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On the growth of solutions of a class of higher order linear differential equations with extremal coefficients. (English) Zbl 1476.34181

Summary: We consider that the linear differential equations \[ f^{(k)} + A_{k - 1}(z) f^{(k - 1)} + \cdots + A_1(z) f' + A_0(z) f = 0\,, \] where \(A_j (j = 0,1, \dots, k - 1)\), are entire functions. Assume that there exists \(l \in \{1,2, \dots, k - 1 \}\), such that \(A_l\) is extremal for Yang’s inequality; then we will give some conditions on other coefficients which can guarantee that every solution \(f(\not\equiv 0)\) of the equation is of infinite order. More specifically, we estimate the lower bound of hyperorder of \(f\) if every solution \(f(\not\equiv 0)\) of the equation is of infinite order.

MSC:

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
34M03 Linear ordinary differential equations and systems in the complex domain
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