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Order and hyper-order of entire solutions of linear differential equations with entire coefficients. (English) Zbl 1029.34076
The authors study the growth of solutions to the linear differential equation $f^{(k)}+ A_{k-1}(z)\cdot f^{(k-1)}+\cdots+ A_1(z)\cdot f^1+ A_0(z)\cdot f= 0,\tag{1}$ where $$A_0(z),\dots, A_{k-1}(z)$$ are entire functions with $$A_0(z)\not\equiv 0$$, $$k\geq 2$$. It is well-known that all solutions to (1) are entire functions, and if some coefficients of (1) are transcendental, then (1) has at least one solution with order $$\sigma(f)= +\infty$$.
The authors investigate the following problem: What conditions on $$A_0(z),\dots, A_{k-1}(z)$$ will guarantee that every solution $$f\not\equiv 0$$ to (1) has infinite order?
Here, an estimate on the lower bounds of the hyper-order is found if every solution $$f\not\equiv 0$$ is of infinite order.
The results here are generalizations of G. Gundersen and K. H. Kwon’s theorems for $$k= 2$$.

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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