Abstract
In this paper, a class Aη,λ(α, β, γ) of analytic functions involving the integral
operator Jη,λf (z) = z + P∞
n=2
(Γ(n+1))2Γ(2+η−λ)Γ(2−η)
Γ(n+η−λ+1)Γ(n−η+1)
anzn,given by Salah and Darus in [5]
is defined. The extreme points for this class are provided, the coefficient bounds and radii of
univalency and starlikeness are also provided.
AMS Subject Classification:
30C45, 30C50
Key Words:
Caputo’s differentiation operator, univalency, starlikeness, Hadamard product
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 2.1.
Theorem 2.1. A function f (z) ∈Aη,λ(α, β, γ) if and only if f (z) can be expressed as f (z) = z + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) Z…
Theorem 2.1. A function f (z) ∈Aη,λ(α, β, γ) if and only if f (z) can be expressed as f (z) = z + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) Z |x|=1 ∞ X n=2 Γ (n + η −λ + 1) Γ (n −η + 1) xn−1 (Γ (n + 1))2 (α + nβ) zn ! dµ (x) ,
Corollary 2.2.
Corollary 2.2. The extreme points of the class Aη,λ(α, β, γ) are fx (z) = z + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) ∞ X n=2 Γ (n + η −λ + 1) Γ…
Corollary 2.2. The extreme points of the class Aη,λ(α, β, γ) are fx (z) = z + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) ∞ X n=2 Γ (n + η −λ + 1) Γ (n −η + 1) xn−1 (Γ (n + 1))2 (α + nβ) zn (|x| = 1) . (2.2)
Corollary 2.3.
Corollary 2.3. If f (z) = z + P∞ n=2 anzn ∈Aη,λ(α, β, γ), then for n ≥2, we have |an| ≤2 (α + β −γ) Γ (n + η −λ + 1) Γ (n −η + 1) Γ (2 −η)…
Corollary 2.3. If f (z) = z + P∞ n=2 anzn ∈Aη,λ(α, β, γ), then for n ≥2, we have |an| ≤2 (α + β −γ) Γ (n + η −λ + 1) Γ (n −η + 1) Γ (2 −η) Γ (2 + η −λ) (Γ (n + 1))2 (α + nβ) .
Corollary 2.4. · radius
Corollary 2.4. If f (z) = z + P∞ n=2 anzn ∈Aη,λ(α, β, γ), then for |z| = r < 1, we have |f (z)| ≤r + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) ∞ X…
Corollary 2.4. If f (z) = z + P∞ n=2 anzn ∈Aη,λ(α, β, γ), then for |z| = r < 1, we have |f (z)| ≤r + 2 (α + β −γ) Γ (2 −η) Γ (2 + η −λ) ∞ X n=2 Γ (n + η −λ + 1) Γ (n −η + 1) rn (Γ (n + 1))2 (α + nβ) . This result follows from (2.3). 3. Radius of Univalency and Starlikeness
Theorem 3.1.
Theorem 3.1. Let f (z) ∈Aη,λ(α, β, γ), then f (z) is univalent(close-to- convex) in |z| < R(α, β, γ), where R(α, β, γ) = inf n ( (nβ + α)…
Theorem 3.1. Let f (z) ∈Aη,λ(α, β, γ), then f (z) is univalent(close-to- convex) in |z| < R(α, β, γ), where R(α, β, γ) = inf n ( (nβ + α) (Γ (n + 1))2 Γ (2 −η) Γ (2 + η −λ) 2n (α + β −γ) Γ (n + η −λ + 1) Γ (n −η + 1) ) 1 n−1 .
Theorem 3.2.
Theorem 3.2. If f (z) ∈Aη,λ(α, β, γ) then f (z) is starlike of order µ, |z| < r0, 0 ≤µ < 1 where r0 = inf n ( (1 −µ) Γ (2 −η) Γ (2 + η −λ)…
Theorem 3.2. If f (z) ∈Aη,λ(α, β, γ) then f (z) is starlike of order µ, |z| < r0 , 0 ≤µ < 1 where r0 = inf n ( (1 −µ) Γ (2 −η) Γ (2 + η −λ) (Γ (n + 1))2 (nβ + α) 2 (n −µ) (α + β −γ) Γ (n + η −λ + 1) Γ (n −η + 1) ) 1 n−1 .
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