Geometric properties of new subclasses of analytic functions defined by Opoola differential operator

Theorem 3. A function f (z) of the form (1) belongs to the class Sm,t β,µ(α, η, γ) if ∞ ∑ n=2 [n −1 + η(1 −n + 2γn −2γα)] Mm n (µ, β, t) |an| ≤2ηγ(1 −α), (7) where Mm n (µ, β, t) = [1 + (n + β −µ −1)t]m . The result in (7) is sharp for functions of the form f (z) = z + 2ηγ(1 −α) [n −1 + η(1 −n + 2γn −2γα)] Mm n (µ, β, t)zn, n ≥2.

Theorem 4. If a function f (z) of the form (2) is in the class Tm,t β,µ(α, η, γ), then ∞ ∑ n=2 [n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t)an ≤2ηγ(1 −α). (11) The result in (11) is sharp for functions of the form f (z) = z − 2ηγ(1 −α) [n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t)zn, n ≥2. (12)

Theorem 5. A function f (z) of the form (1) is in the class Km,t β,µ(α, η, γ) if ∞ ∑ n=2 n[n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t) |an| ≤2ηγ(1 −α). (16)

Theorem 6. If a function f (z) of the form (2) is in the class Cm,t β,µ(α, η, γ), then ∞ ∑ n=2 n[n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t)an ≤2ηγ(1 −α). (20) The result in (20) is sharp for functions of the form f (z) = z − 2ηγ(1 −α) n[n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t)zn, n ≥2. (21)

Theorem 7. If f (z) ∈Tm,t β,µ(α, η, γ), then for |z| = r, r − 2ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m r2 ≤| f (z)| ≤r + 2ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m r2, 1 −r 4ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m ≤| f ′(z)| ≤1 + r 4ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m .

Theorem 8. If f (z) ∈Cm,t β,µ(α, η, γ), then for |z| = r, r − ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m r2 ≤| f (z)| ≤r + ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m r2, 1 −r 2ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m ≤| f ′(z)| ≤1 + r 2ηγ(1 −α) [1 −η + 2ηγ(2 −α)][1 + (β −µ + 1)t]m .

Theorem 9. If f (z) = z −∑∞ n=2 |an| zn and g(z) = z −∑∞ n=2 |bn| zn are in the class Tm,t β,µ(α, η, γ) then, h(z) = z −1 2 ∞ ∑ n=2 |an + bn| zn ∈Tm,t β,µ(α, η, γ).

Theorem 10. If f (z) = z −∑∞ n=2 |an| zn and g(z) = z −∑∞ n=2 |bn| zn are in the class Cm,t β,µ(α, η, γ) then, h(z) = z −1 2 ∞ ∑ n=2 |an + bn| zn ∈Cm,t β,µ(α, η, γ).

Theorem 11. If f (z) ∈Tm,t β,µ(α, η, γ), then f (z) is convex of order ρ in |z| < r(α, η, γ, ρ), where r(α, η, γ, ρ) = inf n≥2 [n −1 + η(1 −n + 2γn −2γα)]Mm n (µ, β, t)(1 −ρ) n(n −ρ)2ηγ(1 −α)  1 n−1 .

Definition 1. A function f (z) of the form (1) is said to belong to the class Sm,t β,µ(α, η, γ) if it satisfies the following condition:

Definition 2. A function f (z) of the form (1) is said to belong to the class Km,t β,µ(α, η, γ) if it satisfies the following condition: