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Abstract
We introduce the new class L(α, β, λ, p) of meromorphic p-valent functions. The aim
of the paper is to obtain coefficient inequalities, growth and distortion, radii of convexity and
starlikeness and the convex linear combinations for the class L(α, β, λ, p).
Results & Lemmas (7)
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Theorem 2.1.
Theorem 2.1. If f ∈P p given by (1.1) satisfies ∞ X n=1 (p + n −1)(1 + αβ)|ap+n−1| ≤β[p(1 −α) + (λ −α)(p −α)]. (2.3) Then f ∈L(α, β, λ, p),…
Theorem 2.1. If f ∈P p given by (1.1) satisfies ∞ X n=1 (p + n −1)(1 + αβ)|ap+n−1| ≤β[p(1 −α) + (λ −α)(p −α)]. (2.3) Then f ∈L(α, β, λ, p), where 0 < β ≤1, 0 < λ + p − p (λ + p)2 −4p(1 + λ) < α < λ + p + p (λ + p)2 −4p(1 + λ) ≤p, (λ + p)2 > 4p(1 + λ) and λ ≥p 2 −2, for all z ∈D.
Corollary 2.1.
Corollary 2.1. Let the function be defined by (1.1). If f ∈L(α, β, λ, p),then |ap+n−1| ≤β[p(1 −α) + (λ −α)(p −α)] (p + n −1)(1 + αβ), (n…
Corollary 2.1. Let the function be defined by (1.1). If f ∈L(α, β, λ, p),then |ap+n−1| ≤β[p(1 −α) + (λ −α)(p −α)] (p + n −1)(1 + αβ) , (n ≥1). The result is sharp for functions f given by (2.4). 3. Distortion Theorem A distortion property for functions in the class f ∈L(α, β, λ, p), is given as follows:
Theorem 3.1.
Theorem 3.1. If the function f given by (1.1) is in the class f ∈L(α, β, λ, p), then for 0 < |z| = r < 1 we have 1 rp −β[p(1 −α) + (λ −α)(p…
Theorem 3.1. If the function f given by (1.1) is in the class f ∈L(α, β, λ, p), then for 0 < |z| = r < 1 we have 1 rp −β[p(1 −α) + (λ −α)(p −α)] p(1 + αβ) rp≤ f(z) ≤1 rp + β[p(1 −α) + (λ −α)(p −α)] p(1 + αβ) rp with equality for f(z) = 1 zp + β[p(1 −α) + (λ −α)(p −α)] p(1 + αβ)
Theorem 4.1.
Theorem 4.1. If the function f defined by (1.1) is in the class L(α, β, λ, p), then f is starlike of order ρ(0 ≤ρ < p), in the disk |z| <…
Theorem 4.1. If the function f defined by (1.1) is in the class L(α, β, λ, p), then f is starlike of order ρ(0 ≤ρ < p), in the disk |z| < r1(α, β, λ, p, ρ), where r1(α, β, λ, p, ρ),is the largest value for which r1 = r1(α, β, λ, p, ρ) = inf n≥1
Theorem 4.2.
Theorem 4.2. If the function f defined by (1.1) is in the class L(α, β, λ, p), then f is convex of order ρ(0 ≤ρ < p), in the disk |z| <…
Theorem 4.2. If the function f defined by (1.1) is in the class L(α, β, λ, p), then f is convex of order ρ(0 ≤ρ < p), in the disk |z| < r2(α, β, λ, p, ρ), where r2(α, β, λ, p, ρ),is the largest value for which r2 = r2(α, β, λ, p, ρ) = inf n≥1
Theorem 5.1
Theorem 5.1 Let fp(z) = z−p, and fp+n−1(z) = z−p + β[p(1 −α) + (λ −α)(p −α)] (p + n −1)(1 + αβ) zp+n−1, (n ≥1). (5.12)
Theorem 5.1 Let fp(z) = z−p, and fp+n−1(z) = z−p + β[p(1 −α) + (λ −α)(p −α)] (p + n −1)(1 + αβ) zp+n−1, (n ≥1). (5.12)
Theorem 5.2.
Theorem 5.2. The class L(α, β, λ, p) is closed under convex linear combinations.
Theorem 5.2. The class L(α, β, λ, p) is closed under convex linear combinations.