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

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 z 2 y ″ ( 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.