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<p>We establish new inequalities for rational analytic functions with prescribed poles and constraints on the location of zeros. Using recent developments in the boundary behavior of holomorphic function−particularly the boundary Schwarz Lemma of Azeroğlu and Örnek, sharper bounds are obtained, that extend known results for rational functions and yield better versions of classical polynomial inequalities for derivatives and polar derivatives. The technique used here, involves utilization of information on certain coefficients and a zero off the boundary of the zero region of the numerator polynomial leading to improved results. Furthermore, our findings have significant impact on the growing interest on the development of classical inequalities in geometric function theory in recent years, and their wide range of applications in various fields of science and engineering such as those in physical systems and discrete dynamical systems.</p>

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Sharp Inequalities for Rational Functions with Fixed Poles and Constrained Zeros

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • Raju Laishangbam,
  • Robinson Soraisam,
  • Barchand Chanam

摘要

We establish new inequalities for rational analytic functions with prescribed poles and constraints on the location of zeros. Using recent developments in the boundary behavior of holomorphic function−particularly the boundary Schwarz Lemma of Azeroğlu and Örnek, sharper bounds are obtained, that extend known results for rational functions and yield better versions of classical polynomial inequalities for derivatives and polar derivatives. The technique used here, involves utilization of information on certain coefficients and a zero off the boundary of the zero region of the numerator polynomial leading to improved results. Furthermore, our findings have significant impact on the growing interest on the development of classical inequalities in geometric function theory in recent years, and their wide range of applications in various fields of science and engineering such as those in physical systems and discrete dynamical systems.