Analysis of certain determinants for a defined subclass of analytic functions using Poisson distribution series in petal shaped domain

Sangarambadi Padmanabhan  Vijayalakshmi; Mehdha Martin; Saurabh Porwal; Raman  Ezhilarasi; Thirumalai Vinjimur  Sudharsan

doi:10.24193/subbmath.2026.2.04

Authors

Sangarambadi Padmanabhan Vijayalakshmi Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai, India https://orcid.org/0000-0003-3978-6760
Mehdha Martin University of Madras and Department of Mathematics, DRBCCC Hindu College, Pattabiram, Chennai, India https://orcid.org/0009-0006-3035-3546
Saurabh Porwal Department of Mathematics, Ram Sahai Governent Degree College, Bairi-Shivrajpur, Kanpur, Uttar Pradesh, India https://orcid.org/0000-0003-0847-3550
Raman Ezhilarasi Department of Mathematics, S.I.V.E.T College, Gowrivakkam, Chennai, India https://orcid.org/0009-0004-9070-3187
Thirumalai Vinjimur Sudharsan Department of Mathematics, S.I.V.E.T College, Gowrivakkam, Chennai, India https://orcid.org/0000-0002-6882-3367

DOI:

https://doi.org/10.24193/subbmath.2026.2.04

Keywords:

Toeplitz determinants, Poisson distribution series, Starlike functions, Petal shaped domain.

Abstract

The current study focuses on obtaining the sharp coefficient estimates and Fekete-Szeg\"{o} inequality for the class \[\Psi_{\vartheta} (m,\lambda)\] and uses the poisson distribution series to obtain the sharp estimates of coefficient inequalities, Fekete-Szeg\"{o} inequality, second order Toeplitz determinants and upper bounds of third order Toeplitz determinants and second order Hankel determinants for a certain analytic function \[\mathbb{U}(z) = z + \delta_2z^2+\delta_3z^3+\cdots,\mathbb{U}(z) \neq 0,\ \ z\in \Delta \] belonging to the class \[\mathbb{P}\Psi_{\vartheta} (m,\lambda, \Upsilon) = \{ \mathbb{U} \in \mathcal{H}: I^q\mathbb{U} \in \Psi_{\vartheta}(m,\lambda)\}, m \geq 0, \lambda, \vartheta \in \mathbb{N}=\lbrace 1,2,... \rbrace \},\] \[ \Upsilon= \Upsilon_i(k) = \frac{k^{i-1}}{(i-1)!}e^{-k}\], defined on the open unit disc \[(z \in \Delta := \{z : |z|
< 1\}).\] This research could motivate others to delve deeper into the coefficient functional problem related to the Poisson distribution series of analytic functions across different categories of univalent functions.

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