Abstract
Let $Co(α)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(α)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(α)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\left(f
Results & Lemmas (15)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 2.1.
Lemma 2.1. Let ψ ∈H1 be such that it is starlike with respect to 1 and suppose that g ∈A satisfies 2 α −1 (α + 1) 2 1 + z 1 −z −1 −zg′′(z)…
Lemma 2.1. Let ψ ∈H1 be such that it is starlike with respect to 1 and suppose that g ∈A satisfies 2 α −1 (α + 1) 2 1 + z 1 −z −1 −zg′′(z) g′(z) = ψ(z), z ∈D, for some α ∈(1, 2]. Then, for f ∈Co(α), the condition (2.2) 2
Theorem 2.3.
Theorem 2.3. Let g be as Lemma 2.1. If f ∈Co(α), then ∥TF∥≤∥TG∥where F(z) = Z z 0 (1 −ζ)α+1f ′(ζ) dζ and G(z) = Z z 0 (1 −ζ)α+1g′(ζ) dζ.…
Theorem 2.3. Let g be as Lemma 2.1. If f ∈Co(α), then ∥TF∥≤∥TG∥where F(z) = Z z 0 (1 −ζ)α+1f ′(ζ) dζ and G(z) = Z z 0 (1 −ζ)α+1g′(ζ) dζ. Now we state the following corollary:
Corollary 2.4.
Corollary 2.4. For f ∈Co(α) and g as in Lemma 2.1, we have (1 −|z|2)f ′′(z) f ′(z) −(α + 1)1 −|z|2 1 −z ≤ (1 −|z|2)g′′(w(z)) g′(w(z)) −(α +…
Corollary 2.4. For f ∈Co(α) and g as in Lemma 2.1, we have (1 −|z|2)f ′′(z) f ′(z) −(α + 1)1 −|z|2 1 −z ≤ (1 −|z|2)g′′(w(z)) g′(w(z)) −(α + 1)1 −|w(z)|2 1 −w(z) , where w : D →D is a holomorphic function with w(0) = 0. Equality holds when w(z) = z.
Theorem 2.5.
Theorem 2.5. Let α ∈(1, 2] be fixed. Then the set of variability of the functional (1 −|z|2)Tf(z), f ∈Co(α), is the closed disk with center…
Theorem 2.5. Let α ∈(1, 2] be fixed. Then the set of variability of the functional (1 −|z|2)Tf(z), f ∈Co(α), is the closed disk with center 2z + (α + 1)(1 −z)/(1 −z) and radius α −1. The points on the boundary of this disk are attained if and only if f is one of the functions gθ, where, gθ(z) = 1 α(1 + eiθ) 1 + eiθz 1 −z α −1 , for θ ∈[0, 2π] \ {π},
Corollary 2.7.
Corollary 2.7. Let f ∈Co(α), α ∈[1, 2]. Then, 4 ≤∥Tf∥≤2α + 2. The equality holds in lower estimate for the function gπ and in upper…
Corollary 2.7. Let f ∈Co(α), α ∈[1, 2]. Then, 4 ≤∥Tf∥≤2α + 2. The equality holds in lower estimate for the function gπ and in upper estimate for the function g0 which are described in the statement of the above theorem.
Theorem 2.8
Theorem 2.8 (Distortion Theorem). Let α ∈(1, 2]. Then, for each f ∈Co(α), we have (1 −r)α−1 (1 + r)α+1 ≤|f ′(z)| ≤(1 + r)α−1 (1 −r)α+1, |z|…
Theorem 2.8 (Distortion Theorem). Let α ∈(1, 2]. Then, for each f ∈Co(α), we have (1 −r)α−1 (1 + r)α+1 ≤|f ′(z)| ≤(1 + r)α−1 (1 −r)α+1, |z| = r < 1. For each z ∈D, z ̸= 0, equality occurs if and only if f = gθ, where θ ∈[0, 2π) \ {π}.
Corollary 2.9.
Corollary 2.9. Co(α) ⊂Hp for 0 < p < 1/α. The result is best possible.
Corollary 2.9. Co(α) ⊂Hp for 0 < p < 1/α. The result is best possible.
Theorem 2.10.
Theorem 2.10. Let f ∈Co(α), α ∈(1, 2]. Then the set of variability of the functional Tf(z)(1−|z|2), f ∈Co(α), whenever f ′′(0) = α+1+(α−1)a…
Theorem 2.10. Let f ∈Co(α), α ∈(1, 2]. Then the set of variability of the functional Tf(z)(1−|z|2), f ∈Co(α), whenever f ′′(0) = α+1+(α−1)a with a ∈D being fixed, is the disk (1 −|z|2)Tf(z) −2z −(α + 1)1 −z 1 −z −(α −1)z(1 + |a|2 + az) + a 1 + |z|2 + 2Re (az)
Corollary 2.16.
Corollary 2.16. Let f ∈Co(α) and f ′′(0) = α + 1 be fixed. Then, 3 + α ≤∥Tf∥≤2 + 2α.
Corollary 2.16. Let f ∈Co(α) and f ′′(0) = α + 1 be fixed. Then, 3 + α ≤∥Tf∥≤2 + 2α.
Theorem 3.1.
Theorem 3.1. Let 1 < α ≤2. Then, f ∈Co(α) if and only if (3.2) 1 z f ⋆(α −1)z −(α + 1 + 2x)z2 (1 −z)3 + f ⋆((α + 1)x + 2)z −(α −1)xz2…
Theorem 3.1. Let 1 < α ≤2. Then, f ∈Co(α) if and only if (3.2) 1 z f ⋆(α −1)z −(α + 1 + 2x)z2 (1 −z)3 + f ⋆((α + 1)x + 2)z −(α −1)xz2 (1 −z)3 ̸= 0 for all |z| < 1 and for all x with |x| = 1. Equivalently, this holds if and only if (3.3)
Theorem 3.7.
Theorem 3.7. Let α ∈(1, 2]. A function f ∈Co(α) if and only if there exists a function s ∈Πα−1 such that (3.8) f(z) = Z z 0 s(t) (1 −t)α+1…
Theorem 3.7. Let α ∈(1, 2]. A function f ∈Co(α) if and only if there exists a function s ∈Πα−1 such that (3.8) f(z) = Z z 0 s(t) (1 −t)α+1 dt.
Theorem 3.10.
Theorem 3.10. A function f ∈Co(α) if and only if s(z) ⋆ z (1 −z)2 + 1 −α x + 1 1 1 −z ̸= 0, z ∈D, |x| = 1, x ̸= −1, for some s ∈Πα−1.
Theorem 3.10. A function f ∈Co(α) if and only if s(z) ⋆ z (1 −z)2 + 1 −α x + 1 1 1 −z ̸= 0, z ∈D, |x| = 1, x ̸= −1, for some s ∈Πα−1.
Theorem 3.15.
Theorem 3.15. Let f ∈Co(α) have the expansion (3.4). Then the following sharp inequality holds (3.16)
Theorem 3.15. Let f ∈Co(α) have the expansion (3.4). Then the following sharp inequality holds (3.16)
Theorem 3.17.
Theorem 3.17. Let α ∈(1, 2]. A function f ∈Co(α) if and only if there exists a φ ∈S∗such that f(z) = Λφ(z), where Λφ(z) = Z z 0 1 (1 −t)α+1…
Theorem 3.17. Let α ∈(1, 2]. A function f ∈Co(α) if and only if there exists a φ ∈S∗such that f(z) = Λφ(z), where Λφ(z) = Z z 0 1 (1 −t)α+1 t φ(t) (α−1)/2 dt.
Corollary 3.19.
Corollary 3.19. Let φ(z) = z + P n≥2 φnzn belong to S∗and α ∈(1, 2]. Then A(φ3, φ2, α) ∈h(D), where h(z) = z + α −2 2(α + 1)z2. It is not…
Corollary 3.19. Let φ(z) = z + P n≥2 φnzn belong to S∗and α ∈(1, 2]. Then A(φ3, φ2, α) ∈h(D), where h(z) = z + α −2 2(α + 1)z2. It is not clear whether the present restriction on α is essential in the last corollary. References 1. F.G. Avkhadiev, Ch. Pommerenke, and K.-J. Wirths: Sharp inequalities for the coeffi- cient of concave schlicht functions, Comment. Math. Helv. 81(2006), 801–807. 2. F.G. Avkhadiev and K.-J. Wirths: Concave schlicht functions with bounded opening angle at infinity, Lobache
Related Papers