On transcendental directions of entire solutions of linear differential equations. (English) Zbl 1485.34211
Summary: This paper is devoted to studying the transcendental directions of entire solutions of \(f^{(n)}+A_{n-1}f^{(n-1)}+\dots+A_0f = 0 \), where \(n(\geq 2)\) is an integer and \(A_i(z)(i = 0, 1,\dots, n-1)\) are entire functions of finite lower order. With some additional conditions, the set of common transcendental directions of non-trivial solutions, their derivatives and their primitives must have a definite range of measure. Moreover, we obtain the lower bound of the measure of the set defined by the common transcendental directions of Jackson difference operator of non-trivial solutions.
MSC:
| 34M10 | Oscillation, growth of solutions to ordinary differential equations in the complex domain |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
Keywords:
transcendental direction; entire function; Jackson difference operator; linear differential equationReferences:
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