Bohr’s Phenomenon for the Solution of Second-Order Differential Equations

Abstract

The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely y″(z)+a(z)y′(z)+b(z)y(z)=0 and z2y″(z)+a(z)y′(z)+b(z)y(z)=d(z). Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F(z) of the above differential equations. We construct several examples by judicious choice of a(z), b(z) and d(z). The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials.

Keywords

• Bohr’s phenomenon
• second-order differential equation
• subordination
• Bessel functions
• Airy functions
• error function