Abstract
For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(α)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,λ)$ for $\log f'(z_0)+αf(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\
Results & Lemmas (8)
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Lemma 2.1.
Lemma 2.1. Let f be an analytic function in D with f(z) = zp + · · ·. If Re 1 + zf ′′(z) f ′(z) > 0, z ∈D, then f ∈(S∗)p. Now, we list…
Lemma 2.1. Let f be an analytic function in D with f(z) = zp + · · · . If Re 1 + zf ′′(z) f ′(z) > 0, z ∈D, then f ∈(S∗)p. Now, we list down some basic properties of V (z0, λ).
Proposition 2.2.
Proposition 2.2. We have (1) V (z0, λ) is a compact subsect of C. (2) V (z0, λ) is a convex subsect of C. (3) For |λ| = 1 or z0 = 0, (2.3)…
Proposition 2.2. We have (1) V (z0, λ) is a compact subsect of C. (2) V (z0, λ) is a convex subsect of C. (3) For |λ| = 1 or z0 = 0, (2.3) V (z0, λ) = {−2 log(1 −λz0)} . (4) For |λ| < 1 and z0 ̸= 0, V (z0, λ) has −2 log(1 −λz0) as an interior point.
Theorem 2.7.
Theorem 2.7. For λ ∈D, α ∈C 0 and z0 ∈D 0, the boundary ∂V (z0, λ) is the Jordan curve given by (−π, π] ∋θ 7→log H′ eiθ,λ(z0) + αHeiθ,λ(z0)…
Theorem 2.7. For λ ∈D, α ∈C \ {0} and z0 ∈D \ {0}, the boundary ∂V (z0, λ) is the Jordan curve given by (−π, π] ∋θ 7→log H′ eiθ,λ(z0) + αHeiθ,λ(z0) = Z z0 0 2δ(eiθζ, λ) 1 −δ(eiθζ, λ)ζ dζ. If f(z0) = Heiθ,λ(z0) for some f ∈Fα(λ) and θ ∈(−π, π], then f(z) = Heiθ,λ(z). 3. Preparation for the proof of Theorem 2.7
Proposition 3.1.
Proposition 3.1. For f ∈Fα(λ) and λ ∈D we have (3.2)
Proposition 3.1. For f ∈Fα(λ) and λ ∈D we have (3.2)
Corollary 3.4.
Corollary 3.4. For f ∈Fα(0) we have (3.5)
Corollary 3.4. For f ∈Fα(0) we have (3.5)
Corollary 3.6.
Corollary 3.6. Let γ: z(t), 0 ≤t ≤1 be a C1-curve in D with z(0) = 0 and z(1) = z0. Then we have V (z0, λ) ⊂ w ∈C: |w −C(λ, γ)| ≤R(λ, γ),
Corollary 3.6. Let γ : z(t), 0 ≤t ≤1 be a C1-curve in D with z(0) = 0 and z(1) = z0. Then we have V (z0, λ) ⊂{w ∈C: |w −C(λ, γ)| ≤R(λ, γ)} ,
Lemma 3.7.
Lemma 3.7. [9] For θ ∈R and λ ∈D the function G(z) = Z z 0 eiθζ 1 + (λeiθ −λ)ζ −eiθζ2 2 dζ, z ∈D, has a double zero at the origin and no…
Lemma 3.7. [9] For θ ∈R and λ ∈D the function G(z) = Z z 0 eiθζ {1 + (λeiθ −λ)ζ −eiθζ2}2 dζ, z ∈D, has a double zero at the origin and no zeros elsewhere in D. Furthermore there exists a starlike univalent function G0 in D such that G = eiθG2 0 and G0(0) = G′ 0(0)−1 = 0.
Proposition 3.8.
Proposition 3.8. Let z0 ∈D 0. Then for θ ∈(−π, π] we have log H′ eiθ,λ(z0) + αHeiθ,λ(z0) ∈∂V (z0, λ). Furthermore, if log f ′(z0) + αf(z0)…
Proposition 3.8. Let z0 ∈D \ {0}. Then for θ ∈(−π, π] we have log H′ eiθ,λ(z0) + αHeiθ,λ(z0) ∈∂V (z0, λ). Furthermore, if log f ′(z0) + αf(z0) = log H′ eiθ,λ(z0) + αHeiθ,λ(z0) for some f ∈Fα(λ) and θ ∈(−π, π], then f = Heiθ,λ.
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