On a class of p -valent functions involving generalized differential operator
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On a class of p-valent functions involving generalized differential operator A. Ta. Yousef1
· Z. Salleh1 · Tariq Al-Hawary2
Received: 17 January 2019 / Accepted: 14 August 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract m In this paper, we introduce a new differential operator T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) in the open unit disc U = {z ∈ C : |z| < 1}. We then, introduce a new subclass of analytic functions ∗ Gw m ( , μ, λ, σ, δ, γ , ϕ, ς, b, p). Moreover, we discuss coefficient estimates, growth and distortion theorems, inclusion properties, closure theorems, and sufficient conditions for close∗ to-convexity and starlikeness for the functions in the class G w m ( , μ, λ, σ, δ, γ , ϕ, ς, b, p). Keywords Analytic functions · p-Valent functions · Starlike functions · Differential operator · Integral operator Mathematics Subject Classification 30C45
1 Introduction Let A( p) denote the class of functions f of the form: f (z) = (z − w) p +
∞
an (z − w)n ,
(1)
n= p+1
which are analytic and normalized in the open unit disk U = {z ∈ C : |z| < 1}. For a function f in A( p), we defined the following differential operator: 0 T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) f (z) = f (z),
B
(2)
A. Ta. Yousef [email protected] Z. Salleh [email protected] Tariq Al-Hawary [email protected]
1
Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus, 21030 Terengganu, Malaysia
2
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
123
A. T. Yousef et al. 1 T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) f (z) μ + λ − p( p − 1)δ − pγ (ς + ϕ) − p(β − σ )(λ − α) = f (z) μ+λ δ (β − σ )(λ − α) + γ (ς + ϕ) (z − w) f (z) + (z − w)2 f (z), + μ+λ μ+λ 2 T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) f (z) μ + λ − p( p − 1)δ − pγ (ς + ϕ) − p(β − σ )(λ − α) (T p1 f (z)) = μ+λ (β − σ )(λ − α) + γ (ς + ϕ) + (z − w)(T p1 f (z)) μ+λ δ + (z − w)2 (T p1 f (z)) , μ+λ
(3)
(4)
and for m = 0, 1, 2, . . ., if f is given by (1) we get: m p T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) f (z) = (z − w) ∞ μ + λ − δ( p( p − 1) − n(n − 1)) + (n − p)(γ (ς + ϕ) + (β − σ )(λ − α)) m + μ+λ n= p+1
an (z − w)n ,
(5)
where f ∈ A( p), p ∈ N, α, β, ϕ, σ , λ, δ, ς, γ ≥ 0, μ > 0, λ = α, and m ∈ N0 . m We observe that the generalized differential operator T p,w,μ,λ,σ,ς,γ ,ϕ,δ (α, β) reduces to several interesting many other differential operators considered earlier for different choices of μ, λ, σ, β, δ, γ , ϕ, ς, w and p: m μ+λ+(n−1)(β−σ )(λ−α) m (i) Tμ,λ,σ (α, β) f (z) = z + ∞ an z n , was introduced and studn=2 μ+λ ied by Amourah and Yousef [9]. m m n (α, β) f (z) = z + ∞ (ii) T1−λ,λ,0 n=2 [1 + (n − 1)(λ − α)(β − σ )] an z , was introduced and studied by Ramadan and Darus [18]. m m n (iii) T1−λ,λ,0 (α, β) f (z) = z + ∞ n=2 [1+(n −1)β(λ−α)] an z , was introduced and studied by Darus and Ibrahim [13]. m m n (iv) T1−λ,λ,0 (0, 1) = z + ∞ n=2 [1 + (n − 1)λ] an z , was introduced and studied by
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