What question did this study set out to answer?

This work focuses on estimating the determinant of second-order Hankel functional matrices for holomorphic functions in Banach spaces.

July 8, 2026Open Access

Hankel Determinants for Holomorphic Functions and Biholomorphic Mappings in Banach Spaces

RDRenata DługoszLodz University of Technology PLPiotr LiczberskiLodz University of Technology ETEdyta TrybuckaRzeszów University of Technology

Key Points

This work focuses on estimating the determinant of second-order Hankel functional matrices for holomorphic functions in Banach spaces.
Estimates are derived for the determinant of the Hankel functional matrix H2^1 based on Q-factors of holomorphic functions.
A sharp estimate for the determinant of the scalar matrix H2^2 is provided.
The study uses properties of homogeneous polynomial series expansions.
A precise estimate for the second-order Hankel determinant is established, enhancing understanding of functional behavior in Banach spaces.
The relationships between the determinant values and holomorphic mappings are explored mathematically.

Abstract

Abstract In the paper we give some estimates of the determinant of the second order Hankel functional matrix H₂^1= arraycc Q₅, ₁ & Q₅, ₂\\ Q₅, ₂ & Q₅, ₃ array H 2 1 = Q f, 1 Q f, 2 Q f, 2 Q f, 3, where Q₅, ₁, Q₅, ₂, Q₅, ₃ Q f, 1, Q f, 2, Q f, 3 are the initial terms in a homogeneous polynomial series expansions of functions from two subfamilies of the family Hol (B, C) H o l (B, C) of all functions f: B C, f: B → C, holomorphic on the open unit ball B B of a Banach space X. X. As application of one of such estimates, we prove a sharp estimate of the determinant of the second order Hankel scalar matrix H₂^2= arraycc Tₗ (Q₅, ₂ (x) ) & Tₗ (Q₅, ₃ (x) ) \\ Tₗ (Q₅, ₃ (x) ) & Tₗ (Q₅, ₄ (x) ) array. H 2 2 = T x (Q

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