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Abstract
Radius constants for several classes of analytic functions on the unit disk are
obtained. These include the radius of starlikeness of a positive order, radius of parabolic
starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In
the main, the radius constants obtained are sharp. Conjectures on the non-sharp constants
are given.
2010 Mathematics Subject Classification: 30C45, 30C80
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 1.1.
Lemma 1.1. [27] Let p ∈P(α), and |z| = r < 1. Then p(z)−1 + (1 −2α)r2 1 −r2 ≤2(1 −α)r 1 −r2.
Lemma 1.1. [27] Let p ∈P(α), and |z| = r < 1. Then p(z)−1 + (1 −2α)r2 1 −r2 ≤2(1 −α)r 1 −r2 .
Lemma 1.2.
Lemma 1.2. [33] Let p ∈P(α), and |z| = r < 1. Then
Lemma 1.2. [33] Let p ∈P(α), and |z| = r < 1. Then
Lemma 1.3.
Lemma 1.3. [6, Lemma 2.4] Let p ∈P(1/2), and |z| = r < 1. Then Re zp′(z) p(z) ≥ ( −r/(1 + r), r < 1/3, −( √ 2− √ 1 −r2)2/(1 −r2), 1/3 ≤r ≤…
Lemma 1.3. [6, Lemma 2.4] Let p ∈P(1/2), and |z| = r < 1. Then Re zp′(z) p(z) ≥ ( −r/(1 + r), r < 1/3, −( √ 2− √ 1 −r2)2/(1 −r2), 1/3 ≤r ≤ p 8 √
Lemma 1.4.
Lemma 1.4. [2] Let 0 < a < √ 2. If ra is given by ra = √ 1 −a2 −(1 −a2) 1/2, 0 < a ≤2 √ 2/3 √
Lemma 1.4. [2] Let 0 < a < √ 2. If ra is given by ra = √ 1 −a2 −(1 −a2) 1/2 , 0 < a ≤2 √ 2/3 √
Lemma 1.5.
Lemma 1.5. [34] Let a > 1/2. If the number Ra satisfies Ra = ( a −1/2, 1/2 < a ≤3/2 √2a −2, a ≥3/2, then w: |w−a| < Ra ⊆ w: |w−1| < Rew.
Lemma 1.5. [34] Let a > 1/2. If the number Ra satisfies Ra = ( a −1/2, 1/2 < a ≤3/2 √2a −2, a ≥3/2, then {w : |w−a| < Ra} ⊆{w : |w−1| < Rew}.
Theorem 2.1.
Theorem 2.1. For the class F1, the following sharp radius results hold: (a) the S L -radius for F1 is RS L = √ 2−1 2 + p 7 −2 √ 2 ≃0.10247,…
Theorem 2.1. For the class F1, the following sharp radius results hold: (a) the S L -radius for F1 is RS L = √ 2−1 2 + p 7 −2 √ 2 ≃0.10247, (b) the M (β)-radius for F1 is RM (β) = β −1 2 +
Theorem 2.2.
Theorem 2.2. For the class F2, the following radius results hold: (a) the S L -radius is RS L = 4 −2 √ 2 √ 2( p 17 −4 √ 2+ 3) ≃0.13009, (b)…
Theorem 2.2. For the class F2, the following radius results hold: (a) the S L -radius is RS L = 4 −2 √ 2 √ 2( p 17 −4 √ 2+ 3) ≃0.13009, (b) the M (β)-radius is RM (β) =
Theorem 2.3.
Theorem 2.3. For the class F3, the following radius results hold: (a) the S L -radius is RS L = 4 −2 √ 2 √ 2( p 17 −4 √ 2+ 3) ≃0.13009, (b)…
Theorem 2.3. For the class F3, the following radius results hold: (a) the S L -radius is RS L = 4 −2 √ 2 √ 2( p 17 −4 √ 2+ 3) ≃0.13009, (b) the M (β)-radius is RM (β) =
Theorem 2.4.
Theorem 2.4. For the class F4, the following sharp radius results hold: (1) the C (α)-radius is RC (α) = 2(1 −α) 3 + p 9 + 4(α −2)(α −1),…
Theorem 2.4. For the class F4, the following sharp radius results hold: (1) the C (α)-radius is RC (α) = 2(1 −α) 3 + p 9 + 4(α −2)(α −1) , (2) the U C V -radius is RU C V = RC (1/2) = 2 √ 3−3 3 ≃0.154701. Conjecture 2.2. The sharp S L -radius and sharp M (β)-radius for the class F3 are given
Theorem 2.5.
Theorem 2.5. For the class F5, the following radius results hold: (a) the S ∗(α)-radius is RS ∗(α) = 1 −α 1 + √ 2 + α2 −2α, (b) the…
Theorem 2.5. For the class F5, the following radius results hold: (a) the S ∗(α)-radius is RS ∗(α) = 1 −α 1 + √ 2 + α2 −2α , (b) the SP-radius is RSP = RS ∗(1/2) = 1 √ 5+ 2 ≃0.236068, (c) the S L -radius is
Lemma 1.2
Repeated OCR label — verify against source
Lemma 1.2 together with (2.9) and (2.10) gives (2.11)
Lemma 1.2 together with (2.9) and (2.10) gives (2.11)
Theorem 3.1.
Theorem 3.1. For the class F6, the following sharp radius results hold: (a) the C (α)-radius is RC (α) = 2(1 −α) 5 + p 25 + 4α(α −1), (b)…
Theorem 3.1. For the class F6, the following sharp radius results hold: (a) the C (α)-radius is RC (α) = 2(1 −α) 5 + p 25 + 4α(α −1) , (b) the U C V -radius is RU C V = RC (1/2) = 5 −2 √ 6 ≃0.101021.
Theorem 3.2.
Theorem 3.2. For the class F7, the following sharp radius results hold: (1) the C (α)-radius is RC (α) = 2(1 −α) 5 + p 25 + 4α(α −1), (2)…
Theorem 3.2. For the class F7, the following sharp radius results hold: (1) the C (α)-radius is RC (α) = 2(1 −α) 5 + p 25 + 4α(α −1) , (2) the U C V -radius is RU C V = RC (1/2) = 5 −2 √ 6 ≃0.101021.
Theorem 3.3.
Theorem 3.3. For the class F8, the following radius results hold: (a) the C (α)-radius is RC (α) = 2(1 −α) 3 + p 9 + 4α(α −1). (b) the U C…
Theorem 3.3. For the class F8, the following radius results hold: (a) the C (α)-radius is RC (α) = 2(1 −α) 3 + p 9 + 4α(α −1) . (b) the U C V -radius is RU C V = RC (1/2) = 3 −2 √ 2 ≃0.171573. The results are sharp.
Theorem 3.1
Repeated OCR label — verify against source
Theorem 3.1, evidently (3.6) 1 + zf ′′(z) f ′(z) −1 + r2 1 −r2 ≤3r + r2 1 −r2, which yields Re 1 + zf ′′(z) f ′(z) ≥1 −3r 1 −r2 ≥α,
Theorem 3.1, evidently (3.6) 1 + zf ′′(z) f ′(z) −1 + r2 1 −r2 ≤3r + r2 1 −r2 , which yields Re 1 + zf ′′(z) f ′(z) ≥1 −3r 1 −r2 ≥α,
Function classes studied:
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