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 $y^{″}\left(z\right)+a\left(z\right)y^{\prime }\left(z\right)+b\left(z\right)y\left(z\right)=0$ and $z^2{y}^{″}\left(z\right)+a\left(z\right)y^{\prime }\left(z\right)+b\left(z\right)y\left(z\right)=d\left(z\right)$. Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution $F\left(z\right)$ of the above differential equations. We construct several examples by judicious choice of $a\left(z\right)$, $b\left(z\right)$ and $d\left(z\right)$. The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials.