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Ma-Minda φ-classes studied in this paper:
Abstract
The aim of the present paper is to obtain the sharp bounds of the Hankel determinants H_2(3) and H_3(1) for the well known class SL^* of starlike functions associated with the right lemniscate of Bernoulli. Further for n=3, we find the sharp bound of the Zalcman functional for the class SL^*. In addition, a couple of interesting results of SL^* is appended at the end.
Results & Lemmas (7)
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. Let and of the form. Then (1.6) and (1.7) for some, and such that, and.
Lemma 1.1. Let $p \in \mathcal{P}$ and of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then
$$2p_2 = p_1^2 + \gamma(4 - p_1^2), \tag{1.5}$$
$$4p_3 = p_1^3 + 2p_1(4 - p_1^2)\gamma - p_1(4 - p_1^2)\gamma^2 + 2(4 - p_1^2)(1 - |\gamma|^2)\eta$$
(1.6)
and
$$8p_4 = p_1^4 + (4 - p_1^2)\gamma(p_1^2(\gamma^2 - 3\gamma + 3) + 4\gamma) - 4(4 - p_1^2)(1 - |\gamma|^2)(p_1(\gamma - 1)\eta + \overline{\gamma}\eta^2 - (1 - |\eta|^2)\rho),$$
(1.7)
for some $\rho$ , $\gamma$ and $\eta$ such that $|\rho| \leq 1$ , $|\gamma| \leq 1$ and $|\eta| \leq 1$ .
Lemma 1.2
Lemma 1.2. [20] Let a, b, c and d satisfy the inequalities 0 < c < 1, 0 < d < 1 and If, then
Lemma 1.2. [20] Let a, b, c and d satisfy the inequalities 0 < c < 1, 0 < d < 1 and
$$8d(1-d)((cb-2a)^2 + (c(d+c)-b)^2) + c(1-c)(b-2dc)^2 \le 4c^2(1-c)^2d(1-d).$$
If $p \in \mathcal{P}$ , then
$$|ap_1^4 + dp_2^2 + 2cp_1p_3 - (3/2)bp_1^2p_2 - p_4| \le 2.$$
Theorem 2.1
Theorem 2.1. If. Then we have The bound is sharp.
Theorem 2.1. If $f \in \mathcal{SL}^*$ . Then we have
$$|H_3(1)| \le 1/36. \tag{2.1}$$
The bound is sharp.
Theorem 2.2
Theorem 2.2. Let. Then we have The result is sharp.
Theorem 2.2. Let $f \in \mathcal{SL}^*$ . Then we have
$$|H_2(3)| \le \frac{1}{36}.\tag{2.17}$$
The result is sharp.
Theorem 2.3 · coeff
Theorem 2.3. Let. Then The estimate is sharp.
Theorem 2.3. Let $f \in \mathcal{SL}^*$ . Then
$$|a_3^2 - a_5| \le \frac{1}{8}.$$
The estimate is sharp.
Theorem 3.1
Theorem 3.1. A function is in the class if and only if (3.1) where and
Theorem 3.1. A function $f \in \mathcal{S}$ is in the class $\mathcal{SL}^*$ if and only if
$$\frac{1}{z}\left(f * H_t(z)\right) \neq 0, \quad (z \in \Delta)$$
(3.1)
where
$$H_t(z) = \frac{z}{(1-z)(1-S(t))} \left(\frac{1}{1-z} - S(t)\right)$$
and
$$S(t) = \sqrt{t} + i\left(\pm\sqrt{\sqrt{1+4t} - (t+1)}\right), \quad (0 < t < 2).$$
Theorem 3.2
Theorem 3.2. The function belongs to the class if.
Theorem 3.2. The function
$$\Theta(z) = \frac{z}{1 - \alpha z}, \quad (z \in \Delta)$$
belongs to the class $\mathcal{SL}^*$ if $|\alpha| \leq 1/4$ .
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_3(1) ≤ 1/36 for class SL* (sharp) [Theorem 2.1]
coefficient_bound
H_2(3) ≤ 1/36 for class SL* (sharp) [Theorem 2.2]
coefficient_bound
|a3^2 - a5| ≤ 1/8 for class SL* (sharp) [Theorem 2.3]
function_family
Class SL*: f in A with zf'(z)/f(z) subordinate to sqrt(1+z)
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