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Ma-Minda φ-classes studied in this paper:
Abstract
Bounds established for second Hankel and Toeplitz determinants with entries determined by logarithmic coefficients of starlike functions mapping the unit disk onto a petal-shaped domain.
Results & Lemmas (6)
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. [ ](#page-11-11): If <sup>p</sup> ∈ P is the form [ ](#page-3-0), then )(1 − |ζ2| )ζ3, 1 <sup>2</sup> + 2(1 − ζ p<sup>3</sup> =…
Lemma 2.1. [\[7,](#page-11-12) [10,](#page-11-13) [11\]](#page-11-11): If <sup>p</sup> ∈ P is the form [\(2.1\)](#page-3-0), then
$$p_1 = 2\zeta_1, \tag{2.2}$$
)(1 − |ζ2|
)ζ3,
$$p_2 = 2\zeta_1^2 + 2(1 - \zeta_1^2)\zeta_2, \tag{2.3}$$
$$-\zeta_1^2 + \zeta_2^2 - 2(1 - \zeta_1^2)\zeta_1 \zeta_2^2 + 2(1 - \zeta_1^2)(1 - |\zeta_2|^2)\zeta_2$$
1
<sup>2</sup> + 2(1 − ζ
p<sup>3</sup> = 2ζ <sup>1</sup> + 4(1 − ζ 1 )ζ1ζ<sup>2</sup> − 2(1 − ζ 1 )ζ1ζ where <sup>ζ</sup><sup>1</sup> <sup>∈</sup> [0, 1] and <sup>ζ</sup>2, ζ<sup>3</sup> <sup>∈</sup> <sup>D</sup> := {<sup>z</sup> <sup>∈</sup> <sup>C</sup> : <sup>|</sup>z| ≤ <sup>1</sup>}.
Lemma 2.2
Lemma 2.2. [ ](#page-11-14): If <sup>A</sup>, <sup>B</sup>, <sup>C</sup> <sup>∈</sup> <sup>R</sup>, let us consider 3 1. If AC ≥ 0, then 2.…
Lemma 2.2. [\[3\]](#page-11-14): If <sup>A</sup>, <sup>B</sup>, <sup>C</sup> <sup>∈</sup> <sup>R</sup>, let us consider
3
$$Y(A, B, C) := \max\{|A + Bz + Cz^2| + 1 - |z|^2, z \in \overline{\mathbb{D}}\}$$
1. If AC ≥ 0, then
$$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$
2. If AC < 0, then
$$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & -4AC(C^{-2} - 1) \le B^2 \wedge |B| < 2(1 - |C|), \\ \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\{4(1 + |C|)^2, -4AC(C^{-2} - 1)\}, \\ \\ R(A,B,C), & Otherwise, \end{cases}$$
where
$$R(A,B,C) = \begin{cases} |A| + |B| - |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & Otherwise. \end{cases}$$
Theorem 3.1
Theorem 3.1. Let, then The above inequality is sharp.
Theorem 3.1. Let $f \in \mathcal{S}_{\rho}^*$ , then
$$\frac{1}{72} \le |H_{2,1}(F_f/2)| \le \frac{1}{16}.\tag{3.4}$$
The above inequality is sharp.
Theorem 3.2
Theorem 3.2. Let, then The above inequality is sharp.
Theorem 3.2. Let $f \in \mathcal{S}_{\rho}^*$ , then
$$\frac{1}{16} \le |H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{9}.\tag{3.12}$$
The above inequality is sharp.
Theorem 4.1
Theorem 4.1. Let, then The above inequality is sharp.
Theorem 4.1. Let $f \in \mathcal{S}_{\rho}^*$ , then
$$\frac{1}{16} \le |T_{2,1}(F_f/2)| \le \frac{1}{2}.$$
The above inequality is sharp.
Theorem 4.2
Theorem 4.2. Let, then The above inequality is sharp.
Theorem 4.2. Let $f \in \mathcal{S}_{\rho}^*$ , then
$$\frac{1}{16} \le |T_{2,1}(F_{f^{-1}}/2)| \le \frac{5}{4}.$$
The above inequality is sharp.
Function classes studied:
Coefficient bounds & claims (9)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_{2,1}(F_f/2) ≤ 1/16 for class S*_rho (sharp) [Theorem 3.1]
coefficient_bound
H_{2,1}(F_f/2) (lower) ≤ 1/72 for class S*_rho (sharp) [Theorem 3.1]
coefficient_bound
H_{2,1}(F_{f^{-1}}/2) ≤ 1/9 for class S*_rho (sharp) [Theorem 3.2]
coefficient_bound
H_{2,1}(F_{f^{-1}}/2) (lower) ≤ 1/16 for class S*_rho (sharp) [Theorem 3.2]
coefficient_bound
T_{2,1}(F_f/2) ≤ 1/2 for class S*_rho (sharp) [Theorem 4.1]
coefficient_bound
T_{2,1}(F_f/2) (lower) ≤ 1/16 for class S*_rho (sharp) [Theorem 4.1]
coefficient_bound
T_{2,1}(F_{f^{-1}}/2) ≤ 5/4 for class S*_rho (sharp) [Theorem 4.2]
coefficient_bound
T_{2,1}(F_{f^{-1}}/2) (lower) ≤ 1/16 for class S*_rho (sharp) [Theorem 4.2]
function_family
Class S*_rho: f in A with zf'/f subordinate to 1+arcsinh(z), mapping D onto petal-shaped domain
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