Possible Intervals forT- andM-Orders of Solutions of Linear Differential Equations in the Unit Disc

Abstract

In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions off(k)+ak-1(z)f(k-1)+⋯+a1(z)f′+a0(z)f=0with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possibleT- andM-orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals forT- andM-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums ofT- andM-orders of functions in the solution bases.