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Results & Lemmas (4)
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Lemma 2.1
Lemma 2.1. If is of the form (2.1), then and for some For, there is a unique function with as in (2.2), namely For and, there is a unique…
Lemma 2.1. If $p \in \mathcal{P}$ is of the form (2.1), then
$$(2.2) c_1 = 2p_1$$
$$(2.3) c_2 = 2p_1^2 + 2(1 - p_1^2)p_2$$
and
$$(2.4) c_3 = 2p_1^3 + 4\left(1 - p_1^2\right)p_1p_2 - 2\left(1 - p_1^2\right)p_1p_2^2 + 2\left(1 - p_1^2\right)\left(1 - |p_2|^2\right)p_3$$
for some $p_1, p_2, p_3 \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \le 1\}.$
For $p_1 \in \mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$ , there is a unique function $p \in \mathcal{P}$ with $c_1$ as in (2.2), namely
$$p(z) = \frac{1 + p_1 z}{1 - p_1 z}, \quad z \in \mathbb{D}.$$
For $p_1 \in \mathbb{D}$ and $p_2 \in \mathbb{T}$ , there is a unique function $p \in \mathcal{P}$ with $c_1$ and $c_2$ as in (2.2) and (2.3), namely
(2.5)
$$p(z) = \frac{1 + (p_1 + \overline{p_1}p_2)z + p_2z^2}{1 - (p_1 - \overline{p_1}p_2)z - p_2z^2}.$$
For $p_1, p_2 \in \mathbb{D}$ and $p_3 \in \mathbb{T}$ , there is unique function $p \in \mathcal{P}$ with $c_1, c_2$ , and $c_3$ as in (2.2)-(2.3), namely
$$p(z) = \frac{1 + (\overline{p_2}p_3 + \overline{p_1}p_2 + p_1)z + (\overline{p_1}p_3 + p_1\overline{p_2}p_3 + p_2)z^2 + p_3z^3}{1 + (\overline{p_2}p_3 + \overline{p_1}p_2 - p_1)z + (\overline{p_1}p_3 - p_1\overline{p_2}p_3 - p_2)z^2 - p_3z^3}, \quad z \in \mathbb{D}.$$
Next we recall the following well-known result due to Choi et al. [9]. Lemma 2.2 plays an important role in the proof of our main results.
Lemma 2.2
Lemma 2.2. Let A, B, C be real numbers and (i) If, then (ii) If AC < 0, then where
Lemma 2.2. Let A, B, C be real numbers and
$$Y(A, B, C) := \max_{z \in \overline{\mathbb{D}}} (|A + Bz + Cz^2| + 1 - |z|^2).$$
(i) If $AC \geq 0$ , then
$$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$
(ii) If AC < 0, then
$$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & -4AC\left(C^{-2} - 1\right) \le B^2 \land |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\left\{4(1+|C|)^2, -4AC\left(C^{-2} - 1\right)\right\}, \\ R(A,B,C), & otherwise, \end{cases}$$
where
$$R(A, B, C) = \begin{cases} |A| + |B| + |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|A| + |C|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$
Theorem 3.1
Theorem 3.1. Let given by (1.1) then The inequality is sharp.
Theorem 3.1. Let $f \in \mathcal{C}$ given by (1.1) then
$$(3.1) |H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{33}.$$
The inequality is sharp.
Theorem 3.2
Theorem 3.2. Let given by (1.1) then The inequality is sharp.
Theorem 3.2. Let $f \in \mathcal{S}$ given by (1.1) then
$$(3.10) |H_{2,1}(F_{f^{-1}}/2)| \le \frac{13}{12}.$$
The inequality is sharp.
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_{2,1}(F_{f^{-1}}/2) ≤ 1/33 for class C (sharp) [Theorem 3.1]
coefficient_bound
H_{2,1}(F_{f^{-1}}/2) ≤ 13/12 for class S* (sharp) [Theorem 3.2]
function_family
Class S*: f in A with Re(zf'/f) > 0
function_family
Class C: f in A with Re(1 + zf''/f') > 0
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