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Abstract
In this paper, the operator denoted by Dm : A →A, is defined by Dm[f](z) = (1 −
λ)Rm[f](z)+λLm[f](z), z ∈U, a differential-integral operator, where Rm is Ruscheweyh differ-
ential operator and Lm is Libera integral operator. By using the operator Dm the class of univalent
functions denoted by M( m,λ,α), 0 ≤λ ≤1, 0 ≤α < 1, is defined and several differential sub-
ordinations are studied.
2010 Mathematics Subject Classification: 30C20; 30C45; 30A40
Results & Lemmas (6)
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Theorem 1.
Theorem 1. The set M( m,λ,α) is convex.
Theorem 1. The set M( m,λ,α) is convex.
Theorem 2.
Theorem 2. If 0 ≤α < 1, 0 ≤λ ≤1 and m ∈N, then we have M( m,λ,α) ⊂M( m,λ,δ), where δ = 2α−1+2(1−α)ln2, δ ≈0,62α+0,38 < 1.
Theorem 2. If 0 ≤α < 1, 0 ≤λ ≤1 and m ∈N, then we have M( m,λ,α) ⊂M( m,λ,δ), where δ = 2α−1+2(1−α)ln2, δ ≈0,62α+0,38 < 1.
Corollary 1.
Corollary 1. If f ∈M( m,λ,δ), then Re Dm[f](z) z > 2α−1+2(1−α)ln2 = δ.
Corollary 1. If f ∈M( m,λ,δ), then Re Dm[f](z) z > 2α−1+2(1−α)ln2 = δ.
Theorem 3.
Theorem 3. Let h be a convex function in U with h(0) = 1. If f ∈A, 0 ≤λ ≤1, m ∈N and satisfies the differential subordination…
Theorem 3. Let h be a convex function in U with h(0) = 1. If f ∈A, 0 ≤λ ≤1, m ∈N and satisfies the differential subordination ((1−λ)Rm[f](z)+λLm[f](z))′ ≺h(z), z ∈U, (23) then (1−λ)Rm[f](z)+λLm[f](z) z ≺q(z) = 1 z Z z 0 h(t)dt and this result is sharp.
Theorem 4.
Theorem 4. Let q be convex function in U, with q(0) = 1, and let θ and ϕ be analytic in a domain D containing q(U). Set Q(z) =…
Theorem 4. Let q be convex function in U, with q(0) = 1, and let θ and ϕ be analytic in a domain D containing q(U). Set Q(z) = zq′(z)ϕ[q(z)] and h(z) = θ[q(z)]+Q(z). Let m ∈N, f ∈A, 0 ≤λ ≤1 and satisfies the differential subordination Dm[f](z) Dm−1[f](z) +z Dm[f](z) Dm−1[f](z) ′ ≺q(z)+zq′(z) (27) then Dm[f](z) Dm−1[f](z) ≺q(z), z ∈U and q is the best dominant. The operator Dm[f] is defined in (6).
Theorem 5.
Theorem 5. Let q be a convex function q(0) = 1, and let θ and ϕ be analytic in a domain D containing q(U). Set Q(z) = zq′(z)ϕ[q(z)] and…
Theorem 5. Let q be a convex function q(0) = 1, and let θ and ϕ be analytic in a domain D containing q(U). Set Q(z) = zq′(z)ϕ[q(z)] and h(z) = θ[q(z)]+Q(z). If f ∈A, λ ≥0, m ∈N, and satisfies the differential subordination (Dm[f](z))′ +z(Dm[f](z))′′ ≺h(z), z ∈U, (34) then (Dm[f](z))′ ≺q(z), z ∈U, and q is the best dominant.
Function classes studied:
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