A Study on the Relationships Between the Norms of Complex Polynomials of Degrees 2, 3, and 4 and Their Derivatives Belonging to a2

In this paper, relations between the norms of complex polynomials of degrees 2, 3, and 4 and their derivatives are studied. Using bound-preserving convolution operators and interpolation formulas, we derive inequalities governing these polynomial norms. Clement Frappier previously found a relation for , but for , a unique relation does not exist. We establish new bounds for these cases by employing determinant analysis and principal minor calculations. The theoretical framework is constructed using Hermitian matrices, semi-bilinear functions, and norm-preserving operators, leading to a refined approach for identifying the smallest positive roots of characteristic equations. The results provide a deeper understanding of polynomial inequalities and contribute to the broader study of functional analysis and complex function theory.