Abstract
The article deals with the class ${\mathcal F}_{α}$ consisting of non-vanishing functions $f$ that are analytic and univalent in $\ID$ such that the complement $\IC\backslash f(\ID) $ is a convex set, $f(1)=\infty ,$ $f(0)=1$ and the angle at $\infty $ is less than or equal to $απ,$ for some $α\in (1,2]$. Related to this class is the class $CO(α)$ of concave univalent mappings in $\ID$, but this differs from ${\mathcal F}_{α}$ with the standard normalization $f(0)=0=f'(0)=1.$ A number of propert
Results & Lemmas (13)
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Lemma 1.
Lemma 1. For n ∈N, define B(x):= Bn(α, x) = F(1 −n, 1 −α; 2; 1 + x). Then for −1 < α < 1, we have |B(x)| ≤|B(1)|.
Lemma 1. For n ∈N, define B(x) := Bn(α, x) = F(1 −n, 1 −α; 2; 1 + x). Then for −1 < α < 1, we have |B(x)| ≤|B(1)|.
Lemma 2.
Lemma 2. If n ∈N, α > 1 and t is such that 0 < αt < 1, then we have (1 + x)−tAn(αt, x) ≤2−tAn(αt, 1), where An(α, x) is defined by (5).
Lemma 2. If n ∈N, α > 1 and t is such that 0 < αt < 1, then we have (1 + x)−tAn(αt, x) ≤2−tAn(αt, 1), where An(α, x) is defined by (5).
Lemma 3.
Lemma 3. Let B(x):= Bn(α, x) = F(1 −n, 1 −α; 2; 1 + x). Then for n ≥1 and α > 1, we have |Bn(α, x)| ≤|Bn(α, 1)|.
Lemma 3. Let B(x) := Bn(α, x) = F(1 −n, 1 −α; 2; 1 + x). Then for n ≥1 and α > 1, we have |Bn(α, x)| ≤|Bn(α, 1)| .
Theorem 1.
Theorem 1. If f ∈Fα for some α ∈(1, 2], then f(z) ≺ 1 + xz 1 −z α for some |x| = 1, x ̸= −1.
Theorem 1. If f ∈Fα for some α ∈(1, 2], then f(z) ≺ 1 + xz 1 −z α for some |x| = 1, x ̸= −1.
Corollary 1.
Corollary 1. If f ∈Fα for some α ∈(1, 2] and f(z) = 1 + P∞ n=1 anzn, then |an| ≤An(α) ≤An(2) = 4n for n ≥1, where An(α, 1) =: An(α) and…
Corollary 1. If f ∈Fα for some α ∈(1, 2] and f(z) = 1 + P∞ n=1 anzn, then |an| ≤An(α) ≤An(2) = 4n for n ≥1, where An(α, 1) =: An(α) and An(α, x) is defined by (5).
Theorem 2.
Theorem 2. Let f ∈CO(α) and b ∈∂f(D) be a nearest point to 0. Then there is an x ∈∂D with x ̸= −1 such that (6) f(z) = −b 1 + xϕ(z) 1…
Theorem 2. Let f ∈CO(α) and b ∈∂f(D) be a nearest point to 0. Then there is an x ∈∂D with x ̸= −1 such that (6) f(z) = −b 1 + xϕ(z) 1 −ϕ(z) α + b, where ϕ ∈B0 is univalent in D, ϕ(1) = 1 and ϕ′(0) = − 1 (1+x)bα. Moreover, each f ∈CO1(α) has the form (7) f(z) = 1
Theorem 2
Theorem 2 is handy in getting many well-known results about the family CO(α). We shall now state and prove some of its consequences. Recall…
Theorem 2 is handy in getting many well-known results about the family CO(α). We shall now state and prove some of its consequences. Recall that the hyperbolic (or Poincar´e) metric for D is the Riemannian metric defined by λD(z)|dz|, where λD(z) = 1 1−|z|2 denotes the hyperbolic density of D with constant curvature −4. Using analytic maps, hyperbolic metrics can be transferred from one domain to another as follows. For a given hyperbolic domain Ω(i.e. its complement contains at least two points)
Corollary 2.
Corollary 2. If f ∈CO(α) is of the form (1) and f(D) = Ω, then we have 1 ≥d(f(0), ∂Ω) = 1 |1 + x|α ≥1 2α for some |x| = 1, x ̸= −1. Here…
Corollary 2. If f ∈CO(α) is of the form (1) and f(D) = Ω, then we have 1 ≥d(f(0), ∂Ω) = 1 |1 + x|α ≥1 2α for some |x| = 1, x ̸= −1. Here d(w, ∂Ω) denotes the Euclidean distance between w ∈Ωand ∂Ω, the boundary of Ω.
Corollary 3.
Corollary 3. If f ∈CO(α) is of the form (1) and f(D) = Ω, then for each a ∈D we have 1 ≥d(f(a), ∂Ω)λΩ(a) ≥ 1 α|x + 1| ≥1 2α for some |x| =…
Corollary 3. If f ∈CO(α) is of the form (1) and f(D) = Ω, then for each a ∈D we have 1 ≥d(f(a), ∂Ω)λΩ(a) ≥ 1 α|x + 1| ≥1 2α for some |x| = 1, x ̸= −1.
Corollary 4.
Corollary 4. If f ∈CO(α) is of the form (1), f(D) = Ω, and α ∈(1, 2], then |an| ≤ 1 2αAn(α) holds for n ≥2, where each coefficient An(α, 1) =…
Corollary 4. If f ∈CO(α) is of the form (1), f(D) = Ω, and α ∈(1, 2], then |an| ≤ 1 2αAn(α) holds for n ≥2, where each coefficient An(α, 1) = An(α) is given by (4). In particular, for f ∈CO(2), we have the sharp estimate (9) an −n + 1 2 ≤n −1 2 for n ≥2.
Lemma 4.
Lemma 4. Suppose that f(z) = F(ϕ(z)) ∈CO(α) for some univalent function ϕ ∈B0, where F(z) = −b 1 + xz 1 −z α + b and b ∈∂f(D) is the…
Lemma 4. Suppose that f(z) = F(ϕ(z)) ∈CO(α) for some univalent function ϕ ∈B0, where F(z) = −b 1 + xz 1 −z α + b and b ∈∂f(D) is the nearest point to 0. Then we have Z Z D
Lemma 5.
Lemma 5. If g is a non-vanishing analytic function in D then Ar = Z Z Dr |g′(z)|2 dσ = 1 2 Z 2π 0 |g(reiθ)|2Re −reiθg′(reiθ) g(reiθ) dθ,…
Lemma 5. If g is a non-vanishing analytic function in D then Ar = Z Z Dr |g′(z)|2 dσ = 1 2 Z 2π 0 |g(reiθ)|2Re −reiθg′(reiθ) g(reiθ) dθ, where dσ denotes the area element dx dy
Theorem 3.
Theorem 3. Assume the hypotheses of Lemma 4. Then we have Z Z Dr
Theorem 3. Assume the hypotheses of Lemma 4. Then we have Z Z Dr
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