Ma-Minda φ-classes studied in this paper:
Abstract
Recently, the subclass of starlike functions associated with exponential function $e^z$, given by ${S}^*_e = \{f(z)\in {S}:{zf'(z)}/{f(z)} \prec e^z, (z\in \mathbb{D}) \}$ was introduced and studied by Mendiratta $et$ $al.$ ~\cite{mendiratta2015subclass}. In this article, we obtain sharp bound for third Hankel determinant for the class $S_e^\ast$, by improving the already known results, in this direction. In our methodology, we use the newly obtained expression for the fourth coefficient of Cara
Results & Lemmas (2)
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. Let and of the form. Then (2.1) <span id="page-3-1"></span> (2.2) and <span id="page-3-2"></span> (2.3) where and.
Lemma 2.1. Let
$$p \in \mathcal{P}$$
and of the form $1 + \sum_{n=1}^{\infty} c_n z^n$ . Then $2c_2 = c_1^2 + \delta(4 - c_1^2),$ (2.1)
<span id="page-3-1"></span>
$$4c_3 = c_1^2 + 2c_1(4 - c_1^2)\delta - c_1(4 - c_1^2)\delta^2 + 2(4 - c_1^2)(1 - |\delta|^2)\alpha$$
(2.2)
and
<span id="page-3-2"></span>
$$8c_4 = c_1^4 + (4 - c_1^2)\delta(c_1^2(\delta^2 - 3\delta + 3) + 4\delta) - 4(4 - c_1^2)(1 - |\delta|^2)(c_1(\delta - 1)\alpha + \overline{\delta}\alpha^2 - (1 - |\alpha|^2)\beta),$$
(2.3)
where $|\alpha| \leq 1, |\beta| \leq 1$ and $|\delta| \leq 1$ .
Theorem 2.1
Theorem 2.1. If. Then we have The bound is sharp.
Theorem 2.1. If $f \in S_e^*$ . Then we have
$$|H_3(1)| \le \frac{1}{9}. (2.4)$$
The bound is sharp.
Function classes studied:
Coefficient bounds & claims (2)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_3(1) ≤ 1/9 for class S*_e (sharp) [Theorem 2.1]
function_family
Class S*_e: f in S such that zf'(z)/f(z) is subordinate to e^z for z in D
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