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Radius constants for analytic functions with fixed second coefficient.

Mahnaz M Nargesi1, Rosihan M Ali2, V Ravichandran3

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This study determines the sharp radius of Janowski starlikeness for analytic functions with specific coefficient bounds. These findings establish new radius constants for starlike functions, advancing geometric function theory.

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Area of Science:

  • Complex Analysis
  • Geometric Function Theory
  • Analytic Number Theory

Background:

  • Explores analytic functions within the unit disk, a fundamental concept in complex analysis.
  • Focuses on the properties of the second coefficient, |a2|, and its relation to the parameter 'b'.
  • Introduces coefficient constraints |a(n)| ≤ cn + d and |a(n)| ≤ c/n for n ≥ 3.

Purpose of the Study:

  • To derive the sharp radius of Janowski starlikeness for analytic functions satisfying given coefficient inequalities.
  • To establish new radius constants for classes of analytic functions.
  • To connect these results with existing literature in geometric function theory.

Main Methods:

  • Utilizes techniques from geometric function theory to analyze the properties of analytic functions.
  • Applies coefficient bounds to determine the radius of starlikeness.
  • Employs mathematical derivations to obtain sharp bounds and constants.

Main Results:

  • Obtained the sharp radius of Janowski starlikeness for functions with coefficients satisfying |a(n)| ≤ cn + d (c, d ≥ 0).
  • Determined the sharp radius of Janowski starlikeness for functions with coefficients satisfying |a(n)| ≤ c/n (c > 0, n ≥ 3).
  • Derived other related radius constants for these function classes.

Conclusions:

  • The study provides precise bounds for Janowski starlike functions based on coefficient constraints.
  • The established results contribute to a deeper understanding of the geometric properties of analytic functions.
  • This research extends and refines previous findings in the field of geometric function theory.