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 <span id="page-2-2"></span><span id="page-2-1"></span> (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
<span id="page-2-2"></span><span id="page-2-1"></span>
$$2p_2 = p_1^2 + \gamma(4 - p_1^2), \tag{1.5}$$
$$4p_3 = p_1^3 + 2p_1(4 - p_1^2)\gamma - p_1(4 - p_1^2)\gamma^2 + 2(4 - p_1^2)(1 - |\gamma|^2)\eta$$
(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 <span id="page-2-6"></span><span id="page-2-5"></span><span id="page-2-3"></span><span id="page-2-0"></span>…
Theorem 2.1. If $f \in \mathcal{SL}^*$ . Then we have
<span id="page-2-6"></span><span id="page-2-5"></span><span id="page-2-3"></span><span id="page-2-0"></span>
$$|H_3(1)| \le 1/36. \tag{2.1}$$
The bound is sharp.
Theorem 2.2
Theorem 2.2. Let. Then we have <span id="page-6-2"></span><span id="page-6-1"></span><span id="page-6-0"></span> The result is sharp.
Theorem 2.2. Let $f \in \mathcal{SL}^*$ . Then we have
<span id="page-6-2"></span><span id="page-6-1"></span><span id="page-6-0"></span>
$$|H_2(3)| \le \frac{1}{36}.\tag{2.17}$$
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 <span id="page-10-2"></span> (3.1) where and
Theorem 3.1. A function $f \in \mathcal{S}$ is in the class $\mathcal{SL}^*$ if and only if
<span id="page-10-2"></span>
$$\frac{1}{z}\left(f * H_t(z)\right) \neq 0, \quad (z \in \Delta)$$
(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 <span id="page-11-0"></span> belongs to the class if.
Theorem 3.2. The function
<span id="page-11-0"></span>
$$\Theta(z) = \frac{z}{1 - \alpha z}, \quad (z \in \Delta)$$
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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