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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 (11)
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 <sup>p</sup> ∈ P be of the form <sup>p</sup>(z) = 1 + <sup>X</sup><sup>∞</sup> n=1 cnz n, then and for some x, z with…
Lemma 1.1. ([16]) If <sup>p</sup> ∈ P be of the form <sup>p</sup>(z) = 1 + <sup>X</sup><sup>∞</sup> n=1 cnz n , then
$$2c_2 = c_1^2 + x(4 - c_1^2),$$
and
$$4c_3 = c_1^3 + 2c_1x(4 - c_1^2) - c_1x^2(4 - c_1^2) + 2(4 - c_1^2)(1 - |x|^2)z,$$
for some x, z with |x| ≤ 1 and |z| ≤ 1.
Lemma 1.2 · coeff
Lemma 1.2. ([22, Theorem 1]) If f ∈ N be given by (1.1), then The result is sharp for the function f<sup>n</sup> such that f 0 n (z) = (1 −…
Lemma 1.2. ([22, Theorem 1]) If f ∈ N be given by (1.1), then
$$|a_n| \le \frac{1}{n(n-1)}, \quad n \ge 2.$$
The result is sharp for the function f<sup>n</sup> such that f 0 n (z) = (1 − z n−1 ) <sup>1</sup>/(n−1), n ≥ 2.
As it is known that, if f(z) ∈ N then zf<sup>0</sup> (z) ∈ M, therefore from Lemma 1.2, we conclude that
Lemma 1.3 · coeff
Lemma 1.3. If f(z) ∈ M be given by (1.1), then The result is sharp for the function gn(z) = z(1 − z n−1 ) <sup>1</sup>/(n−1), n ≥ 2.
Lemma 1.3. If f(z) ∈ M be given by (1.1), then
$$|a_n| \le \frac{1}{n-1}, \quad n \ge 2.$$
The result is sharp for the function gn(z) = z(1 − z n−1 ) <sup>1</sup>/(n−1), n ≥ 2.
Lemma 1.4 · coeff
Lemma 1.4. ([22, Corollary 2]) If f ∈ N be given by (1.1), then Equality is attained for the function f such that f 0 (z) = (1 − z 2 e iθ)…
Lemma 1.4. ([22, Corollary 2]) If f ∈ N be given by (1.1), then
$$|a_3 - a_2^2| \le 1/4.$$
Equality is attained for the function f such that f 0 (z) = (1 − z 2 e iθ) 1/2 , θ ∈ [0, 2π].
Theorem 2.1 · coeff
Theorem 2.1. Let the function f ∈ M be given by (1.1), then The result (2.1) is sharp and equality in (2.1) is attained for the function
Theorem 2.1. Let the function f ∈ M be given by (1.1), then
$$|a_3 - a_2^2| \le 1. (2.1)$$
The result (2.1) is sharp and equality in (2.1) is attained for the function
$$e_1(z) = z - z^2.$$
Theorem 2.2 · coeff
Theorem 2.2. Let the function be given by (1.1), then The result (2.4) is sharp and equality is attained for the function and.
Theorem 2.2. Let the function $f \in \mathcal{M}$ be given by (1.1), then
$$|a_2 a_4 - a_3^2| \le \frac{1}{4}. (2.4)$$
The result (2.4) is sharp and equality is attained for the function
$$e_2(z) = z - \frac{1}{2}z^3$$
and $e_3(z) = z(1-z^2)^{1/2}$ .
Theorem 2.3 · coeff
Theorem 2.3. Let the function be given by (1.1), then
Theorem 2.3. Let the function $f \in \mathcal{M}$ be given by (1.1), then
$$|a_2 a_3 - a_4| \le \frac{2\sqrt{3}}{9}. (2.5)$$
Theorem 2.4
Theorem 2.4. Let the function be given by (1.1), then
Theorem 2.4. Let the function $f \in \mathcal{M}$ be given by (1.1), then
$$|H_{3,1}(f)| \le \frac{81 + 16\sqrt{3}}{216}.$$
Theorem 2.5 · coeff
Theorem 2.5. Let the function be given by (1.1), then The result (2.8) is sharp and equality in (2.8) is attained for the function where.
Theorem 2.5. Let the function $f \in \mathcal{N}$ be given by (1.1), then
$$|a_2 a_3 - a_4| \le \frac{1}{12}. (2.8)$$
The result (2.8) is sharp and equality in (2.8) is attained for the function $e_4$ where $e'_4(z) = (1-z^3)^{1/3}$ .
Theorem 2.6 · coeff
Theorem 2.6. Let the function be given by (1.1), then
Theorem 2.6. Let the function $f \in \mathcal{N}$ be given by (1.1), then
$$|a_2a_4 - a_3^2| \le \frac{9}{320}. (2.10)$$
Theorem 2.8 · coeff
Theorem 2.8. Let the function f ∈ N be given by (1.1), then Proof. Using Lemma 1.2, Lemma 1.4, Theorem 2.5, Theorem 2.6 and the triangle…
Theorem 2.8. Let the function f ∈ N be given by (1.1), then
$$|H_{3,1}(f)| \le \frac{139}{5760}.$$
Proof. Using Lemma 1.2, Lemma 1.4, Theorem 2.5, Theorem 2.6 and the triangle inequality on H3,1(f), we get
$$|H_{3,1}(f)| \leq |a_3||a_2a_4 - a_3^2| + |a_4||a_2a_3 - a_4| + |a_5||a_3 - a_2^2|$$
$$\leq \frac{1}{6} \frac{9}{320} + \frac{1}{12} \frac{1}{12} + \frac{1}{20} \frac{1}{4} = \frac{139}{5760}.$$
This completes the proof of Theorem 2.8.
Function classes studied:
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