Geometric Properties of Analytic Functions Defined by the Miller–Ross-Type Poisson Distribution Series

Theorem 1. Let u be defined by (1). Then u ∈Sm ν,c(σ, υ) if ∞ X n=2 [1 + σ(n −1)](2n −υ −1)Φν c(n, m)an ≤1 −υ, (14) where −1 ≤υ < 1, 0 ≤σ ≤1, and ∞ X n=2 [1 + σ(n −1)]Φν c(n, m)an < 1. (15)

Theorem 2. A necessary and sufficient condition for u(z) of the form (6) to belong to the class TSm ν,c(σ, υ) is that ∞ X n=2 [1 + σ(n −1)](2n −υ −1)Φν c(n, m)an ≤1 −υ, (16) where −1 ≤υ < 1 and 0 ≤σ ≤1, together with ∞ X n=2 [1 + σ(n −1)]Φν c(n, m)an < 1.

Theorem 3. Let u(z) defined by (6) and g(z) = z − ∞ P n=2 bnzn be in the class TSm ν,c(σ, υ). Then the function h(z) defined by h(z) = (1 −ζ)u(z) + ζg(z) = z − ∞ X n=2 cnzn, where cn = (1 −ζ)an + ζbn, 0 ≤ζ < 1 is also in the class TSm ν,c(σ, υ).

Theorem 4. Let u1(z) = z and un(z) = z − 1 −υ [1 + σ(n −1)][2n −υ −1]Φνc(n, m)zn, (19) for n = 2, 3, · · · . Then u(z) ∈TSm ν,c(σ, υ) if and only if u(z) can be expressed in the form u(z) = ∞ P n=2 ζnun(z), where ζn ≥0 and ∞ P n=1

Theorem 5. Let the function uj(z), j = 1, 2, · · · , l defined by (18) be in the classes TSm ν,c(σ, υj), j = 1, 2, · · · , l respectively. Then the function h(z) defined by h(z) = z −1 l ∞ X n=2   l X j=1 an,j

Theorem 6. Let u ∈TSm ν,c(σ, υ). Then

Theorem 7. Let u(z) = z + ∞ X n=2 anzn ∈Sm ν,c(σ, υ), and define the partial sums by uq(z) = z + q X n=2 anzn, q ∈N. Suppose that

Theorem 8. Under the same hypotheses, ℜ  u′(z) u′q(z)  > 1 −q + 1 dq+1 , z ∈U. The result is sharp.

Theorem 9. Under the same hypotheses, ℜ u′ q(z) u′(z)  > dq+1 q + 1 + dq+1 , z ∈U. The result is sharp.

Theorem 10. Let g ∈Sm ν,c(σ, υ) and suppose that dn = [1 + σ(n −1)][2n −υ −1]Φν c(n, m) 1 −υ , n ≥2. If ξ = 1 − δ(1 −υ) 2 h (1 −υ) −(1 + σ)(3 −υ)Φνc(2, m) i, and

Theorem 11. Let u(z) = z + ∞ X n=2 anzn ∈Sm ν,c(σ, υ). Then for any real number µ, |a3 −µa2 2| ≤max  1 d3 , |µ| d2 2

Theorem 12. Let u(z) = z + ∞ X n=2 anzn ∈Sm ν,c(σ, υ), and let µ ∈R. Then |a3 −µa2 2| ≤1 −υ d3 max {1, |2Θ −1|} , where Θ = 1 2

Definition 1. For −1 ≤υ < 1, and 0 ≤σ < 1, we let Sm ν,c(σ, υ) be the subclass of A consisting of functions of the form (1) and satisfying the analytic criterion ℜ ( z(Fm ν,cu(z))′ + σz2(Fm ν,cu(z))′′ (1 −σ)Fm

Definition 2. A function u ∈A is said to belong to the class Sm,ξ ν,c (σ, υ) if there exists a function g ∈Sm ν,c(σ, υ) such that