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

Author

Listed:
  • Mohammad Khan Haidary

    (Kabul University)

  • Naweedullah Hashimi

    (Kabul University)

  • Sayed Abdul Bashir Osmani

    (Kabul University)

Abstract

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.

Suggested Citation

  • Mohammad Khan Haidary & Naweedullah Hashimi & Sayed Abdul Bashir Osmani, 2025. "A Study on the Relationships Between the Norms of Complex Polynomials of Degrees 2, 3, and 4 and Their Derivatives Belonging to a2," Journal of Mathematics and Statistics Studies, Al-Kindi Center for Research and Development., vol. 6(3), pages 50-55, July.
    • Handle: RePEc:alk:jmss00:v:6:y:2025:i:3:id:9504
    DOI: 10.32996/jmss.2025.6.3.4
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