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Related Experiment Video

Updated: Aug 13, 2025

Biaxial Mechanical Characterizations of Atrioventricular Heart Valves
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The second and third Hankel determinants for starlike and convex functions associated with Three-Leaf function.

Amina Riaz1, Mohsan Raza2, Muhammad Ahsan Binyamin2

  • 1Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.

Heliyon
|January 23, 2023
PubMed
Summary

This study provides precise upper limits for Hankel determinants of coefficients in starlike and convex functions associated with a three-leaf-like domain, enhancing geometric function theory understanding.

Keywords:
Convex functionHankel determinantsStarlike functionThree-leaf functionUnivalent function

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Area of Science:

  • Geometric Function Theory
  • Complex Analysis
  • Harmonic Analysis

Background:

  • The study of analytic functions and their geometric properties is a fundamental area within complex analysis.
  • Hankel determinants are significant in characterizing the behavior of coefficients for various function classes.
  • Previous research has explored bounds for Hankel determinants in different function families, but specific domains require dedicated analysis.

Purpose of the Study:

  • To determine sharp bounds for the second Hankel determinant ( ) for functions belonging to the class of starlike functions related to a three-leaf-like domain.
  • To establish sharp bounds for the third Hankel determinant ( ) and the fourth Hankel determinant ( ) for functions within the class of convex functions associated with the same three-leaf-like domain.

Main Methods:

  • Utilizing established inequalities and techniques from geometric function theory.
  • Applying coefficient bounds specific to the defined starlike and convex function classes.
  • Careful computation and analysis of the Hankel determinant expressions for the relevant function classes.

Main Results:

  • Sharp upper bounds for the second Hankel determinant ( ) were derived for starlike functions related to the three-leaf-like domain.
  • Sharp upper bounds for the third Hankel determinant ( ) and the fourth Hankel determinant ( ) were obtained for convex functions associated with the three-leaf-like domain.

Conclusions:

  • The derived bounds provide precise quantitative measures for the coefficients of functions in these specific geometric classes.
  • This research contributes to a deeper understanding of the coefficient behavior and geometric properties of functions related to the three-leaf-like domain.