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Third Hankel Determinant For A Class Of Starlike Functions Associated With Exponential Function

Kunal Joshi, S. Sivaprasad Kumar · 2022 📄 arXiv →

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Ma-Minda φ-classes studied in this paper:
Exponential starlike — φ(z) = e^z ✓
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) (2.2) and (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) $$4c_3 = c_1^2 + 2c_1(4 - c_1^2)\delta - c_1(4 - c_1^2)\delta^2 + 2(4 - c_1^2)(1 - |\delta|^2)\alpha$$ (2.2) and $$8c_4 = c_1^4 + (4 - c_1^2)\delta(c_1^2(\delta^2 - 3\delta + 3) + 4\delta) - 4(4 - c_1^2)(1 - |\delta|^2)(c_1(\delta - 1)\alpha + \overline{\delta}\alpha^2 - (1 - |\alpha|^2)\beta),$$ (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:
q-starlike bi-univalent close-to-convex strongly starlike bounded turning typically real

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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