Abstract
In this paper, the estimate for the third Hankel determinant H3,1(f)
of Taylor coefficients of function f(z) = z +
∞
X
n=2
anzn, belonging to certain classes
of analytic functions in the open unit disk D, are investigated.
Mathematics Subject Classification (2010): 30C45, 30C50.
Results & Lemmas (12)
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. ([16]) If p ∈P be of the form p(z) = 1 + ∞ X n=1 cnzn, then 2c2 = c2 1 + x(4 −c2 1), and 4c3 = c3 1 + 2c1x(4 −c2 1) −c1x2(4 −c2…
Lemma 1.1. ([16]) If p ∈P be of the form p(z) = 1 + ∞ X n=1 cnzn, then 2c2 = c2 1 + x(4 −c2 1), and 4c3 = c3 1 + 2c1x(4 −c2 1) −c1x2(4 −c2 1) + 2(4 −c2 1)(1 −|x|2)z, for some x, z with |x| ≤1 and |z| ≤1.
Lemma 1.2.
Lemma 1.2. ([22, Theorem 1]) If f ∈N be given by (1.1), then |an| ≤ 1 n(n −1), n ≥2. The result is sharp for the function fn such that f ′…
Lemma 1.2. ([22, Theorem 1]) If f ∈N be given by (1.1), then |an| ≤ 1 n(n −1), n ≥2. The result is sharp for the function fn such that f ′ n(z) = (1 −zn−1)1/(n−1), n ≥2. As it is known that, if f(z) ∈N then zf ′(z) ∈M, therefore from Lemma 1.2, we conclude that
Lemma 1.3.
Lemma 1.3. If f(z) ∈M be given by (1.1), then |an| ≤ 1 n −1, n ≥2. The result is sharp for the function gn(z) = z(1 −zn−1)1/(n−1), n ≥2.
Lemma 1.3. If f(z) ∈M be given by (1.1), then |an| ≤ 1 n −1, n ≥2. The result is sharp for the function gn(z) = z(1 −zn−1)1/(n−1), n ≥2.
Lemma 1.4.
Lemma 1.4. ([22, Corollary 2]) If f ∈N be given by (1.1), then |a3 −a2 2| ≤1/4. Equality is attained for the function f such that f ′(z) =…
Lemma 1.4. ([22, Corollary 2]) If f ∈N be given by (1.1), then |a3 −a2 2| ≤1/4. Equality is attained for the function f such that f ′(z) = (1 −z2eiθ)1/2, θ ∈[0, 2π]. 2. Main results Our first main result is contained in the following theorem:
Theorem 2.1.
Theorem 2.1. Let the function f ∈M be given by (1.1), then |a3 −a2 2| ≤1. (2.1) The result (2.1) is sharp and equality in (2.1) is attained…
Theorem 2.1. Let the function f ∈M be given by (1.1), then |a3 −a2 2| ≤1. (2.1) The result (2.1) is sharp and equality in (2.1) is attained for the function e1(z) = z −z2.
Theorem 2.2.
Theorem 2.2. Let the function f ∈M be given by (1.1), then |a2a4 −a2 3| ≤1 4. (2.4) The result (2.4) is sharp and equality is attained for…
Theorem 2.2. Let the function f ∈M be given by (1.1), then |a2a4 −a2 3| ≤1 4. (2.4) The result (2.4) is sharp and equality is attained for the function e2(z) = z −1 2z3 and e3(z) = z(1 −z2)1/2.
Theorem 2.2.
Theorem 2.2. □
Theorem 2.2. □
Theorem 2.3.
Theorem 2.3. Let the function f ∈M be given by (1.1), then |a2a3 −a4| ≤2 √ 3 9. (2.5)
Theorem 2.3. Let the function f ∈M be given by (1.1), then |a2a3 −a4| ≤2 √ 3 9 . (2.5)
Theorem 2.4.
Theorem 2.4. Let the function f ∈M be given by (1.1), then |H3,1(f)| ≤81 + 16 √ 3 216.
Theorem 2.4. Let the function f ∈M be given by (1.1), then |H3,1(f)| ≤81 + 16 √ 3 216 .
Theorem 2.5.
Theorem 2.5. Let the function f ∈N be given by (1.1), then |a2a3 −a4| ≤1 12. (2.8) The result (2.8) is sharp and equality in (2.8) is…
Theorem 2.5. Let the function f ∈N be given by (1.1), then |a2a3 −a4| ≤1 12. (2.8) The result (2.8) is sharp and equality in (2.8) is attained for the function e4 where e′ 4(z) = (1 −z3)1/3.
Theorem 2.6.
Theorem 2.6. Let the function f ∈N be given by (1.1), then |a2a4 −a2 3| ≤ 9 320. (2.10)
Theorem 2.6. Let the function f ∈N be given by (1.1), then |a2a4 −a2 3| ≤ 9 320. (2.10)
Theorem 2.8.
Theorem 2.8. Let the function f ∈N be given by (1.1), then |H3,1(f)| ≤139 5760.
Theorem 2.8. Let the function f ∈N be given by (1.1), then |H3,1(f)| ≤139 5760.
Function classes studied:
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