### Abstract

Let A(n) denote the class of analytic functions f in the open unit disk U = {z : |z| < 1} normalized by f (0) = f′(0)-1 = 0. In this paper, we introduce and study the classes S _{n,μ}(γ,α,β, λ,Ω) and R _{n,μ}(γ,α,β,λ, Ω) of functions f ∈l A(n) with (μ)z(D _{λ} ^{Ω+2}(α,ω) f (z))′ +(1-μ)z(D _{λ} ^{Ω+1 λ} (α,ω) f (z))′ ≠ 0 and satisfy some conditions available in literature, where f ∈ A(n),α,ω,μ ≥ 0,Ω ∈ N∪{0}, z ∈U, and D _{λ} ^{m}(α,ω) f (z) :A → A, is the linear fractional differential operator, newly defined as follows D _{λ} ^{m}(α,ω) f (z)= z + ∑ _{k=2} ^{∞} a _{k}(1+(k -1) λω ^{α}) ^{m}z ^{k}·Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes S _{n,μ}(γ,α,β,λ, Ω,ω) and R _{n,μ}(γ,α,β, λ,Ω,ω) are given.

Original language | English |
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Pages (from-to) | 223-242 |

Number of pages | 20 |

Journal | Tamkang Journal of Mathematics |

Volume | 43 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2012 |

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### Keywords

- (n,σ)-neighborhoods
- Analytic functions
- Coefficient estimates
- Convex function
- Differential operator
- Extreme points
- Growth and distortion theorems
- Hadamard product
- Identity function
- Inclusion properties
- Neighborhoods properties

### ASJC Scopus subject areas

- Metals and Alloys
- Materials Science(all)

### Cite this

**Some subclasses of analytic functions of complex order defined by new differential operator.** / Darus, Maslina; Faisal, Imran.

Research output: Contribution to journal › Article

*Tamkang Journal of Mathematics*, vol. 43, no. 2, pp. 223-242. https://doi.org/10.5556/j.tkjm.43.2012.223-242

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TY - JOUR

T1 - Some subclasses of analytic functions of complex order defined by new differential operator

AU - Darus, Maslina

AU - Faisal, Imran

PY - 2012/6

Y1 - 2012/6

N2 - Let A(n) denote the class of analytic functions f in the open unit disk U = {z : |z| < 1} normalized by f (0) = f′(0)-1 = 0. In this paper, we introduce and study the classes S n,μ(γ,α,β, λ,Ω) and R n,μ(γ,α,β,λ, Ω) of functions f ∈l A(n) with (μ)z(D λ Ω+2(α,ω) f (z))′ +(1-μ)z(D λ Ω+1 λ (α,ω) f (z))′ ≠ 0 and satisfy some conditions available in literature, where f ∈ A(n),α,ω,μ ≥ 0,Ω ∈ N∪{0}, z ∈U, and D λ m(α,ω) f (z) :A → A, is the linear fractional differential operator, newly defined as follows D λ m(α,ω) f (z)= z + ∑ k=2 ∞ a k(1+(k -1) λω α) mz k·Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes S n,μ(γ,α,β,λ, Ω,ω) and R n,μ(γ,α,β, λ,Ω,ω) are given.

AB - Let A(n) denote the class of analytic functions f in the open unit disk U = {z : |z| < 1} normalized by f (0) = f′(0)-1 = 0. In this paper, we introduce and study the classes S n,μ(γ,α,β, λ,Ω) and R n,μ(γ,α,β,λ, Ω) of functions f ∈l A(n) with (μ)z(D λ Ω+2(α,ω) f (z))′ +(1-μ)z(D λ Ω+1 λ (α,ω) f (z))′ ≠ 0 and satisfy some conditions available in literature, where f ∈ A(n),α,ω,μ ≥ 0,Ω ∈ N∪{0}, z ∈U, and D λ m(α,ω) f (z) :A → A, is the linear fractional differential operator, newly defined as follows D λ m(α,ω) f (z)= z + ∑ k=2 ∞ a k(1+(k -1) λω α) mz k·Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes S n,μ(γ,α,β,λ, Ω,ω) and R n,μ(γ,α,β, λ,Ω,ω) are given.

KW - (n,σ)-neighborhoods

KW - Analytic functions

KW - Coefficient estimates

KW - Convex function

KW - Differential operator

KW - Extreme points

KW - Growth and distortion theorems

KW - Hadamard product

KW - Identity function

KW - Inclusion properties

KW - Neighborhoods properties

UR - http://www.scopus.com/inward/record.url?scp=84863194380&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863194380&partnerID=8YFLogxK

U2 - 10.5556/j.tkjm.43.2012.223-242

DO - 10.5556/j.tkjm.43.2012.223-242

M3 - Article

VL - 43

SP - 223

EP - 242

JO - Tamkang Journal of Mathematics

JF - Tamkang Journal of Mathematics

SN - 0049-2930

IS - 2

ER -