Properties of solutions of a heterogeneous differential equation of the second order

Abstract

Suppose that a power series $A(z)=\sum_{n=0}^{\infty}a_n z^{n}$ has the radius of convergence $R[A]\in [1,+\infty]$. For a heterogeneous differential equation $$z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=A(z)$$ with complex parameters geometrical properties of its solutions (convexity, starlikeness and close-to-convexity) in the unit disk are investigated. Two cases are considered: if $\gamma_2\neq0$ and $\gamma_2=0$. We also consider cases when parameters of the equation are real numbers. Also we prove that for a solution $f$ of this equation the radius of convergence $R[f]$ equals to $R[A]$ and the recurrent formulas for the coefficients of the power series of $f(z)$ are found. For entire solutions it is proved that the order of a solution $f$ is not less then the order of $A$ ($\varrho[f]\ge\varrho[A]$) and the estimate is sharp. The same inequality holds for generalized orders ($\varrho_{\alpha\beta}[f]\ge \varrho_{\alpha\beta}[A]$). For entire solutions of this equation the belonging to convergence classes is studied. Finally, we consider a linear differential equation of the endless order $\sum\limits_{n=0}^{\infty}\dfrac{a_n}{n!}w^{(n)}=\Phi(z),$ and study a possible growth of its solutions.