Third-order Hankel determinant sharp estimates for the inverse of complex valued holomorphic functions.

Area of Science:

• Complex Analysis
• Geometric Function Theory

Background:

• Estimating coefficients of inverse complex-valued functions is challenging.
• Research on inverse functions, particularly concerning sharp estimates of the third-order Hankel determinant, is limited.

Purpose of the Study:

• To estimate initial coefficients for inverse functions that are bounded turning and subordinated with the cosine function.
• To investigate inverse coefficient problems and second and third-order Hankel determinants for these functions.

Main Methods:

• Utilizing Carathéodory functions for coefficient estimation.
• Applying techniques from geometric function theory to analyze inverse functions.

Main Results:

• Derived sharp estimates for initial coefficients of the inverse functions.
• Provided sharp estimates for the second and third-order Hankel determinants.
• Identified the extremal functions and their corresponding inverses.

Conclusions:

• The study successfully provides sharp coefficient estimates for a specific class of inverse functions.
• The findings contribute to the understanding of inverse problems in geometric function theory, particularly for bounded turning and cosine-subordinated functions.