Abstract
Using the third-order differential subordination basic results, we introduce certain
classes of admissible functions and investigate some applications of third-order
differential subordination for p-valent functions associated with generalized fractional
differintegral operator.
MSC: 30C45; 30C50
Results & Lemmas (8)
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Lemma 1.1
Lemma 1.1 ([2, p. 449]) Let Ω be a subset of C, ψ ∈Ψn[Ω,q] and p ∈H[a,n] with n ≥2. If q ∈Q(a) and satisfies the following conditions Re…
Lemma 1.1 ([2, p. 449]) Let Ω be a subset of C, ψ ∈Ψn[Ω,q] and p ∈H[a,n] with n ≥2. If q ∈Q(a) and satisfies the following conditions Re wq′′(w) q′(w) ≥0 and zp′(z) q′(w) ≤n, z ∈U,w ∈∂U \ E(q), then ψ p(z),zp′(z),z2p′′(z),z3p′′′(z);z
Theorem 2.1
Theorem 2.1 Let Ω be a subset of C and φ ∈Φp[Ω,q]. If q ∈Q0 satisfies the following conditions: Re wq′′(w) q′(w) ≥0 and z(N k,δ,ζ…
Theorem 2.1 Let Ω be a subset of C and φ ∈Φp[Ω,q]. If q ∈Q0 satisfies the following conditions: Re wq′′(w) q′(w) ≥0 and z(N k,δ,ζ p,λ,μ,ηf (z))′ q′(w) ≤p, (2.1) then φ N k,δ,ζ
Corollary 2.1
Corollary 2.1 Let Ω be a subset of C and q be univalent in U with q ∈Q0. Let φ ∈ Φp[Ω,qρ] for some ρ ∈(0,1), where qρ(z) = q(ρz). If qρ…
Corollary 2.1 Let Ω be a subset of C and q be univalent in U with q ∈Q0. Let φ ∈ Φp[Ω,qρ] for some ρ ∈(0,1), where qρ(z) = q(ρz). If qρ satisfies the following conditions: Re wq′′ ρ(w) q′ρ(w) ≥0 and z(N k,δ,ζ p,λ,μ,ηf (z))′ q′ρ(w) ≤p, w ∈∂U \ E(qρ), (2.9) then
Corollary 2.2
Corollary 2.2 Let φ ∈Φp[h,q], where h is univalent in U and suppose that q ∈Q0 satisfies conditions (2.1). Then φ N k,δ,ζ p,λ,μ,ηf (z),N…
Corollary 2.2 Let φ ∈Φp[h,q], where h is univalent in U and suppose that q ∈Q0 satisfies conditions (2.1). Then φ N k,δ,ζ p,λ,μ,ηf (z),N k+1,δ,ζ p,λ,μ,η f (z),N k+2,δ,ζ p,λ,μ,η f (z),N k+3,δ,ζ p,λ,μ,η f (z);z ≺h(z) (2.10) implies N k,δ,ζ p,λ,μ,ηf (z) ≺q(z).
Corollary 2.3
Corollary 2.3 Let q be univalent in U with q ∈Q0 and φ ∈Φp[h,qρ] for some ρ ∈(0,1), where qρ(z) = q(ρz). If qρ satisfies conditions (2.9),…
Corollary 2.3 Let q be univalent in U with q ∈Q0 and φ ∈Φp[h,qρ] for some ρ ∈(0,1), where qρ(z) = q(ρz). If qρ satisfies conditions (2.9), then the subordination (2.10) implies that N k,δ,ζ p,λ,μ,ηf (z) ≺q(z). We next show the relation between the best dominant of a differential subordination and the solution of a corresponding differential equation.
Corollary 2.4
Corollary 2.4 Let h be univalent in U and ψ be given by (2.8) where φ ∈Φp[h,q]. Suppose that the differential equation ψ …
Corollary 2.4 Let h be univalent in U and ψ be given by (2.8) where φ ∈Φp[h,q]. Suppose that the differential equation ψ q(z),zq′(z),z2q′′(z),z3q′′′(z);z = h(z)
Corollary 3.1
Corollary 3.1 Let Ω be a subset of C and φ ∈Φp[Ω,M]. If we suppose that z N k,δ,ζ p,λ,μ,ηf (z) ′ ≤pM, z ∈U, (3.3) and the function q…
Corollary 3.1 Let Ω be a subset of C and φ ∈Φp[Ω,M]. If we suppose that z N k,δ,ζ p,λ,μ,ηf (z) ′ ≤pM, z ∈U, (3.3) and the function q is given by (3.1), then φ N k,δ,ζ p,λ,μ,ηf (z),N k+1,δ,ζ p,λ,μ,η f (z),N k+2,δ,ζ p,λ,μ,η f (z),N k+3,δ,ζ p,λ,μ,η f (z);z
Corollary 3.2
Corollary 3.2 Let φ ∈Φp[q(U),M] and suppose that the function q given by (3.1) satisfies condition (3.3). Then φ N k,δ,ζ p,λ,μ,ηf (z),N…
Corollary 3.2 Let φ ∈Φp[q(U),M] and suppose that the function q given by (3.1) satisfies condition (3.3). Then φ N k,δ,ζ p,λ,μ,ηf (z),N k+1,δ,ζ p,λ,μ,η f (z),N k+2,δ,ζ p,λ,μ,η f (z),N k+3,δ,ζ p,λ,μ,η f (z);z < M, z ∈U, implies N k,δ,ζ p,λ,μ,ηf (z) ≺Mz. Let φ(α1,β1,γ1,ε1;z) = α1 + β1 and Ω = h(U), where h(z) = 2Mz. We will show that φ ∈ Φp[h(U),M] by proving that condition (3.2) is satisfied. Thus,
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