Differential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications

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International Journal of Advances in Applied Mathematics and Mechanics

Differential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications

Research Article

Abbas Kareem Wanas^∗

Department of Mathematics,College of Computer Science and Mathematics, University of Al-Qadisiya,Diwaniya, Iraq

Received 15 December 2014; accepted (in revised version) 09 March 2015

Abstract: In the present paper, we study new classes of meromorphic multivalent quasi-convex functions, we obtain some subordination theorems for such classes in punctured unit disk. Also we give some applications of firstâ ˘A¸Sorder differential subordination.

MSC: 34G10• 26A33 • 30C45

Keywords: Meromorphic multivalent quasi-convex functions• Differential subordination • Derivative operator.

1. Introduction

Let L_p(λ) denotes the class of all functions f of the form:

f(z ) = z^−p+ X∞ n=0

a_nz^n−λ 0< λ < 1,p ∈ N = {1,2,···} , (1)

which are analytic in the punctured unit disk U = {z ∈ C : 0 < |z | < 1}.

Also, let T_p(λ) denotes the class of all functions f of the form:

f(z ) = z^−p− X∞ n=0

a_nz^n−λ a_n> 0,0 < λ < 1,p ∈ N = {1,2,···} , (2)

which are analytic in the punctured unit disk U .

For two functions f and g analytic in∆ = {z ∈ C : |z | < 1}, we say f is subordinate to g in ∆, denote by f ≺ g or f(z ) ≺ g (z )(z ∈ ∆), if there exists a Schwarz function w analytic in U with w (0) = 0 and |w (z )| < 1(z ∈ ∆) such that f(z ) = g (w (z )),(z ∈ ∆). In particular, if the function g is univalent in ∆ , then f ≺ g if and only if f (0) = g (0) and f(∆) ⊂ g (∆).

Letψ : C³× U → C . and let h be univalent in ∆. Assume that k , ψ are analytic and univalent in ∆ if k satisfies the differential subordination

ψ(k(z ), z k⁰(z ), z²k⁰⁰(z ); z ) ≺ h(z ), (3)

then k is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply dominant if k≺ q for all k satisfying (3) . A dominant ˇq

∗ Corresponding author.

E-mail address:[email protected]

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Abbas Kareem Wanas/ Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 126 - 133 that satisfies ˇq≺ q for all dominants q of (3) is said to be the best dominant of (3).

Let L_pbe the class of all functionsΦ of the form:

Φ(z ) = z^−p+ X∞ n=0

a_nzⁿ p∈ N = {1, 2, · · · } , which are analytic in the punctured unit disk U . Also, let Tpbe the class of all functionsΦ of the form:

Φ(z ) = z^−p− X∞ n=0

a_nzⁿ a_n> 0,p ∈ N = {1,2,···} ,

which are analytic in the punctured unit disk U .

A function f ∈ Lp(λ) Tp(λ) is meromorphic multivalent starlike if f (z ) 6= 0 and

−R e

§z f⁰(z ) f(z )

ª

> 0, z ∈ U .

Similar, f ∈ Lp(λ) Tp(λ) is meromorphic multivalent convex if f⁰(z ) 6= 0 and

−R e

§

1+z f⁰⁰(z ) f⁰(z )

ª

> 0, z ∈ U .

A function f ∈ Lp(λ) Tp(λ) is called meromorphic multivalent Quasi-convex function if there exists a meromorphic multivalent convex function g such that g(z ) 6= 0 and

−R e

¨ z f⁰(z )₀ g⁰(z )

«

> 0, z ∈ U .

A functionΦ ∈ Lp T_p is meromorphic multivalent starlike if Φ(z ) 6= 0 and

−R e

§zΦ⁰(z ) Φ(z )

ª

> 0, z ∈ U .

Similar,a functionΦ is meromorphic multivalent convex if Φ⁰(z ) 6= 0 and

−R e

§

1+zΦ⁰⁰(z ) Φ⁰(z )

ª

> 0, z ∈ U .

Moreover, a functionΦ is called meromorphic multivalent Quasi-convex function if there exists a meromorphic multivalent convex functionΨ such that Ψ⁰(z ) 6= 0 and

−R e

(z Φ⁰(z ))⁰ Ψ⁰(z )

> 0, z ∈ U .

2. Preliminaries

Definition 2.1 (Srivastava and Owa[11]).

The fractional derivative of orderλ,(0 < λ < 1) of a function f is defined by D_z^λf(z ) = 1

Γ (1 − λ) d d z

Z z 0

f(")

(z − ")^λ d", (4)

where f is an analytic function in a simply-connected region of the z-plane containing the origin, and the multi- plicity of(z − ")^−λis removed by requiring log(z − ") to be real, when (z − ") > 0.

Let a , b , c∈ C with c 6= 0, −1, −2, · · · . The Gaussian hypergeometric function2F₁( see[12] ) is defined by

2F₁(z ) = 2F₁(a, b, c ; z ) = X∞ n=0

(a)n(b )n

(c )n

zⁿ

n ! (z ∈ ∆),

where(x )nis the Pochhammer symbol defined, in terms of the Gamma function, by (x )n = Γ (x + n)

Γ (x ) = § 1 (n = 0),

x(x + 1)···(x + n − 1) (n ∈ N ).

127

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Definition 2.2 (Goyal and Goyal[4]).

Let 0≤ λ < 1 and µ, ν ∈ R . Then, in terms of familiar (Gauss) hypergeometric function2F₁, the generalized fractional derivative operator J_0,z^λ,µ,νof a function f is defined by:

J_0,z^λ,µ,νf(z ) =







Γ (1−λ)1 d

d zz^λ−µRz

0(z − ")^−λf(").2F₁(µ − λ,−ν;1 − λ;1 −_z^")d " ,(0 ≤ λ < 1)

dⁿ

d zⁿJ_0,z^λ−n,µ,νf(z ) (n ≤ λ < n + 1,n ∈ N ).

(5) where the function f is analytic in a simply-connected region of the z-plane containing the origin, with the order

f(z ) = O(|z |^ε), (z → 0)

forε > ma x {0,µ − ν} − 1 , and the multiplicity of (z − ε)^−λis removed by requiring l o g(z − ε) > 0 to be real, when (z − ε) > 0 .

By comparing (4) with (5), we find J_0,z^λ,λ,νf(z ) = D_z^λf(z ),(0 ≤ λ < 1).

In terms of gamma function, we have J_0,z^λ,µ,νzⁿ = Γ (n + 1)Γ (n − µ + ν + 1)

Γ (n − µ + 1)Γ (n − λ + ν + 1)z^n−µ,(0 ≤ λ < 1,µ,ν ∈ R,n > ma x0,µ − ν − 1). (6)

Lemma 2.1 (Miller and Mocanu[8]).

Let q be univalent in the unit disk4 and θ and φ be analytic in a domain D containing q (4) with φ (w ) 6= 0 when w∈ q (4). Set Q (z ) = z q⁰(z )φ q (z ) and h (z ) = θ q (z ) +Q (z ). Suppose that

1- Q(z ) is starlike univalent in 4, and 2- R e

nz h⁰(z ) Q(z )

o> 0 for z ∈ 4.

Ifθ (k (z )) + z k⁰(z )φ (k (z )) ≺ θ q (z ) + z q⁰(z )φ q (z ) , then k ≺ q and q is the best dominant.

Lemma 2.2 (Shanmugam and et al.[9]).

Let q be convex univalent in the unit disk4 and ψ and γ ∈ C with R e n

1+^{z q}_q0⁰⁰(z )^{(z )}+^ψ_γo

> 0. If k is analytic in 4 and ψk (z ) + γz k⁰(z ) ≺ ψq (z ) + γz q⁰(z ), then k (z ) ≺ q (z ) and q is the best dominant.

Such type of study was carried out by various authors for another classes, like, Ibrahim and Darus[5–7] , Darus and Ibrahim[3], Singh et al. [10], Billing [2] and Atshan and Wanas [1].

3. Subordination results

In this section, we obtain some sufficient conditions for subordination of analytic functions in the classes L_p(λ) and T_p(λ).

Theorem 3.1.

Let the function q be univalent in U , q(z ) 6= 0 and assume that R e

1+ p r +s(1 − t )

t q(z )s−r

+ (r − 1)z q⁰(z )

q(z ) +z q⁰⁰(z ) q⁰(z )

> 0, (7)

where r, s ∈ C , t ∈ C \ {0}. Suppose that z q (z )r−1

q⁰(z ) is starlike univalent in U . If f ∈ Lp(λ) satisfies the subordi- nation

(1 − t ) − z^pf⁰(z )⁰ g⁰(z )

!αs

+ t − z^pf⁰(z )⁰ g⁰(z )

!αr

p+ α z z^pf⁰(z )⁰⁰

z^pf⁰(z )⁰ −z g⁰⁰(z ) g⁰(z )

!



≺ (1 − t ) q (z )s

+ t q (z )r

p+z q⁰(z ) q(z )

, (8)

then

− z^pf⁰(z )⁰ g⁰(z )

!α

≺ q (z ) , (z ∈ U , α ∈ C \ {0}) and q is the best dominant.

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Abbas Kareem Wanas/ Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 126 - 133

Proof. Define the function k by

k(z ) = − z^pf⁰(z )⁰ g⁰(z )

!α

, z∈ U . (9)

Note that

(1 − t )(k (z ))^s+ t (k (z ))^r

p+z k⁰(z ) k(z )

= (1 − t ) − z^pf⁰(z )⁰ g⁰(z )

!αs

+ t − z^pf⁰(z )⁰ g⁰(z )

!αr

×



p+ α z z^pf⁰(z )⁰⁰

z^pf⁰(z )0 −z g⁰⁰(z ) g⁰(z )

!

. (10)

From (8) and (10), we have (1 − t )(k (z ))^s+ t (k (z ))^r

p+z k⁰(z ) k(z )

≺ (1 − t ) q (z )s

+ t q (z )r

p+z q⁰(z ) q(z )

. (11)

By setting

θ (w ) = (1 − t ) w^s+ t p w^r a n d φ (w ) = t w^r−1, w6= 0,

we see thatθ (w ) is analytic in C , φ (w ) is analytic in C \{0} and that φ (w ) 6= 0, w ∈ C \{0}.

Also, we get

Q(z ) = z q⁰(z )φ q (z ) = t z q (z )^r⁻¹q⁰(z ) and

h(z ) = θ q (z ) +Q (z ) = (1 − t ) q (z )^s+ t q (z )r

p+z q⁰(z ) q(z )

. It is clear that Q(z ) is starlike univalent in U ,

R e z h⁰(z ) Q(z )

= Re

1+ p r +s(1 − t )

t q(z )_s−r

+ (r − 1)z q⁰(z )

q(z ) +z q⁰⁰(z ) q⁰(z )

> 0. (12)

From (7) and (12), we have R e z h⁰(z )

Q(z )

> 0.

Therefore, by Lemma2.1, we get k(z ) ≺ q (z ). By using (9), we obtain the result.

By fixingα = p = 1 in Theorem3.1, we obtain the following corollary:

Corollary 3.1.

Let the function q be univalent in U , q(z ) 6= 0 and assume that R e

1+ p r +s(1 − t )

t q(z )_s−r

+ (r − 1)z q⁰(z )

q(z ) +z q⁰⁰(z ) q⁰(z )

> 0,

where r, s ∈ C , t ∈ C \ {0}. Suppose that z q (z )r−1

q⁰(z ) is starlike univalent in U . If f ∈ Lp(λ) satisfies the subordi- nation

(1 − t ) − z f⁰(z )⁰ g⁰(z )

!s

+ t − z f⁰(z )⁰ g⁰(z )

!r

1+ α z z f⁰(z )⁰⁰

z f⁰(z )⁰ −z g⁰⁰(z ) g⁰(z )

!



≺ (1 − t ) q (z )s

+ t q (z )r

1+z q⁰(z ) q(z )

, then

− z f⁰(z )⁰ g⁰(z ) ≺ q (z ) and q is the best dominant.

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By taking q(z ) = − _{1+B z}^1+Az (−1 ≤ B < A ≤ 1) in Corollary3.1, we obtain the following corollary:

Corollary 3.2.

Let the function q be convex univalent in U , and assume that

R e

1+ p r +s(1 − t ) t

−

1+ Az 1+ B z

_s−r

+1+ r (A − B) z − AB z² (1 + Az )(1 + B z )

> 0,

where r, s∈ C , t ∈ C \ {0}. If f ∈ Lp(λ) satisfies the subordination

(1 − t ) − z f⁰(z )⁰ g⁰(z )

!s

+ t − z f⁰(z )⁰ g⁰(z )

!r

1+ α z z f⁰(z )⁰⁰

z f⁰(z )⁰ −z g⁰⁰(z ) g⁰(z )

!



≺ (1 − t )

−

1+ Az 1+ B z

s

+ t

−

1+ Az 1+ B z

r 1+ 2Az + AB z² (1 + Az )(1 + B z )

, then

− z f⁰(z )⁰ g⁰(z ) ≺ −

1+ Az 1+ B z

,(−1 ≤ B < A ≤ 1) and q(z ) = − _{1+B z}^1+Az is the best dominant.

Theorem 3.2.

Let the function q be convex univalent in U , q⁰(z ) 6= 0 and assume that R e

1+z q⁰⁰(z ) q⁰(z ) +1

γ

> 0, (13)

whereγ ∈ C \{0}.

Suppose that

−

z^pf⁰(z )⁰ g⁰(z )

_α

is analytic in U . If f ∈ Tp(λ) satisfies the subordination

− z^pf⁰(z )⁰ g⁰(z )

!α

+ αγ − z^pf⁰(z )⁰ g⁰(z )

!α

z z^pf⁰(z )⁰⁰

z^pf⁰(z )0 −z g⁰⁰(z ) g⁰(z )

!

≺ q (z ) + γz q⁰(z ), (14)

then

− z^pf⁰(z )⁰ g⁰(z )

!α

≺ q (z ) , (z ∈ U , α ∈ C \ {0})

and q is the best dominant.

Proof. Define the function k by

k(z ) = − z^pf⁰(z )⁰ g⁰(z )

!α

, z∈ U . (15)

Note that

k(z ) + γz k⁰(z ) = − z^pf⁰(z )⁰ g⁰(z )

!α

+ αγ − z^pf⁰(z )⁰ g⁰(z )

!α

z z^pf⁰(z )⁰⁰

z^pf⁰(z )0 −z g⁰⁰(z ) g⁰(z )

!

. (16)

From (14) and (16), we have

k(z ) + γz k⁰(z ) ≺ q (z ) + γz q⁰(z ). (17)

By settingψ = 1 in Lemma2.2, we get k(z ) ≺ q (z ). By using (15), we obtain the result.

By fixingα = p = 1 in Theorem3.2, we obtain the following corollary:

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Corollary 3.3.

Let the function q be convex univalent in U , q⁰(z ) 6= 0 and assume that (3.2). Suppose that−

z^pf⁰(z )⁰

g⁰(z ) is analytic in U . If f ∈ Tp(λ) satisfies the subordination

− z f⁰(z )⁰

g⁰(z ) + γ − z f⁰(z )⁰ g⁰(z )

! z z f⁰(z )⁰⁰

z f⁰(z )⁰ −z g⁰⁰(z ) g⁰(z )

!

≺ q (z ) + γz q⁰(z ),

then

− z f⁰(z )⁰

g⁰(z ) ≺ q (z ) , (z ∈ U ).

By taking q(z ) = − ^1+z_1−z in Corollary3.3, we obtain the following corollary:

Corollary 3.4.

Let the function q be convex univalent in U and assume that

R e

z²+ 1 (1 − z )(1 + z )+1

γ

> 0.

If f ∈ Tp(λ) satisfies the subordination

− z f⁰(z )⁰

g⁰(z ) + γ − z f⁰(z )⁰ g⁰(z )

! z z f⁰(z )⁰⁰

z f⁰(z )0 −z g⁰⁰(z ) g⁰(z )

!

≺ −

1+ z 1− z

− 2γz

(1 − z )(1 + z ), then

− z f⁰(z )⁰ g⁰(z ) ≺ −

1+ z 1− z

, (z ∈ U ) and q is the best dominant.

4. Applications of fractional derivative operator

In this section, we introduce some applications of section3containing fractional derivative operators. Assume that Φ(z ) =

X∞ n=0

σnzⁿ.

By Definition2.1, we have D_z^λΦ(z ) =

X∞ n=0

Γ (n + 1)

Γ (n + 1 − λ)σnz^n−λ= X∞ n=0

a_nz^n−λ,

where

a_n= Γ (n + 1)

Γ (n + 1 − λ)σn, n= 0,1,2,... .

Thus z^−p+ D_z^λΦ(z ) ∈ Lp(λ) and z^−p− D_z^λΦ(z ) ∈ Tp(λ)(σn≥ 0), then we have the following results:

Theorem 4.1.

Let the assumptions of Theorem3.1hold. Then





−

z^p z^−p+ D_z^λΦ(z )⁰⁰ z^−p+ D_z^λΨ (z )⁰







α

≺ q (z ) , z ∈ U

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Proof. Define the function f by f (z ) = z^−p+ D_z^λΦ(z ) (z ∈ U ),

it can easily observed that f ∈ Lp(λ). Thus by using Theorem3.1, we obtain the result.

Theorem 4.2.





−

z^p z^−p− D_z^λΦ(z )⁰⁰ z^−p− D_z^λΨ (z )0







α

≺ q (z ) , z ∈ U

Proof. Define the function f by f (z ) = z^−p− D_z^λΦ(z ) (z ∈ U ),

it can easily observed that f ∈ Tp(λ). Thus by using Theorem3.2, we obtain the result.

By using (6), we have J_0,z^λ,µ,νΦ(z ) =

X∞ n=0

Γ (n + 1)Γ n − µ + ν + 1

Γ n − µ + 1 Γ (n − λ + ν + 1)σnz^n−µ= X∞ n=0

a_nz^n−µ, where

a_n= Γ (n + 1)Γ n − µ + ν + 1

Γ n − µ + 1 Γ (n − λ + ν + 1)σn, n= 0,1,2,... .

Letµ = λ. Then z^−p+ J_0,z^λ,µ,νΦ(z ) ∈ Lp(λ) and z^−p− J_0,z^λ,µ,νΦ(z ) ∈ Tp(λ) (σn≥ 0), then we have the following results:

Theorem 4.3.





−

z^p

z^−p+ J_0,z^λ,µ,νΦ(z )⁰⁰

z^−p+ J_0,z^λ,µ,νΨ (z )0







α

≺ q (z ) , z ∈ U

Proof. Define the function f by f (z ) = z^−p+ J_0,z^λ,µ,νΦ(z ) (z ∈ U ),

it can easily observed that f ∈ Lp(λ). Thus by using Theorem3.1, we obtain the result.

Theorem 4.4.







−

z^p

z^−p− J_0,z^λ,µ,νΦ(z )⁰⁰

z^−p− J_0,z^λ,µ,νΨ (z )⁰







α

≺ q (z ) , z ∈ U

Proof. Define the function f by f (z ) = z^−p− J_0,z^λ,µ,νΦ(z ) (z ∈ U ),

it can easily observed that f ∈ Tp(λ). Thus by using Theorem3.2, we obtain the result.

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References

[1] W. G. Atshan, A. K. Wanas, Differential subordination theorems of analytic functions and some applications, American Journal of Scientific Research 49(2012) 91-101.

[2] B. B. Billing, A subordination theorem with applications to analytic functions, Bulletin of Mathematical Analysis and Applications 3(3) (2011) 1-8.

[3] M. Darus, R. W. Ibrahim, Coefficient inequalities for a new class of univalent functions, Lobachevskii J. Math.

29(4) (2008) 221â ˘A¸S229.

[4] S. P. Goyal, R. Goyal, On a class of multivalent functions defined by a generalized Ruscheweyh derivatives in- volving a general fractional derivative operator, J. Indian Acad. Math. 27(2) (2005) 439-456.

[5] R. W. Ibrahim, M. Darus, On subordination theorems for new classes of normalize analytic functions, Appl.

Math. Sci. 2(56) (2008) 2785â ˘A¸S2794.

[6] R. W. Ibrahim, M. Darus, Subordination results for new classes of meromorphic functions, American Journal of Scientific Research 2(2009) 31-36.

[7] R. W. Ibrahim, M. Darus, Differential subordination for classes of normalized analytic functions, General Math- ematics 18(3) (2010) 41-50.

[8] S. S. Miller, P. T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics Vol. 225, Marcel Dekker Inc., New York and Basel, 2000.

[9] T. N. Shanmugam, V. Ravichangran, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Aust. J. Math. Anal. Appl. 3(1) (2006) 1-11.

[10] S. Singh, S. Gupta, S. Singh, Differential subordination and superordination theorems for certain analytic functions 1, General Mathematics 18(2) (2010) 143-159.

[11] H. M. Srivastava, S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1989.

[12] H. M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Com- pany, Singapore, New Jersey, London and Hong Kong, 1992.

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