Estimates bounds for Carathéodory functions in the complex domain are applied to demonstrate sharp limits for the inverse of analytic functions. Determining these values is considered a more difficult task compared to finding the values of analytic functions themselves. The challenge lies in finding the sharp estimate for the functionals. While some recent studies have made progress in calculating the sharp boundary values of Hankel determinants associated with inverse functions, the Toeplitz determinant is yet to be addressed. Our research aims to estimate the determinants of the Toeplitz matrix, which is also linked to inverse functions. We also focus on computing these determinants for familiar analytical functions (pre-starlike, starlike, convex, symmetric-starlike) while investigating coefficient values. The study also provides an improvement to the estimation of the determinants of the pre-starlike class presented by Li and Gou.

When extreme values or outliers occur in asymmetric datasets, conventional mean estimation methods suffer from low accuracy and reliability. This study introduces a novel class of robust Särndal-type mean estimators utilizing re-descending M-estimator coefficients. These estimators effectively combine the benefits of robust regression techniques and the integration of extreme values to improve mean estimation accuracy under simple random sampling. The proposed methodology leverages distinct re-descending coefficients from prior studies. Performance evaluation is conducted using three real-world datasets and three synthetically generated datasets containing outliers, with results indicating superior performance of the proposed estimators in terms of mean squared error (MSE) and percentage relative efficiency (PRE). Hence, the robustness, adaptability, and practical importance of these estimators are illustrated by these findings for survey sampling and more generally for data-intensive contexts.

In the open unit disk, we introduce a new subclass of analytic and bi-univalent functions described by the Neutrosophic Poisson distribution series. The estimates for the Fekete--Szeg\"{o} type and the Taylor coefficients. Then, using the Horadam polynomials, the inequalities corresponding to the functions that belong to this recently discovered subclass are investigated. Along with some corollaries, we also go over the implications of the findings presented in this essay. Numerous branches of mathematics, science, and technology are anticipated to heavily rely on the neutrophilic Poisson distribution.

Applying the state-of-the-art Bell polynomials to the open unit disk, a differentialoperator θmξ, P is produced. In this paper, we shall provide a family of analytic functions related to the differential operator indicated above. The upper bound for the nonlinear functional |a2a4−a23|, otherwise known as the Hankel determinant, is our primary finding. Aside from using the Bell polynomial, the differential operator is gained using the Hadamard product. Coefficient equating and other fundamentals of classical calculus will be used in the primary finding of the upper bound.

This work presents and investigates a new class of meromorphically uniformly convex functions with positive coefficients that a differential operator defines. It also derives properties such as coefficient bounds, distortion properties, $\delta$-neighborhoods, convex linear combination, and convolution properties.

In this paper, we introduce and investigate a novel subfamily of Bazilevič functions, defined using the $ m $-leaf polynomial and the principle of subordination. For this subfamily, we establish several properties, including subordination results, inclusion relationships, and sharp coefficient bounds. Furthermore, we derive a general Fekete–Szegö inequality, providing a solution to this classical problem for functions within the new subfamily.