DIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E(Φ, Ψ)

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Differential Subordination Results Rabha W. Ibrahim and

Maslina Darus vol. 10, iss. 1, art. 8, 2009

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DIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E (Φ, Ψ)

RABHA W. IBRAHIM AND MASLINA DARUS

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600, Selangor Darul Ehsan Malaysia

EMail:rabhaibrahim@yahoo.com maslina@ukm.my

Received: 05 July, 2008

Accepted: 02 December, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 34G10, 26A33, 30C45.

Key words: Fractional calculus; Subordination; Hadamard product.

Abstract: We define new classes of the family E(Φ,Ψ), in a unit diskU := {z ∈ C,|z| < 1},as follows: for analytic functionsF(z),Φ(z)and Ψ(z)so that

<{^F(z)∗Φ(z)_F(z)∗Ψ(z)}>0, z ∈U, F(z)∗Ψ(z)6= 0where the operator∗denotes the convolution or Hadamard product. Moreover, we establish some subordination results for these new classes.

Acknowledgement: The work presented here was supported by SAGA: STGL-012-2006, Academy of Sciences, Malaysia.

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1. Introduction and preliminaries.

Let B_α⁺ be the class of all analytic functions F(z) in the open disk U := {z ∈ C, |z|<1},of the form

F(z) = 1 +

∞

X

n=1

a_nz^n+α−1, 0< α≤1,

satisfyingF(0) = 1. And let B⁻_α be the class of all analytic functions F(z)in the open diskU of the form

F(z) = 1−

∞

X

n=1

a_nz^n+α−1, 0< α≤1, a_n≥0; n= 1,2,3, . . . , satisfyingF(0) = 1.With a view to recalling the principle of subordination between analytic functions, let the functionsf and g be analytic in U.Then we say that the functionf is subordinate tog if there exists a Schwarz functionw(z),analytic inU such that

f(z) =g(w(z)), z ∈U.

We denote this subordination byf ≺ g orf(z) ≺g(z), z ∈ U.If the functiong is univalent inU the above subordination is equivalent to

f(0) =g(0) and f(U)⊂g(U).

Letφ : C³ ×U → Cand let hbe univalent inU.Assume thatp, φ are analytic and univalent inU andpsatisfies the differential superordination

(1.1) h(z)≺φ(p(z)), zp⁰(z), z²p⁰⁰(z);z), thenpis called a solution of the differential superordination.

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An analytic functionq is called a subordinant ifq ≺ pfor allpsatisfying (1.1).

A univalent functionqsuch thatp≺qfor all subordinantspof (1.1) is said to be the best subordinant.

LetB⁺be the class of analytic functions of the form f(z) = 1 +

∞

X

n=1

a_nzⁿ, a_n ≥0.

Given two functionsf, g ∈ B⁺, f(z) = 1 +

∞

X

n=1

a_nzⁿ and g(z) = 1 +

∞

X

n=1

b_nzⁿ their convolution or Hadamard productf(z)∗g(z)is defined by

f(z)∗g(z) = 1 +

∞

X

n=1

a_nb_nzⁿ, a_n≥0, b_n ≥0, z ∈U.

Juneja et al. [1] define the familyE(Φ,Ψ),so that

<

f(z)∗Φ(z) f(z)∗Ψ(z)

>0, z ∈U where

Φ(z) =z+

∞

X

n=2

ϕ_nzⁿ and Ψ(z) = z+

∞

X

n=2

ψ_nzⁿ

are analytic in U with the conditions ϕn ≥ 0, ψn ≥ 0, ϕn ≥ ψn for n ≥ 2 and f(z)∗Ψ(z)6= 0.

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Definition 1.1. LetF(z) ∈ B⁺_α,we define the familyE_α⁺(Φ,Ψ)so that

(1.2) <

F(z)∗Φ(z) F(z)∗Ψ(z)

>0, z ∈U, where

Φ(z) = 1 +

∞

X

n=1

ϕ_nz^n+α−1 and Ψ(z) = 1 +

∞

X

n=1

ψ_nz^n+α−1

are analytic in U under the conditions ϕn ≥ 0, ψn ≥ 0, ϕn ≥ ψn forn ≥ 1 and F(z)∗Ψ(z)6= 0.

Definition 1.2. LettingF(z) ∈ B_α⁻,we define the familyE_α⁻(Φ,Ψ) which satisfies (1.2) where

Φ(z) = 1−

∞

X

n=1

ϕ_nz^n+α−1 and Ψ(z) = 1−

∞

X

n=1

ψ_nz^n+α−1

are analytic in U under the conditions ϕ_n ≥ 0, ψ_n ≥ 0, ϕ_n ≥ ψ_n forn ≥ 1 and F(z)∗Ψ(z)6= 0.

In the present paper, we establish some sufficient conditions for functions F ∈ B⁺_α andF ∈ B_α⁻to satisfy

(1.3) F(z)∗Φ(z)

F(z)∗Ψ(z) ≺q(z), z ∈U,

where q(z) is a given univalent function in U such that q(0) = 1. Moreover, we give applications for these results in fractional calculus. We shall need the following known results.

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Lemma 1.3 ([2]). Letq(z)be convex in the unit disk U withq(0) = 1and<{q} >

1

2, z ∈U.If0≤µ <1, pis an analytic function inU withp(0) = 1and if (1−µ)p²(z) + (2µ−1)p(z)−µ+ (1−µ)zp⁰(z)

≺(1−µ)q²(z) + (2µ−1)q(z)−µ+ (1−µ)zq⁰(z), thenp(z)≺q(z)andq(z)is the best dominant.

Lemma 1.4 ([3]). Letq(z)be univalent in the unit diskU and letθ(z)be analytic in a domainDcontainingq(U).Ifzq⁰(z)θ(q)is starlike inU,and

zp⁰(z)θ(p(z))≺zq⁰(z)θ(q(z)) thenp(z)≺q(z)andq(z)is the best dominant.

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2. Main Results

In this section, we verify some sufficient conditions of subordination for analytic functions in the classesB⁺_α andB⁻_α.

Theorem 2.1. Let the functionq(z)be convex in the unit diskU such thatq(0) = 1 and <{q} > ¹₂. If F ∈ B⁺_α and ^F_F(z)∗Ψ(z)^(z)∗Φ(z) an analytic function in U satisfies the subordination

(1−µ)

2

+ (2µ−1)

−µ + (1−µ)

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺(1−µ)q²(z) + (2µ−1)q(z)−µ+ (1−µ)zq⁰(z),

then F(z)∗Φ(z)

F(z)∗Ψ(z) ≺q(z) andq(z)is the best dominant.

Proof. Let the functionp(z)be defined by p(z) := F(z)∗Φ(z)

F(z)∗Ψ(z), z ∈U.

It is clear thatp(0) = 1.Then straightforward computation gives us (1−µ)p²(z) + (2µ−1)p(z)−µ+ (1−µ)zp⁰(z)

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= (1−µ)

2

+ (2µ−1)

−µ + (1−µ)z

0

= (1−µ)

2

+ (2µ−1)

−µ + (1−µ)

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺(1−µ)q²(z) + (2µ−1)q(z)−µ+ (1−µ)zq⁰(z).

By the assumption of the theorem we have that the assertion of the theorem follows by an application of Lemma1.3.

Corollary 2.2. IfF ∈ B_α⁺ and ^F_F^(z)∗Φ(z)_(z)∗Ψ(z) is an analytic function in U satisfying the subordination

(1−µ)

2

+ (2µ−1)

−µ + (1−µ)

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺(1−µ)

1 +Az 1 +Bz

2

+ (2µ−1)

1 +Az 1 +Bz

−µ+ (1−µ)

1 +Az 1 +Bz

(A−B)z (1 +Az)(1 +Bz),

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then F(z)∗Φ(z)

F(z)∗Ψ(z) ≺

1 +Az 1 +Bz

, −1≤B < A≤1 and(^1+Az_1+Bz)is the best dominant.

Proof. Let the functionq(z)be defined by q(z) :=

1 +Az 1 +Bz

, z ∈U.

It is clear thatq(0) = 1and<{q} > ¹₂ for arbitraryA, B, z ∈ U,then in view of Theorem2.1we obtain the result.

Corollary 2.3. IfF ∈ B_α⁺ and ^F_F^(z)∗Φ(z)_(z)∗Ψ(z) is an analytic function in U satisfying the subordination

(1−µ)

2

+ (2µ−1)

−µ + (1−µ)

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺(1−µ)

1 +z 1−z

2

+ (2µ−1)

1 +z 1−z

−µ + (1−µ)

1 +z 1−z

2z 1−z²

,

then F(z)∗Φ(z)

F(z)∗Ψ(z) ≺

1 +z 1−z

, and(^1+z_1−z)is the best dominant.

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Define the functionϕ_α(a, c;z)by ϕα(a, c;z) := 1 +

∞

X

n=1

(a)_n

(c)_nz^n+α−1, (z ∈U; a ∈R, c ∈R\{0,−1,−2, . . .}), where(a)nis the Pochhammer symbol defined by

(a)_n := Γ(a+n) Γ(a) =







1, (n= 0);

a(a+ 1)(a+ 2)· · ·(a+n−1), (n∈N).

Corresponding to the functionϕ_α(a, c;z),define a linear operatorL_α(a, c)by L_α(a, c)F(z) := ϕ_α(a, c;z)∗F(z), F(z) ∈ B⁺_α

or equivalently by

Lα(a, c)F(z) := 1 +

∞

X

n=1

(a)_n

(c)_nanz^n+α−1. For details see [4]. Hence we have the following result:

Corollary 2.4. Let the functionq(z)be convex in the unit diskU such thatq(0) = 1 and<{q}> ¹₂.If_L^L^α^(a,c)Φ(z)

α(a,c)Ψ(z) is an analytic function inU satisfying the subordination (1−µ)

L_α(a, c)Φ(z) Lα(a, c)Ψ(z)

2

+ (2µ−1)

−µ + (1−µ)

z(L_α(a, c)Φ(z))⁰

Lα(a, c)Φ(z) − z(L_α(a, c)Ψ(z))⁰ Lα(a, c)Ψ(z)

≺(1−µ)q²(z) + (2µ−1)q(z)−µ+ (1−µ)zq⁰(z),

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then Lα(a, c)Φ(z)

L_α(a, c)Ψ(z) ≺q(z) andq(z)is the best dominant.

Theorem 2.5. Let the functionq(z)be univalent in the unit diskU such thatq⁰(z)6=

0and ^zq_q(z)⁰^(z) is starlike inU.IfF ∈ B_α⁻satisfies the subordination a

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺azq⁰(z) q(z) ,

then F(z)∗Φ(z)

F(z)∗Ψ(z) ≺q(z), z ∈U, andq(z)is the best dominant.

Proof. Let the functionp(z)be defined by p(z) :=

, z ∈U.

By setting

θ(ω) := a

ω, a6= 0,

it can easily observed thatθ(ω)is analytic inC− {0}.Then we obtain azp⁰(z)

p(z) =a

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺azq⁰(z) q(z) .

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By the assumption of the theorem we have that the assertion of the theorem follows by an application of Lemma1.4.

Corollary 2.6. IfF ∈ B_α⁻satisfies the subordination a

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) − z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺a (A−B)z (1 +Az)(1 +Bz)

then F(z)∗Φ(z)

F(z)∗Ψ(z) ≺

1 +Az 1 +Bz

, −1≤B < A≤1 and(^1+Az_1+Bz)is the best dominant.

Corollary 2.7. IfF ∈ B_α⁻satisfies the subordination a

z(F(z)∗Φ(z))⁰

F(z)∗Φ(z) −z(F(z)∗Ψ(z))⁰ F(z)∗Ψ(z)

≺a 2z

1−z²

, then

F(z)∗Φ(z) F(z)∗Ψ(z) ≺

1 +z 1−z

, and(^1+z_1−z)is the best dominant.

Define the functionφα(a, c;z)by φ_α(a, c;z) := 1−

∞

X

n=1

(a)_n

(c)_nz^n+α−1, (z ∈U; a ∈R, c ∈R\{0,−1,−2, . . .}), where(a)n is the Pochhammer symbol. Corresponding to the functionφα(a, c;z), define a linear operatorLα(a, c)by

Lα(a, c)F(z) := φα(a, c;z)∗F(z), F(z) ∈ B_α⁻

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or equivalently by

L_α(a, c)F(z) := 1−

∞

X

n=1

(a)_n

(c)_na_nz^n+α−1. Hence we obtain the next result.

Corollary 2.8. Let the functionq(z)be univalent in the unit diskU such thatq⁰(z)6=

0and ^zq_q(z)⁰^(z) is starlike inU.IfF ∈ B_α⁻satisfies the subordination a

z(L_α(a, c)Φ(z))⁰

Lα(a, c)Φ(z) − z(L_α(a, c)Ψ(z))⁰ Lα(a, c)Ψ(z)

≺azq⁰(z) q(z) ,

then L_α(a, c)Φ(z)

Lα(a, c)Ψ(z) ≺q(z), z ∈U, andq(z)is the best dominant.

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3. Applications

In this section, we introduce some applications of Section2containing fractional integral operators. Assume thatf(z) =P∞

n=1ϕ_nzⁿand let us begin with the following definitions

Definition 3.1 ([5]). The fractional integral of orderα for the function f(z)is de- fined by

I_z^αf(z) := 1 Γ(α)

Z z 0

f(ζ)(z−ζ)^α−1dζ; 0≤α <1,

where the function f(z) is analytic in a simply-connected region of the complex z−plane (C) containing the origin. The multiplicity of (z −ζ)^α−1 is removed by requiringlog(z−ζ)to be real when(z−ζ)>0.Note that,I_z^αf(z) =h

z^α−1 Γ(α)

i f(z), forz >0and0forz ≤0(see [6]).

From Definition3.1, we have I_z^αf(z) =

z^α−1 Γ(α)

f(z) = z^α−1 Γ(α)

∞

X

n=1

ϕ_nzⁿ=

∞

X

n=1

a_nz^n+α−1

wherea_n := _Γ(α)^ϕⁿ for alln = 1,2,3, . . . ,thus1 +I_z^αf(z) ∈ B_α⁺and1−I_z^αf(z) ∈ B⁻_α (ϕ_n≥0).Then we have the following results.

Theorem 3.2. Let the assumptions of Theorem2.1hold. Then (1 +I_z^αf(z))∗Φ(z)

(1 +I_z^αf(z))∗Ψ(z)

≺q(z), z 6= 0, z ∈U andq(z)is the best dominant.

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Proof. Let the functionF(z)be defined by

F(z) := 1 +I_z^αf(z), z ∈U.

Theorem 3.3. Let the assumptions of Theorem2.5hold. Then (1−I_z^αf(z))∗Φ(z)

(1−I_z^αf(z))∗Ψ(z)

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

F(z) := 1−I_z^αf(z), z ∈U.

LetF(a, b;c;z)be the Gauss hypergeometric function (see [7]) defined, forz ∈ U,by

F(a, b;c;z) =

∞

X

n=0

(a)_n(b)_n (c)_n(1)_nzⁿ.

We need the following definitions of fractional operators in Saigo type fractional calculus (see [8,9]).

Definition 3.4. Forα > 0andβ, η ∈ R,the fractional integral operator I_0,z^α,β,η is defined by

I_0,z^α,β,ηf(z) = z^−α−β Γ(α)

Z z 0

(z−ζ)^α−1F

α+β,−η;α; 1− ζ z

f(ζ)dζ,

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

f(z) = O(|z|)(z →0), >max{0, β−η} −1

and the multiplicity of(z−ζ)^α−1 is removed by requiringlog(z−ζ)to be real when z−ζ >0.

From Definition3.4, withβ <0,we have I_0,z^α,β,ηf(z) = z^−α−β

Γ(α) Z z

0

(z−ζ)^α−1F(α+β,−η;α; 1− ζ

z)f(ζ)dζ

=

∞

X

n=0

(α+β)_n(−η)_n (α)_n(1)_n

z^−α−β Γ(α)

Z z 0

(z−ζ)^α−1

1− ζ z

n

f(ζ)dζ

=

∞

X

n=0

B_nz^{−α−β−n} Γ(α)

Z z 0

(z−ζ)^n+α−1f(ζ)dζ

=

∞

X

n=0

Bn

z^−β−1 Γ(α) f(ζ)

= B

Γ(α)

∞

X

n=1

ϕ_nz^n−β−1, where

B_n:= (α+β)_n(−η)_n (α)n(1)n

and B :=

∞

X

n=0

B_n. Denotea_n := ^Bϕ_Γ(α)ⁿ for all n = 1,2,3, . . . ,and letα=−β.Thus,

1 +I_0,z^α,β,ηf(z) ∈ B_α⁺ and 1−I_0,z^α,β,ηf(z) ∈ B_α⁻ (ϕn≥0),

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andn we have the following results

Theorem 3.5. Let the assumptions of Theorem2.1hold. Then

"

(1 +I_0,z^α,β,ηf(z))∗Φ(z) (1 +I_0,z^α,β,ηf(z))∗Ψ(z)

#

F(z) := 1 +I_0,z^α,β,ηf(z), z ∈U.

Theorem 3.6. Let the assumptions of Theorem2.5hold. Then

"

(1−I_0,z^α,β,ηf(z))∗Φ(z) (1−I_0,z^α,β,ηf(z))∗Ψ(z)

#

F(z) := 1−I_0,z^α,β,ηf(z), z ∈U.

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