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Ma-Minda φ-classes studied in this paper:
Abstract
In this paper we give improved, probably not sharp, upper bounds of the Hankel determinant of third order for various classes of univalent functions and conjecture the sharp one.
Results & Lemmas (3)
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
Lemma 1. Let be a Schwarz function. Then, for any real numbers and the following sharp estimate holds where is given in complete form in…
Lemma 1. Let $\omega(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function. Then, for any real numbers $\mu$ and $\nu$ the following sharp estimate holds
$$\Psi(\omega) = |c_3 + \mu c_1 c_2 + \nu c_1^3| \le \Phi(\mu, \nu),$$
where $\Phi(\mu,\nu)$ is given in complete form in [16, Lemma 2], and here we will use only
$$\Phi(\mu,\nu) = \begin{cases} 1, & (\mu,\nu) \in D_1 \cup D_2 \cup \{(2,1)\} \\ |\nu|, & (\mu,\nu) \in \cup_{k=3}^7 D_k \end{cases},$$
where
$$\begin{split} D_1 &= \left\{ (\mu, \nu) : |\mu| \leq \frac{1}{2}, \, -1 \leq \nu \leq 1 \right\}, \\ D_2 &= \left\{ (\mu, \nu) : \frac{1}{2} \leq |\mu| \leq 2, \, \frac{4}{27} (|\mu| + 1)^3 - (|\mu| + 1) \leq \nu \leq 1 \right\}, \\ D_3 &= \left\{ (\mu, \nu) : |\mu| \leq \frac{1}{2}, \, \nu \leq -1 \right\}, \\ D_4 &= \left\{ (\mu, \nu) : |\mu| \geq \frac{1}{2}, \, \nu \leq -\frac{2}{3} (|\mu| + 1) \right\}, \\ D_5 &= \left\{ (\mu, \nu) : |\mu| \leq 2, \, \nu \geq 1 \right\}, \\ D_6 &= \left\{ (\mu, \nu) : 2 \leq |\mu| \leq 4, \, \nu \geq \frac{1}{12} (\mu^2 + 8) \right\}, \\ D_7 &= \left\{ (\mu, \nu) : |\mu| \geq 4, \, \nu \geq \frac{2}{3} (|\mu| - 1) \right\}. \end{split}$$
We will also use the following, almost forgotten result of Carleson ([3]).
Lemma 2
Lemma 2. Let be a Schwarz function. Then, and.
Lemma 2. Let $\omega(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function. Then
$$|c_2| \le 1 - |c_1|^2$$
, $|c_3| \le 1 - |c_1|^2 - \frac{|c_2|^2}{1 + |c_1|}$ and $|c_4| \le 1 - |c_1|^2 - |c_2|^2$ .
Theorem 1 · coeff
Theorem 1. Let is of the form. - (i) If, then - (ii) If, then - (iii) If, then - (iv) If, then
Theorem 1. Let $f \in A$ is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ .
- (i) If $f \in S^*$ , then $|H_3(1)| \le 0.777987...$
- (ii) If $f \in \mathcal{S}_s^*$ , then $|H_3(1)| \le \frac{1}{4} + \frac{1}{3\sqrt{3}} = 0.44245...$
- (iii) If $f \in \mathcal{S}_e^*$ , then $|H_3(1)| \le \frac{17}{72} = 0.23611...$
- (iv) If $f \in \mathcal{S}_q^*$ , then $|H_3(1)| \le \frac{17}{72} = 0.23611...$
Coefficient bounds & claims (8)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|H_3(1)| ≤ 0.777987 for class S* [Theorem 1(i)]
coefficient_bound
|H_3(1)| ≤ 1/4 + 1/(3*3**(1/2)) for class S*_s [Theorem 1(ii)]
coefficient_bound
|H_3(1)| ≤ 17/72 for class S*_e [Theorem 1(iii)]
coefficient_bound
|H_3(1)| ≤ 17/72 for class S*_q [Theorem 1(iv)]
function_family
Class S*: f in A : z*f'(z)/f(z) subordinate to (1+z)/(1-z)
function_family
Class S*_s: f in A : 2*z*f'(z)/(f(z)-f(-z)) subordinate to (1+z)/(1-z); starlike with respect to symmetric points
function_family
Class S*_e: f in A : z*f'(z)/f(z) subordinate to e^z
function_family
Class S*_q: f in A : z*f'(z)/f(z) subordinate to z+sqrt(1+z^2)
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