Properties of Structured Matrix and Their Relation to Fractional Mappings

The special types of matrices whose entries are functions and their derivatives, namely Jacobian, Hessian, Wronskian, Hankel and Toeplitz determinants occur persistenly in the solution of certain general mathematical problems. In recent years, the determinants of Hankel and Toeplitz matrices attracted several appli- cations arising in the field of Geometric function theory (GFT in short). GFT is a branch of mathematics that stands at the intersection of complex analysis and geometry, seeking to understand the intricate connections between the analytic properties of functions and their geometric behavior. In the context of GFT, Han- kel and Toeplitz determinants emerge naturally in the study of certain extremal problems involving analytic functions, particularly those conformally mapping re- gions in the complex plane. Some noteworthy applications of these determinants in GFT are as follows: 1. To characterize and locate the poles of meromorphic functions ([75], p.329). 2. To characterize meromorphic function to be of bounded characteristic in the open unit disc of the complex plane, a function which is a ratio of two bounded analytic functions ([16]). 3. To determine the coefficient bounds based on Hankel matrix.([80]). 4. The characterization of convergence of a power series p(z) = 1 + ∞ n=1 cnzn to a function with positive real part in the the open unit disc of the complex plane depends on the Toeplitz determinant associated with coefficients of such power series ([57]).