Double Subordination-Preserving Properties for Certain Integral Operators

Lemma 2.1 [10]. Let β,γ ∈C with β ̸= 0 and let h ∈(U) with h(0) = c. If Re{βh(z) + γ} > 0 (z ∈U), then the solution of the differential equation q(z)+ zq′(z) βq(z)+γ = h(z) (z ∈U) (2.1) with q(0) = c is analytic in U and satisfies Re{βq(z) +γ} > 0 (z ∈U).

Lemma 2.2 [1]. Let p ∈ with p(0) = a and let q(z) = a + anzn + ··· be analytic in U with q(z) ̸≡a and n ≥1. If q is not subordinate to p, then there exist points z0 = r0eiθ ∈U and ζ0 ∈∂U\E( f ), for which q(Ur0) ⊂p(U), q z0  = p ζ0 , z0q′z0  = mζ0p′ζ0  (m ≥n). (2.2)

Lemma 2.3 [3]. Let q ∈[a,1], let ϕ : C2 →C, and set ϕ(q(z),zq′(z)) ≡h(z). If L(z,t) = ϕ(q(z),tzq′(z)) is a subordination chain and p ∈[a,1] ∩, then h(z) ≺ϕ p(z),zp′(z)  (z ∈U) (2.3) implies that q(z) ≺p(z) (z ∈U). (2.4) Furthermore, if ϕ(q(z),zp′(z)) = h(z) has a univalent solution q ∈, then q is the best subordinant. We now recall that the Gauss hypergeometric function 2F1(a,b;c;z) is defined by ([11], see also [12, Chapter 14])

Lemma 2.4 [13]. Let β > 0, β + γ > 0 and let Iβ,γ be the integral operator defined by (1.7). If α ∈[−γ/β,1), then the order of starlikeness of the class Iβ,γ(∗(α)), that is, the largest number δ = δ(α;β,γ) such that Iβ,γ ∗(α)  ⊂∗(δ), (2.7) is given by the number δ(α;β,γ) = inf{Req(z) : z ∈U}, where q(z) = 1 βQ(z) −γ β, Q(z) = 1

Lemma 2.5 [14]. The function L(z,t) = a1(t)z + ···, with a1(t) ̸= 0 and limt→∞|a1(t)| = ∞, is a subordination chain if and only if Re z∂L(z,t)/∂z ∂L(z,t)/∂t

Theorem 3.1. Let f ,g ∈β,γ with β > 0 and 0 < β +γ ≤1. Suppose that Re

Theorem 3.2. Let f ,g ∈β,γ with β > 0 and 0 < β +γ ≤1. Suppose that Re

Theorem 3.3. Let f ,gk ∈β,γ (k = 1,2) with β > 0 and 0 < β +γ ≤1. Suppose that Re

Corollary 3.4. Let f ,gk ∈β,γ (k = 1,2) with β > 0 and 0 < β + γ ≤1. Suppose that the condition (3.31) is satisfied and Re