Generalized Briot-Bouquet differential equation based on new differential operator with complex connections

Lemma 1.3. If β ∈[0, 1], z ∈U then each function of the form • σ(z) = β + (1 −β)√ 1 + z, • σ(z) = β + (1 −β)ez, • σ(z) = β + (1 −β)(1 + sin(z)), • σ(z) = β + (1 −β)eez−1, has the upper and lower bound for all r ∈(0, 1), θ ∈[0, 2π) as follows: min |z|=r ℜ(σ(z)) = σ(−r) = min |z|=r |σ(ξ)| and max |z|=r ℜ(σ(z)) = σ(r) = max |z|=r |σ(z)|.

Lemma 1.4. If τ > 0 and σ ∈H[1, n], then there are constants ℘> 0 and ν > 0 with ν = ν(℘, τ, n) so that σ(z) + τzσ′(z) ≺ 1 + z 1 −z ν ⇒σ(z) ≺ 1 + z 1 −z ℘ .

Lemma 1.5. Let ϕ(z) be a convex function in U and h(z) = ϕ(z) + nν(zϕ′(z)) for ν > 0 and n is a positive integer. If ϱ ∈H[ϕ(0), n], and ϱ(z) + νzϱ′(z) ≺h(z), z ∈U, then ϱ(z) ≺ϕ(z), and this result is sharp. 2 Results In this section, we deal with the class S∗ m(α, κ, σ) for special types of σ(z) given in

Lemma 1.3.

Theorem 2.1. The class S∗ m(α, κ, σ) achieves the following inclusion: S∗ m(α, κ, σ) ⊂S∗ m(α, κ, γ) ⊂S∗ m(α, κ), where σ is one of the type in Lemma 1.3 and S∗ m(α, κ, γ) := { f ∈Λℜ z(Jm α,κ f (z)))′ Jm α,κ f (z)  > γ};

Theorem 2.3. The class S∗ m(α, κ, σ) achieves the following inclusion: S∗ m(α, κ, σ) ⊂Mm(α, κ, γ) := { f ∈Λℜ z(Jm α,κ f (z)))′ Jm α,κ f (z)  < γ, γ > 1}. where σ is given in Lemma 1.3.

Theorem 2.1, we have ℜ z(Jm α,λ f (z)))′ Jm α,λ f (z)  < β + (1 −β) √ 2 := γ, ℜ z(Jm α,λ f (z)))′ Jm α,λ f (z)

Theorem 2.5. If f ∈Λ satisfies the subordination z(Jm α,κ f (z))′ Jm α,κ f (z)  2 + z(Jm α,κ f (z))′′ (Jm α,κ f (z))′ −z(Jm α,κ f (z))′ Jm α,κ f (z)  ≺

Theorem 2.6. Let ϕ be a convex function such that ϕ(0) = 0 and let ℏbe the function ℏ(z) = ϕ(z) + z 1 −ℓϕ′(z), z ∈U, ℓ∈(0, 1). If for a function f ∈Λ satisfies the subordination z Jm+1 α,κ f (z)
ℓJm α,κ f (z) 1 −ℓ Jm+1 α,κ