Results & Lemmas (13)
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Theorem 1.1
Theorem 1.1 see 25 . Let f ∈CV. Then, 1 fλz: 1 −λz λfz ∈ST if and only if λ ∈C and |λ −1| ≤1/3; 2 if f′′0 0, then fλ…
Theorem 1.1 see 25 . Let f ∈CV. Then, 1 fλz : 1 −λz λfz ∈ST if and only if λ ∈C and |λ −1| ≤1/3; 2 if f′′0 0, then fλ ∈ST for λ ∈0, 1 . For α > 0 and 0 < λ ≤1, the class CPα, λ consists of functions f ∈S satisfying Re zf′z ′ f′ λz α >
Lemma 2.1.
Lemma 2.1. Let α > 0 and 0 < λ ≤1. A function f is in the class SPα, λ if and only if 1 z
Lemma 2.1. Let α > 0 and 0 < λ ≤1. A function f is in the class SPα, λ if and only if 1 z
Lemma 2.2.
Lemma 2.2. Let α > 0 and 0 < λ ≤1. If Ht,λz: z ∞ k2 hk,λtzk ∈SP′α, λ, 2.8 then hk,λt ≤ ⎧ ⎪ ⎪ ⎨ ⎪
Lemma 2.2. Let α > 0 and 0 < λ ≤1. If Ht,λz : z ∞ k2 hk,λtzk ∈SP′α, λ, 2.8 then hk,λt ≤ ⎧ ⎪ ⎪ ⎨ ⎪
Lemma 2.3.
Lemma 2.3. For each complex number ϵ and f ∈A, define the function Fϵ by Fϵz: fz ϵz 1 ϵ. 2.15 Let α > 0, 0 < λ ≤1, and Fϵ ∈SPα,…
Lemma 2.3. For each complex number ϵ and f ∈A, define the function Fϵ by Fϵz : fz ϵz 1 ϵ . 2.15 Let α > 0, 0 < λ ≤1, and Fϵ ∈SPα, λ for |ϵ| < δ for some δ > 0. Then 1 z
Theorem 2.4.
Theorem 2.4. Let α > 0 and 0 < λ ≤1. Let f ∈A and δ > 0. For a complex number ϵ with |ϵ| < δ, let the function Fϵ, defined by 2.15, be in…
Theorem 2.4. Let α > 0 and 0 < λ ≤1. Let f ∈A and δ > 0. For a complex number ϵ with |ϵ| < δ, let the function Fϵ, defined by 2.15, be in SPα, λ. Then, Nδ′f ⊂SPα, λ for δ′ : ⎧ ⎪ ⎨ ⎪ ⎩ 2δ α1 −α, 0 < α < 1 2, δ, α ≥1
Lemma 2.5
Lemma 2.5 see 31 . Let f ∈CV, g ∈ST, and suppose F is any analytic function defined on Δ. Then fz∗gzFz fz∗gz ⊂coFΔ, z ∈Δ,…
Lemma 2.5 see 31 . Let f ∈CV, g ∈ST, and suppose F is any analytic function defined on Δ. Then fz∗gzFz fz∗gz ⊂coFΔ, z ∈Δ, 2.25 where co stands for the closed convex hull.
Lemma 2.6.
Lemma 2.6. If f ∈CV, g ∈SPα, λ, and gλ ∈ST, then f∗g ∈SPα, λ.
Lemma 2.6. If f ∈CV, g ∈SPα, λ, and gλ ∈ST, then f∗g ∈SPα, λ.
Theorem 2.7.
Theorem 2.7. Let α > 0 and 0 ≤λ ≤1. If f ∈CPα, λ and fλ ∈CV, then the function Fϵ defined by 2.15 belongs to SPα, λ for |ϵ| < 1/4.
Theorem 2.7. Let α > 0 and 0 ≤λ ≤1. If f ∈CPα, λ and fλ ∈CV, then the function Fϵ defined by 2.15 belongs to SPα, λ for |ϵ| < 1/4.
Theorem 2.8.
Theorem 2.8. Let α > 0 and 0 ≤λ ≤1. If f ∈CPα, λ and fλ ∈CV, then Nδ′f ⊂SPα, λ, , where δ′: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2 α1 −α, 0 < α < 1
Theorem 2.8. Let α > 0 and 0 ≤λ ≤1. If f ∈CPα, λ and fλ ∈CV, then Nδ′f ⊂SPα, λ, , where δ′ : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2 α1 −α, 0 < α < 1
Lemma 2.10
Lemma 2.10 see 32 . Let Ω be a set in the complex plane C and suppose that the mapping Φ: C2 × Δ →C satisfies Φiρ, σ; z /∈Ω for z ∈Δ,…
Lemma 2.10 see 32 . Let Ω be a set in the complex plane C and suppose that the mapping Φ : C2 × Δ →C satisfies Φiρ, σ; z /∈Ω for z ∈Δ, and for all real ρ, σ such that σ ≤−n1 ρ2/2. If the function pz 1 cnzn · · · is analytic in Δ and Φpz, zp′z; z ∈Ω for all z ∈Δ, then Re pz > 0.
Lemma 2.11.
Lemma 2.11. Let 0 ≤λ ≤1/3. If pz 1 cz · · · is analytic in Δ and Re pz zp′z 1 −λ λpz > 0, 2.33 then Re pz > 0.
Lemma 2.11. Let 0 ≤λ ≤1/3. If pz 1 cz · · · is analytic in Δ and Re pz zp′z 1 −λ λpz > 0, 2.33 then Re pz > 0.
Theorem 2.12.
Theorem 2.12. Let 0 ≤λ ≤1/3. If f ∈SPα, λ, then fλ ∈ST.
Theorem 2.12. Let 0 ≤λ ≤1/3. If f ∈SPα, λ, then fλ ∈ST.
Corollary 2.13.
Corollary 2.13. Let 0 ≤λ ≤1/3. If f ∈CPα, λ, then fλ ∈CV. In view of this corollary, the statement that fλ ∈CV can be omitted from…
Corollary 2.13. Let 0 ≤λ ≤1/3. If f ∈CPα, λ, then fλ ∈CV. In view of this corollary, the statement that fλ ∈CV can be omitted from Theorems 2.7 and 2.8 if 0 ≤λ ≤1/3. Also clearly that f ∈CPα, 1 implies f1 f ∈CV. Thus, Theorem 2.8 reduces to the corresponding result in 30 for λ 1. Acknowledgments The first two authors acknowledge the support from the USM’s RU grant, while the fourth author acknowledges the support from the University Research Council of Kent State University and Universi
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