On the hyper-order of solutions to some second-order linear differential equations.

*(English)*Zbl 1014.34078Consider the second-order linear differential equation
\[
f'' + A(z)f' + B(z)f = 0 \tag{1}
\]
with entire functions \(A\) and \(B\). It is well known that all solutions are entire functions, and if \(A\) or \(B\) are transcendental, then there exists a solution \(f\) with \(\sigma(f)=\infty\), where
\[
\sigma(f) = \varlimsup_{r\to\infty}{\frac{\log{T(r,f)}}{\log{r}}}
\]
denotes the order of growth of \(f\). Several authors investigated sufficient conditions on \(A\) and \(B\) such that all nontrivial solutions to (1) have infinite order of growth. For such solutions, it is an important aspect to give more precise estimates for their rates of growth. For that purpose, the concept of hyper-order was introduced. This is defined by
\[
\sigma_2(f) = \varlimsup_{r\to\infty}{\frac{\log{\log{T(r,f)}}}{\log{r}}} .
\]

The author considers the differential equation \[ f'' + h_1(z)e^{P(z)}f' + h_0(z)e^{Q(z)}f = 0 , \tag{2} \] where \(P\), \(Q\) are nonconstant polynomials and \(h_0\), \(h_1\) are entire functions such that \(h_0 \not\equiv 0\), \(\sigma(h_0)<\deg Q\) and \(\sigma(h_1)<\text{deg}{P}\). He generalizes results of Z. Chen [Chin. Ann. Math., Ser. A 20, 7-14 (1999; Zbl 0946.34072)] and K. H. Kwon [Kodai Math. J. 19, 378-387 (1996; Zbl 0879.34006)] by using a different method of proof. The main result reads as follows: Let \(P\) and \(Q\) be nonconstant polynomials with \[ P(z)= a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0,\quad Q(z)= b_nz^n+b_{n-1}z^{n-1}+\cdots+b_1z+b_0, \] with \(a_n \neq 0\), \(a_n=cb_n\), \(c>1\) and \(\text{deg}(P-cQ) = m \geq 1\). Furthermore, let \(h_j\), \(j=0,1\), be entire functions such that \(\sigma(h_j)<m\). Then every nontrivial solution \(f\) to the differential equation (2) satisfies \(\sigma_2(f)=n\).

The author considers the differential equation \[ f'' + h_1(z)e^{P(z)}f' + h_0(z)e^{Q(z)}f = 0 , \tag{2} \] where \(P\), \(Q\) are nonconstant polynomials and \(h_0\), \(h_1\) are entire functions such that \(h_0 \not\equiv 0\), \(\sigma(h_0)<\deg Q\) and \(\sigma(h_1)<\text{deg}{P}\). He generalizes results of Z. Chen [Chin. Ann. Math., Ser. A 20, 7-14 (1999; Zbl 0946.34072)] and K. H. Kwon [Kodai Math. J. 19, 378-387 (1996; Zbl 0879.34006)] by using a different method of proof. The main result reads as follows: Let \(P\) and \(Q\) be nonconstant polynomials with \[ P(z)= a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0,\quad Q(z)= b_nz^n+b_{n-1}z^{n-1}+\cdots+b_1z+b_0, \] with \(a_n \neq 0\), \(a_n=cb_n\), \(c>1\) and \(\text{deg}(P-cQ) = m \geq 1\). Furthermore, let \(h_j\), \(j=0,1\), be entire functions such that \(\sigma(h_j)<m\). Then every nontrivial solution \(f\) to the differential equation (2) satisfies \(\sigma_2(f)=n\).

Reviewer: Rainer Brück (Dortmund)

##### 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 |

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\textit{Z. X. Chen}, Acta Math. Sin., Engl. Ser. 18, No. 1, 79--88 (2002; Zbl 1014.34078)

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