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.
Highlights
This research is a continuation of recent activity in the field of complex differential equations
Even though a recent unit disc counterpart of Wiman-Valiron theory 5 has been successfully applied to differential equations, the possible orders of solutions of 1.1 in D have been obtained only by assuming that coefficients are α-polynomial regular
Growth of Solutions with Respect to Nevanlinna Characteristic The T -order of growth of f ∈ M D is defined as log T r, f σT f
Summary
This research is a continuation of recent activity in the field of complex differential equations. Gundersen-Steinbart-Wang showed that this finite set consists of rational numbers obtained from simple arithmetic with the degrees of the polynomial coefficients in 1.1 4, Theorem 1 Their proof relies on classical Wiman-Valiron theory in C. Even though a recent unit disc counterpart of Wiman-Valiron theory 5 has been successfully applied to differential equations, the possible orders of solutions of 1.1 in D have been obtained only by assuming that coefficients are α-polynomial regular. These α-polynomial regular functions have similar growth properties than polynomials in the sense that maximal growth is attained in every direction. These findings are applied to estimate the sums of T - and M-orders of functions in the solution bases of 1.1 from below
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