Bol. Soc. Paran. Mat.

Theorem 2.1. A function f ∈A given by (1.1) is in the class Ω∗ m(δ, λ, α, β, b) if and only if ∞ X n=2 [1 + δ(n −1)] µ + λ + (n −1)(λ −α)(β −σ) µ + λ m an ≤|b| , (2.1) where α, σ ≥0, β, λ, µ > 0, λ ̸= α and m ∈N0.

Corollary 2.2. If f ∈Ω∗ m(δ, λ, α, β, b) is given by (1.1), then an ≤ |b| [1 + δ(n −1)] h µ+λ+(n−1)(λ−α)(β−σ) µ+λ im , n ≥2. 3. Growth and Distortion Theorems A growth and distortion property for function f to be in the class Ω∗ m(δ, λ, α, β, b) is contained in the following theorem.

Theorem 3.1. If the function f defined by (1.6) is in the class Ω∗ m(δ, λ, α, β, b), then for |z| = r < 1, we have r − |b| [1 + δ] h µ+λ+(λ−α)(β−σ) µ+λ im r2 ≤|f(z)| ≤r + |b| [1 + δ] h µ+λ+(λ−α)(β−σ) µ+λ

Theorem 4.1. Let the functions fj(z) defined by (4.1) be in the class Ω∗ m(δ, λ, α, β, b), α, σ ≥0, β, λ, µ > 0, λ ̸= α and m ∈N0, for every j = 1, 2, · · · , I. Then the function G(z) defined by G(z) = z − ∞ X n=2 pnzn, pn ≥0 (4.2) is a member of the class Ω∗ m(δ, λ, α, β, b), where pn = 1

Theorem 4.2. The class Ω∗ m(δ, λ, α, β, b) is closed under convex linear combina- tion, where α, σ ≥0, β, λ, µ > 0, λ ̸= α and m ∈N0 .

Theorem 5.1. If the function f defined by (1.6) is in the class Ω∗ m(δ, λ, α, β, b), where α, σ ≥0, β, λ, µ > 0, λ ̸= α, m ∈N0. Then the function F(z) defined by F(z) = c + 1 zc z Z 0 tc−1f(t)dt, (c > −1) (5.1) also belongs to the class Ω∗ m(δ, λ, α, β, b).

Theorem 6.1. If f ∈Ω∗ m(δ, λ, α, β, b), then f(z) is close-to-convex of order η in |z| < h1(µ, δ, b, η), where h1(µ, δ, b, η) = inf n    (1 −η) [1 + δ(n −1)] h µ+λ+(n−1)(λ−α)(β−σ) µ+λ im n |b| 

Theorem 6.2. If f ∈Ω∗ m(δ, λ, α, β, b), then f(z) is starlike of order η in |z| < h2(µ, δ, b, η), where h2(µ, δ, b, η) = inf n    (1 −η) [1 + δ(n −1)] h µ+λ+(n−1)(λ−α)(β−σ) µ+λ im (n −η) |b| 

Corollary 6.3. If f ∈Ω∗ m(δ, λ, α, β, b), then f(z) is convex of order η in |z| < h3(µ, δ, b, η), where h3(µ, δ, b, η) = inf n    (1 −η) [1 + δ(n −1)] h µ+λ+(n−1)(λ−α)(β−σ) µ+λ im n(n −η) |b| 