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

Abstract

Suppose that a power seriesA(z) =∑∞n=0anznhas the radius of convergenceR[A]∈[1,+∞].For a heterogeneous differential equationz2w′′+ (β0z2+β1z)w′+ (γ0z2+γ1z+γ2)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: ifγ26=0 andγ2=0. Wealso consider cases when parameters of the equation are realnumbers. Also we prove that for asolutionfof this equation the radius of convergenceR[f]equals toR[A]and the recurrent formulasfor the coefficients of the power series off(z)are found. For entire solutions it is proved that theorder of a solutionfis not less then the order ofA(̺[f]≥̺[A]) and the estimate is sharp. The sameinequality holds for generalized orders (̺αβ[f]≥̺αβ[A]). For entire solutions of this equation thebelonging to convergence classes is studied. Finally, we consider a linear differential equation of theendless order∞∑n=0ann!w(n)=Φ(z), and study a possible growth of its solutions.