) if
( H p , η , μ λ , δ f ( z ) ) p z p 1 1 + A z 1 + B z ( z U ; p N ) ,
which is equivalent to:
( H p , η , μ λ , δ f ( z ) ) p z p 1 1 B ( H p , η , μ λ , δ f ( z ) ) p z p 1 A < 1 ( z U ) .
For convenience, we write T p , η , μ λ , δ 1 2 ξ p , 1 = T p , η , μ λ , δ ( ξ ) ( 0 ξ < p ) , which satisfies the inequality:
( H p , η , μ λ , δ f ( z ) ) z p 1 > ξ ( 0 ξ < p ) .
In this paper, we investigate some subordination and convolution properties for classes of multivalent functions, which are defined by a generalized fractional differintegral operator. The theory of subordination received great attention, particularly in many subclasses of univalent and multivalent functions (see, for example, [13,15,16,17]).

2. Preliminaries

To prove our main results, we shall need the following lemmas.
Lemma 1.
[18]. Let h be an analytic and convex (univalent) function in U with h ( 0 ) = 1 . Additionally, let φ given by:
ϕ ( z ) = 1 + c n z n + c n + 1 z n + 1 +
be analytic in U . If:
ϕ ( z ) + z ϕ ( z ) σ h ( z ) ( ( σ ) 0 ; σ 0 ) ,
then:
ϕ ( z ) ψ ( z ) = σ n z σ n 0 z t σ n 1 h ( t ) d t h ( z ) ,
and ψ is the best dominant of (6).
Denote by P ( ς ) the class of functions Φ given by:
Φ ( z ) = 1 + c 1 z + c 1 z 2 + ,
which are analytic in U and satisfy the following inequality:
Φ ( z ) > ς ( 0 ς < 1 ) .
Using the well-known growth theorem for the Carathéodory functions (cf., e.g., [19]), we may easily deduce the following result:
Lemma 2.
[19]. If Φ P ( ς ) . Then
Φ ( ς ) 2 ς 1 + 2 ( 1 ς ) 1 + z ( 0 ς < 1 ) .
Lemma 3.
[20]. For 0 ς 1 , ς 2 < 1 ,
P ( ς 1 ) P ( ς 2 ) P ( ς 3 ) ( ς 3 = 1 2 ( 1 ς 1 ) ( 1 ς 2 ) ) .
The result is the best possible.
Lemma 4.
[21] . Let ϕ be such that φ ( 0 ) = 1 and φ ( z ) 0 and A , B C , with A B , B 1 , ν C * .
( i ) If ν ( A B ) B 1 1 or ν ( A B ) B + 1 1 , B 0 and φ ( z ) satisfies:
1 + z φ ( z ) ν φ ( z ) 1 + A z 1 + B z ,
then:
φ ( z ) ( 1 + B z ) ν A B B
and this is the best dominant.
( i i ) If B = 0 and ν A < π and if φ satisfies:
1 + z φ ( z ) ν φ ( z ) 1 + A z ,
then:
φ ( z ) e ν A z ,
and this is the best dominant.
Lemma 5.
[2]. Let Ω C , b C , b > 0 and ψ : C 2 × U C satisfy ψ i x , y ; z Ω for all x , y b i x 2 2 b and all z U . If p ( z ) = 1 + p 1 z + p 2 z 2 + , is analytic in U and if:
ψ p ( z ) , z p z ; z Ω ,
then p ( z ) > 0 in U .
Lemma 6.
[22]. Let ψ z be analytic in U with ψ ( 0 ) = 1 and ψ ( z ) 0 for all z . If there exist two points z 1 , z 2 U such that:
π 2 ρ 1 = arg { ψ ( z 1 ) } < arg { ψ ( z ) } < π 2 ρ 2 = arg { ψ ( z 2 ) } ,
for some ρ 1 and ρ 2 ρ 1 , ρ 2 > 0 and for all z z < z 1 = z 2 , then:
z 1 ψ ( z 1 ) ψ ( z 1 ) = i ρ 1 + ρ 2 2 κ a n d z 2 ψ ( z 2 ) ψ ( z 2 ) = i ρ 1 + ρ 2 2 κ ,
where:
κ 1 a 1 + a a n d a = i tan ρ 2 ρ 1 ρ 2 + ρ 1 .

3. Properties Involving H p , η , μ λ , δ

Unless otherwise mentioned, we assume throughout this paper that p N , δ > p , μ , η R , μ < p + 1 , < λ < η + p + 1 , 1 B < A 1 , θ > 0 , and the powers are considered principal ones.
Theorem 1.
Let f A ( p ) satisfy:
( 1 θ ) H p , η , μ λ , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ + 1 f ( z ) p z p 1 1 + A z 1 + B z .
Then:
H p , η , μ λ , δ f ( z ) p z p 1 1 τ > δ + p θ 0 1 u δ + p θ 1 1 A u 1 B u d u 1 τ , τ 1 .
The estimate in (13) is sharp.
Proof. 
Let:
ϕ ( z ) = H p , η , μ λ , δ f ( z ) p z p 1 ( z U ) .
Then, φ is analytic in U . After some computations, we get:
( 1 θ ) H p , η , μ λ , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ + 1 f ( z ) p z p 1 = ϕ ( z ) + θ z ϕ ( z ) δ + p 1 + A z 1 + B z .
Now, by using Lemma 1, we deduce that:
H p , η , μ λ , δ f ( z ) p z p 1 δ + p θ z δ + p θ 0 z t δ + p θ 1 1 + A t 1 + B t d t ,
or, equivalently,
H p , η , μ λ , δ f ( z ) p z p 1 = δ + p θ 0 1 u δ + p θ 1 1 + A u w ( z ) 1 + B u w ( z ) d u ,
and so:
H p , η , μ λ , δ f ( z ) p z p 1 > δ + p θ 0 1 u δ + p θ 1 1 A u 1 B u d u .
Since:
χ 1 τ χ 1 τ χ C , χ 0 , τ 1 .
The inequality (13) now follows from (16) and (17). To prove that the result is sharp, let:
H p , η , μ λ , δ f ( z ) p z p 1 = δ + p θ 0 1 u δ + p θ 1 1 + A u z 1 + B u z d u .
Now, for f ( z ) defined by (18), we have:
( 1 θ ) H p , η , μ λ , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ + 1 f ( z ) p z p 1 = 1 + A z 1 + B z z U ,
Letting z 1 , we obtain:
H p , η , μ λ , δ f ( z ) p z p 1 δ + p θ 0 1 u δ + p θ 1 1 A u 1 B u d u ,
which ends our proof. ☐
Putting θ = 1 and using Lemma 1 for Equation (15) in Theorem 1, we obtain the following example.
Example 1.
Let the function f ( z ) A ( p ) . Then, following containment property holds,
T p , μ , η λ , δ + 1 ( A , B ) T p , μ , η λ , δ ( A , B ) .
Using (4) instead of (3) in Theorem 1, one can prove the following theorem.
Theorem 2.
Let f A ( p ) satisfy
( 1 θ ) H p , η , μ λ + 1 , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ f ( z ) p z p 1 1 + A z 1 + B z .
Then:
H p , η , μ λ + 1 , δ f ( z ) p z p 1 1 τ > p + η λ θ 0 1 u p + η λ θ 1 1 A u 1 B u d u 1 τ , τ 1 .
The result is sharp.
Putting θ = 1 in Theorem 2, we obtain the following example.
Example 2.
Let the function f ( z ) A ( p ) . Then, following inclusion property holds
T p , μ , η λ , δ ( A , B ) T p , μ , η λ + 1 , δ ( A , B ) .
For a function f A ( p ) , the generalized Bernardi–Libera–Livingston integeral operator F p , γ is defined by (see [23]):
F p , γ f ( z ) = γ + p z p 0 z t γ 1 f ( t ) d t = z p + k = 1 γ + p γ + p + k z p + k f ( z ) ( γ > p ) = z p 3 F 2 1 , 1 , γ + p ; 1 , γ + p + 1 ; z f ( z ) .
Lemma 7.
If f A p , prove that:
(i) 
H p , η , μ λ , δ F p , γ f = F p , γ H p , η , μ λ , δ f ,
(ii) 
z H p , η , μ λ , δ F p , γ f ( z ) = ( p + γ ) H p , η , μ λ , δ f ( z ) γ H p , η , μ λ , δ F p , γ f ( z ) .
Proof. 
Since
H p , η , μ λ , δ F p , γ f = z p 3 F 2 δ + p , p + 1 μ , p + 1 λ + η ; p + 1 , p + 1 μ + η ; z F p , γ f = z p 3 F 2 δ + p , p + 1 μ , p + 1 λ + η ; p + 1 , p + 1 μ + η ; z z p 3 F 2 1 , 1 , γ + p ; 1 , γ + p + 1 ; z f ( z ) ,
and:
F p , γ H p , η , μ λ , δ f = z p 3 F 2 1 , 1 , γ + p ; 1 , γ + p + 1 ; z H p , η , μ λ , δ f = z p 3 F 2 1 , 1 , γ + p ; 1 , γ + p + 1 ; z z p 3 F 2 δ + p , p + 1 μ , p + 1 λ + η ; p + 1 , p + 1 μ + η ; z f ( z ) .
Now, the first part of this lemma follows. Furthermore,
z F p , γ f ( z ) = ( p + γ ) f ( z ) γ F p , γ f ( z ) .
If we replace f ( z ) by H p , η , μ λ , δ f ( z ) and using the first part of this lemma, we get (21). ☐
Theorem 3.
Suppose that p + γ > 0 , f T p , η , μ λ , δ ( A , B ) and F p , γ defined by (20). Then:
H p , η , μ λ , δ F p , γ f ( z ) p z p 1 1 τ > p + γ 0 1 u p + γ 1 1 A u 1 B u d u 1 τ , τ 1 .
The result is sharp.
Proof. 
Let:
ϕ ( z ) = H p , η , μ λ , δ F p , γ f ( z ) p z p 1 ( z U ) .
Then, φ is analytic in U . After some calculations, we have:
( H p , η , μ λ , δ f ( z ) ) p z p 1 = ϕ ( z ) + z ϕ ( z ) p + γ 1 + A z 1 + B z .
Employing the same technique that was used in proving Theorem 1, the remaining part of the theorem can be proven. ☐
Theorem 4.
Let 1 B i < A i 1 ( i = 1 , 2 ) . If each of the functions f i A ( p ) satisfies:
( 1 θ ) H p , η , μ λ , δ f i ( z ) z p + θ H p , η , μ λ , δ + 1 f i ( z ) z p 1 + A i z 1 + B i z ( i = 1 , 2 ) ,
then:
( 1 θ ) H p , η , μ λ , δ F ( z ) z p + θ H p , η , μ λ , δ + 1 F ( z ) z p 1 + ( 1 2 ϱ ) z 1 z ,
where:
F ( z ) = H p , η , μ λ , δ ( f 1 f 2 ) ( z )
and:
ϱ = 1 4 ( A 1 B 1 ) ( A 2 B 2 ) ( 1 B 1 ) ( 1 B 2 ) 1 1 2 2 F 1 1 , 1 ; δ + p θ + 1 ; 1 2 .
The result is possible when B 1 = B 2 = 1 .
Proof. 
Suppose that f i A ( p ) ( i = 1 , 2 ) satisfy the condition (25). Setting:
p i ( z ) = ( 1 θ ) H p , η , μ λ , δ f i ( z ) z p + θ H p , η , μ λ , δ + 1 f i ( z ) z p ( i = 1 , 2 ) ,
we have:
p i ( z ) P ( ς i ) ς i = 1 A i 1 B i , i = 1 , 2 .
Thus, by making use of the identity (3) in (29), we get:
H p , η , μ λ , δ f i ( z ) = δ + p θ z p δ + p θ 0 z t δ + p θ 1 p i ( t ) d t ( i = 1 , 2 ) ,
which, in view of F given by (27) and (30), yields:
H p , η , μ λ , δ F ( z ) = δ + p θ z p δ + p θ 0 z t δ + p θ 1 F ( t ) d t ,
where:
F ( z ) = ( 1 θ ) H p , η , μ λ , δ F ( z ) z p + θ H p , η , μ λ , δ + 1 F ( z ) z p = δ + p θ z δ + p θ 0 z t δ + p θ 1 ( p 1 p 2 ) ( t ) d t .
Since p i ( z ) P ( ς i ) ( i = 1 , 2 ) , it follows from Lemma 3 that:
( p 1 p 2 ) ( z ) P ( ς 3 ) ( ς 3 = 1 2 ( 1 ς 1 ) ( 1 ς 2 ) ) .
Now, by using (33) in (32) and then appealing to Lemma 2, we have:
F ( z ) = δ + p θ 0 1 u δ + p θ 1 ( p 1 p 2 ) ( u z ) d u δ + p θ 0 1 u δ + p θ 1 2 ς 3 1 + 2 ( 1 ς 3 ) 1 + u z d u > δ + p θ 0 1 u δ + p θ 1 2 ς 3 1 + 2 ( 1 ς 3 ) 1 + u d u = 1 4 ( A 1 B 1 ) ( A 2 B 2 ) ( 1 B 1 ) ( 1 B 2 ) 1 δ + p θ 0 1 u δ + p θ 1 ( 1 + u ) 1 d u = 1 4 ( A 1 B 1 ) ( A 2 B 2 ) ( 1 B 1 ) ( 1 B 2 ) 1 1 2 2 F 1 1 , 1 ; δ + p θ + 1 ; 1 2 = ϱ .
When B 1 = B 2 = 1 , we consider the functions f i ( z ) A ( p ) ( i = 1 , 2 ) , which satisfy (25), are defined by:
H p , η , μ λ , δ f i ( z ) = δ + p θ z p δ + p θ 0 z t δ + p θ 1 1 + A i t 1 t d t ( i = 1 , 2 ) .
Thus, it follows from (32) that:
F ( z ) = δ + p θ 0 1 u δ + p θ 1 1 ( 1 + A 1 ) ( 1 + A 2 ) + ( 1 + A 1 ) ( 1 + A 2 ) ( 1 u z ) d u
= 1 ( 1 + A 1 ) ( 1 + A 2 ) + ( 1 + A 1 ) ( 1 + A 2 ) ( 1 z ) 1 2 F 1 1 , 1 ; δ + p θ + 1 ; z z 1
1 ( 1 + A 1 ) ( 1 + A 2 ) + 1 2 ( 1 + A 1 ) ( 1 + A 2 ) 2 F 1 1 , 1 ; δ + p θ + 1 ; 1 2 as z 1 ,
which evidently ends the proof. ☐
Theorem 5.
Let υ C , and let A , B C with A B and B 1 . Suppose that:
υ ( δ + p ) ( A B ) B 1 1 o r υ ( δ + p ) ( A B ) B + 1 1 i f B 0 , υ ( δ + p ) A π i f B = 0 .
If f A ( p ) with H p , η , μ λ , δ f ( z ) 0 for all z U = U { 0 } , then:
H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) 1 + A z 1 + B z ,
implies:
H p , η , μ λ , δ f ( z ) z p υ q ( z ) ,
where:
q ( z ) = ( 1 + B z ) υ ( δ + p ) ( A B ) / B i f B 0 , e υ ( δ + p ) A z i f B = 0 ,
is the best dominant.
Proof. 
Putting:
Δ ( z ) = H p , η , μ λ , δ f ( z ) z p υ ( z U ) .
Then, Δ is analytic in U , Δ ( 0 ) = 1 and Δ ( z ) 0 for all z U . Taking the logarithmic derivatives on both sides of (34) and using (3), we have:
1 + z Δ ( z ) υ ( δ + p ) Δ ( z ) = H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) 1 + A z 1 + B z .
Now, the assertions of Theorem 5 follow by Lemma 4. ☐
Theorem 6.
Let 0 α 1 , ζ > 1 . If f ( z ) A p satisfies:
1 α H p , η , μ λ , δ + 2 f ( z ) H p , η , μ λ , δ + 1 f ( z ) + α H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) < ζ ,
then:
H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) < β ,
where β 1 , is the positive root of the equation:
2 δ + p + α β 2 2 ζ δ + p + 1 1 α β 1 α = 0 .
Proof. 
Let:
H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) = β + 1 β φ ( z ) .
Then, φ is analytic in U , φ ( 0 ) = 1 and φ ( z ) 0 for all z U . Taking the logarithmic derivatives on both sides of (37) and using the identity (3), we have:
δ + p + 1 H p , η , μ λ , δ + 2 f ( z ) H p , η , μ λ , δ + 1 f ( z ) δ + p H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) = 1 + 1 β z φ ( z ) β + 1 β φ ( z ) ,
and so:
1 α H p , η , μ λ , δ + 2 f ( z ) H p , η , μ λ , δ + 1 f ( z ) + α H p , η , μ λ , δ + 1 f ( z ) H p , η , μ λ , δ f ( z ) = α β + 1 α δ + p β δ + p + 1
+ 1 β α + 1 α δ + p δ + p + 1 φ ( z ) + 1 α 1 β β + 1 β φ ( z ) δ + p + 1 z φ z .
Let:
Ψ r , s ; z = α β + 1 α δ + p β δ + p + 1 + 1 β α + 1 α δ + p δ + p + 1 r + 1 α 1 β β + 1 β φ ( z ) δ + p + 1 s ,
and:
Ω = w C : ( w ) < ζ .
Then, for x , y 1 + x 2 2 , we have:
Ψ i x , y ; z = α β + 1 α δ + p β δ + p + 1 + 1 α 1 β β y β 2 + 1 β 2 x 2 δ + p + 1 α β + 1 α δ + p β δ + p + 1 1 α 1 β 2 β δ + p + 1 = ζ ,
where β is the positive root of Equation (36). Suppose that:
R β = 2 δ + p + α β 2 2 ζ δ + p + 1 1 α β 1 α = 0 .
For β = 0 , R 0 = 1 α 0 and for β = 1 , R 1 = 2 δ + p 1 ζ + 2 α ζ 0 . This proves that β 1 , . Thus, for z U , Ψ i x , y ; z Ω , and so, we obtain the required result by an application of Lemma 5. ☐
Theorem 7.
Suppose that 0 < ε 1 , ε 2 1 . If:
π 2 ε 1 < arg ( 1 θ ) H p , η , μ λ , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ + 1 f ( z ) p z p 1 < π 2 ε 2 ,
then:
π 2 ξ 1 < arg H p , η , μ λ , δ f ( z ) p z p 1 < π 2 ξ 2 ,
where:
ε 1 = ξ 1 + 2 π arctan ξ 1 + ξ 2 θ 2 δ + p 1 a 1 + a , ε 2 = ξ 2 + 2 π arctan ξ 1 + ξ 2 θ 2 δ + p 1 a 1 + a .
Proof. 
Let:
ϕ ( z ) = H p , η , μ λ , δ f ( z ) p z p 1 ( z U ) .
Then, from Theorem 1, we have:
( 1 θ ) H p , η , μ λ , δ f ( z ) p z p 1 + θ H p , η , μ λ , δ + 1 f ( z ) p z p 1 = ϕ ( z ) + θ z ϕ ( z ) δ + p .
Let U ( z ) be the function that maps U onto the domain:
w C : π 2 ε 1 < arg ( w ) < π 2 ε 2 ,
with U ( 0 ) = 1 , then:
ϕ ( z ) + θ z ϕ ( z ) δ + p U ( z ) .
Assume that z 1 , z 2 are two points in U such that the condition (9) is satisfied, then by Lemma 6, we obtain (10) under the constraint (11). Therefore,
arg δ + p ϕ ( z 1 ) + θ z 1 ϕ ( z 1 ) = arg ϕ ( z 1 ) δ + p + θ z 1 ϕ ( z 1 ) ϕ ( z 1 ) = arg ϕ ( z 1 ) + arg δ + p + θ z 1 ϕ ( z 1 ) ϕ ( z 1 ) = π 2 ξ 1 + arg δ + p i θ ξ 1 + ξ 2 κ 2 = π 2 ξ 1 arctan ξ 1 + ξ 2 θ κ 2 δ + p π 2 ξ 1 arctan ξ 1 + ξ 2 θ 2 δ + p 1 a 1 + a ,
and:
arg δ + p ϕ ( z 2 ) + θ z 2 ϕ ( z 2 ) π 2 ξ 2 + arctan ξ 1 + ξ 2 θ 2 δ + p 1 a 1 + a .
which contradicts the assumption (38). This evidently completes the proof of Theorem 7. ☐

Acknowledgments

The authors would like to thank all referees for their valuable comments which led to the improvement of this paper.

Author Contributions

All the authors read and approved the final manuscript as a consequence of the authors meetings.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Bulboacă, T. Differential Subordinations and Superordinations, Recent Results; House of Scientific Book Publ.: Cluj-Napoca, Romania, 2005. [Google Scholar]
  2. Miller, S.S.; Mocanu, P.T. Differential Subordination: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker Inc.: New York, NY, USA, 2000; Volume 225. [Google Scholar]
  3. Owa, S. On the distortion theorems I. Kyungpook Math. J. 1978, 18, 53–59. [Google Scholar]
  4. Owa, S.; Srivastava, H.M. Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39, 1057–1077. [Google Scholar] [CrossRef]
  5. Srivastava, H.M.; Saigo, M.; Owa, S. A class of distortion theorems involving certain operators of fractional calculus. J. Math. Anal. Appl. 1988, 131, 412–420. [Google Scholar] [CrossRef]
  6. Prajapat, J.K.; Raina, R.K̇.; Srivastava, H.M. Some inclusion properties for certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operators. Integral Transform. Spec. Funct. 2007, 18, 639–651. [Google Scholar] [CrossRef]
  7. Goyal, S.P.; Prajapat, J.K. A new class of analytic p-valent functions with negative coefficients and fractional calculus operators. Tamsui Oxf. J. Math. Sci. 2004, 20, 175–186. [Google Scholar]
  8. Prajapat, J.K.; Aouf, M.K. Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator. Comput. Math. Appl. 2012, 63, 42–47. [Google Scholar] [CrossRef]
  9. Tang, H.; Deng, G.; Li, S.; Aouf, M.K. Inclusion results for certain subclasses of spiral-like multivalent functions involving a generalized fractional differintegral operator. Integral Transform. Spec. Funct. 2013, 24, 873–883. [Google Scholar] [CrossRef]
  10. Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. Some characterizations of integral operators associated with certain classes of p-valent functions defined by the Srivastava-Saigo-Owa fractional differintegral operator. Complex Anal. Oper. Theory 2016, 10, 1267–1275. [Google Scholar] [CrossRef]
  11. Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator. Quaest. Math. 2016, 39, 545–560. [Google Scholar] [CrossRef]
  12. Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator. Afr. Mat. 2017, 28, 99–107. [Google Scholar] [CrossRef]
  13. Mostafa, A.O.; Aouf, M.K.; Zayed, H.M.; Bulboacă, T. Multivalent functions associated with Srivastava-Saigo-Owa fractional differintegral operator. RACSAM 2017. [Google Scholar] [CrossRef]
  14. Mostafa, A.O.; Aouf, M.K.; Zayed, H.M. Inclusion relations for subclasses of multivalent functions defined by Srivastava–Saigo–Owa fractional differintegral operator. Afr. Mat. 2018. [Google Scholar] [CrossRef]
  15. Mostafa, A.O.; Aouf, M.K.; Zayed, H.M. Subordinating results for p-valent functions associated with the Srivastava–Saigo–Owa fractional differintegral operator. Afr. Mat. 2018. [Google Scholar] [CrossRef]
  16. Mostafa, A.O.; Aouf, M.K. Some applications of differential subordination of p-valent functions associated with Cho-Kwon-Srivastava operator. Acta Math. Sin. (Engl. Ser.) 2009, 25, 1483–1496. [Google Scholar] [CrossRef]
  17. Wang, Z.; Shi, L. Some properties of certain extended fractional differintegral operator. RACSAM 2017, 1–11. [Google Scholar] [CrossRef]
  18. Hallenbeck, D.Z.; Ruscheweyh, S. Subordination by convex functions. Proc. Am. Math. Soc. 1975, 52, 191–195. [Google Scholar] [CrossRef]
  19. Pommerenke, C. Univalent Functions; Vandenhoeck & Ruprecht: Göttingen, Germany, 1975. [Google Scholar]
  20. Stankiewicz, J.; Stankiewicz, Z. Some applications of Hadamard convolution in the theory of functions. Ann. Univ. Mariae Curie-Sklodowska Sect. A 1986, 40, 251–265. [Google Scholar]
  21. Obradović, M.; Owa, S. On certain properties for some classes of starlike functions. J. Math. Anal. Appl. 1990, 145, 357–364. [Google Scholar] [CrossRef]
  22. Takahashi, N.; Nunokawa, M. A certain connection between starlike and convex functions. Appl. Math. Lett. 2003, 16, 653–655. [Google Scholar] [CrossRef]
  23. Choi, J.H.; Saigo, M.; Srivastava, H.M. Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276, 432–445. [Google Scholar] [CrossRef]

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