Finite order solutions of complex linear differential equations.

*(English)*Zbl 1063.30031The authors investigate the growth of transcendental solutions \(f\) of complex linear homogeneous differential equations
\[
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_1(z)f'+A_0(z)f=0\,, \quad k\geq 2\,, \tag{*}
\]
with entire coefficients \(A_j(z)\) \((A_0(z)\not\equiv 0)\). The main result of this paper states that if exactly one of the intermediate coefficients exponentially dominates all the other coefficients in a sector and if \(f\) is of finite order, then a derivative \(f^{(j)}\) is asymptotically constant in a slightly smaller sector. The authors apply this result in order to give conditions on the coefficients to ensure that every transcendental solution is of infinite order. This type of results seems to originate in the work of G. G. Gundersen, [Trans. Am. Math. Soc. 305, 415-429 (1988; Zbl 0669.34010)]. In fact, there he considered the second order linear differential equation, that is, (*) when \(k=2\), and posed especially two questions:

Q1: What condition on \(A_0(z)\) and \(A_1(z)\) will guarantee that every solution \(f\not\equiv 0\) has infinite order?

Q2: If the equation possesses a solution \(f\not\equiv 0\) of finite order, then how do the properties of \(A_0(z)\) and \(A_1(z)\) affect the properties of \(f\)?

Here we say that an entire function \(A(z)\) exponentially dominates another entire function \(B(z)\) in a sector \(S(0):=\{z\mid \theta_1\leq \arg z \leq \theta_2\}\), if there are two real positive constants \(\alpha\) and \(\beta\) such that \(| A(z)| \geq \exp\{(1+o(1))\alpha | z| ^{\beta}\}\) and \(| B(z)| \leq \exp\{o(1)| z| ^{\beta}\}\) hold as \(z\to\infty\) in \(S(0)\). When \(k=2\), Gundersen himself answers questions Q1 and Q2, respectively as follows:

A1: If \(A_0(z)\) exponentially dominates \(A_1(z)\) in some \(S(0)\), every solution \(f\not\equiv 0\) of (*) has infinite order.

A2: Assume that \(A_1(z)\) exponentially dominates \(A_0(z)\) in some \(S(0)\).

If \(f\not\equiv 0\) is a finite order solution of (*), then there exists a nonzero constant \(b_0\) such that \(| f(z)- b_0| \), as well as \(| f^{(m)}(z)| \) for any \(m\in \mathbb{N}\), grows possibly \(\leq \exp \{-(1+o(1))\alpha | z| ^{\beta}\}\) as \(z\to \infty\) in a smaller angle \(S(\varepsilon):=\{z\mid\theta_1+\varepsilon \leq \arg z \leq \theta_2-\varepsilon\}\).

These two results have been generalized respectively by B. Belaïdi and S. Hamouda [Electron. J. Differ. Equ. 2001, Paper No. 61, 5 p., electronic only (2001; Zbl 1039.30012), Kodai Math. J. 25, No. 3, 240–245 (2002; Zbl 1037.34082)] and by B. Belaïdi and K. Hamani [Electron. J. Differ. Equ. 2003, Paper No. 17, 12 p., electronic only (2003; Zbl 1029.34076)] into a higher order case: the first two papers show that A1 is true when \(A_0(z)\) exponentially dominates all the other \(A_j(z)\) \((1\leq j\leq k-1)\) in some \(S(0)\) or even under certain weaker dominance conditions, and the third one extends A2 into the case when \(A_1(z)\) exponentially dominates all the other \(A_j(z)\) \((j=0,2, 3, \ldots, k-1)\).

The paper under review extends also A2 when one intermediate coefficient \(A_s(z)\) \((1\leq s\leq k-1)\) exponentially dominates all the other \(A_j(z)\)’s (\(j\neq s\)) in some \(S(0)\). The authors however express the growth conditions for the coefficients and some estimates for the growth of \(f\) or its derivatives ‘more explicitly’ without making use of \(o(1)\): They assume that \(| A_{s}(z)| \geq \exp ((1+\delta )\alpha | z| ^{\beta})\) and \(| A_{j}(z)| \leq \exp (\delta \alpha | z| ^{\beta})\) hold for all \(j\neq s\) whenever \(| z| =r\geq r_{\delta}\) in the sector \(S(0)\) for a \(\delta >0\) with \(k\delta < 1\). And they conclude that given \(\varepsilon>0\) small enough, if \(f\) is a transcendental solution of finite order of (*) then (i) there exists \(j\in \{ 0,\dots ,s-1\}\) and a complex constant \(b_{j}\neq 0\) such that \(| f^{(j)}(z)-b_{j}| \leq \exp (-(1-k\delta )\alpha | z| ^{\beta})\) in \(S(\varepsilon )\), provided \(| z| \) is large enough. (ii) For each integer \(m\geq j+1\), \(| f^{(m)}(z)| \leq \exp (-(1-k\delta )\alpha | z| ^{\beta})\) in \(S(3\varepsilon )\) for all \(| z| \) large enough. According to the proof, the assumption on a solution \(f\) to be transcendental can be replaced by \(f^{(s)}\not\equiv 0\). Especially when \(s=1\), we may assume \(f\not\equiv 0\) as in A2 since \(A_0(z)\not\equiv 0\). With an example it is remarked that \(j<s-1\) or \(j=0\) may happen in (i).

The authors prove those theorems with their skilled estimates and a careful induction on \(s\) based on Gundersen’s reasoning which uses the Phragmén-Lindelöf theorem. The authors’ final result also relates to Q1. They provide a lower bound on the iterated order of such infinite order solutions.

(Reviewer’s notes: 1) Sectors are often identified with the corresponding angles in this paper. 2) One can easily adjust the inequality in (3.4) when \(\beta< 1\), as well as the integrals in (3.2) and (4.7). 3) As the case when \(j=0\), \(j=s-1\) may also happen in (i), so that the estimate \(0\leq j\leq s-1\) is sharp in this sense. For example, \(f=e^{-z}+z\) solves \(f^{(4)}+f^{\prime\prime\prime}+(z+1)e^zf^{\prime\prime}-f'-f=0\). Take \(\alpha=5/12\), \(\beta=1\), \(\delta=1/5\) and \(S(0)=\{0\leq \arg z \leq \pi/3\}\). Then for any \(\varepsilon\) small enough, we have \(j=1\) in this case.)

Q1: What condition on \(A_0(z)\) and \(A_1(z)\) will guarantee that every solution \(f\not\equiv 0\) has infinite order?

Q2: If the equation possesses a solution \(f\not\equiv 0\) of finite order, then how do the properties of \(A_0(z)\) and \(A_1(z)\) affect the properties of \(f\)?

Here we say that an entire function \(A(z)\) exponentially dominates another entire function \(B(z)\) in a sector \(S(0):=\{z\mid \theta_1\leq \arg z \leq \theta_2\}\), if there are two real positive constants \(\alpha\) and \(\beta\) such that \(| A(z)| \geq \exp\{(1+o(1))\alpha | z| ^{\beta}\}\) and \(| B(z)| \leq \exp\{o(1)| z| ^{\beta}\}\) hold as \(z\to\infty\) in \(S(0)\). When \(k=2\), Gundersen himself answers questions Q1 and Q2, respectively as follows:

A1: If \(A_0(z)\) exponentially dominates \(A_1(z)\) in some \(S(0)\), every solution \(f\not\equiv 0\) of (*) has infinite order.

A2: Assume that \(A_1(z)\) exponentially dominates \(A_0(z)\) in some \(S(0)\).

If \(f\not\equiv 0\) is a finite order solution of (*), then there exists a nonzero constant \(b_0\) such that \(| f(z)- b_0| \), as well as \(| f^{(m)}(z)| \) for any \(m\in \mathbb{N}\), grows possibly \(\leq \exp \{-(1+o(1))\alpha | z| ^{\beta}\}\) as \(z\to \infty\) in a smaller angle \(S(\varepsilon):=\{z\mid\theta_1+\varepsilon \leq \arg z \leq \theta_2-\varepsilon\}\).

These two results have been generalized respectively by B. Belaïdi and S. Hamouda [Electron. J. Differ. Equ. 2001, Paper No. 61, 5 p., electronic only (2001; Zbl 1039.30012), Kodai Math. J. 25, No. 3, 240–245 (2002; Zbl 1037.34082)] and by B. Belaïdi and K. Hamani [Electron. J. Differ. Equ. 2003, Paper No. 17, 12 p., electronic only (2003; Zbl 1029.34076)] into a higher order case: the first two papers show that A1 is true when \(A_0(z)\) exponentially dominates all the other \(A_j(z)\) \((1\leq j\leq k-1)\) in some \(S(0)\) or even under certain weaker dominance conditions, and the third one extends A2 into the case when \(A_1(z)\) exponentially dominates all the other \(A_j(z)\) \((j=0,2, 3, \ldots, k-1)\).

The paper under review extends also A2 when one intermediate coefficient \(A_s(z)\) \((1\leq s\leq k-1)\) exponentially dominates all the other \(A_j(z)\)’s (\(j\neq s\)) in some \(S(0)\). The authors however express the growth conditions for the coefficients and some estimates for the growth of \(f\) or its derivatives ‘more explicitly’ without making use of \(o(1)\): They assume that \(| A_{s}(z)| \geq \exp ((1+\delta )\alpha | z| ^{\beta})\) and \(| A_{j}(z)| \leq \exp (\delta \alpha | z| ^{\beta})\) hold for all \(j\neq s\) whenever \(| z| =r\geq r_{\delta}\) in the sector \(S(0)\) for a \(\delta >0\) with \(k\delta < 1\). And they conclude that given \(\varepsilon>0\) small enough, if \(f\) is a transcendental solution of finite order of (*) then (i) there exists \(j\in \{ 0,\dots ,s-1\}\) and a complex constant \(b_{j}\neq 0\) such that \(| f^{(j)}(z)-b_{j}| \leq \exp (-(1-k\delta )\alpha | z| ^{\beta})\) in \(S(\varepsilon )\), provided \(| z| \) is large enough. (ii) For each integer \(m\geq j+1\), \(| f^{(m)}(z)| \leq \exp (-(1-k\delta )\alpha | z| ^{\beta})\) in \(S(3\varepsilon )\) for all \(| z| \) large enough. According to the proof, the assumption on a solution \(f\) to be transcendental can be replaced by \(f^{(s)}\not\equiv 0\). Especially when \(s=1\), we may assume \(f\not\equiv 0\) as in A2 since \(A_0(z)\not\equiv 0\). With an example it is remarked that \(j<s-1\) or \(j=0\) may happen in (i).

The authors prove those theorems with their skilled estimates and a careful induction on \(s\) based on Gundersen’s reasoning which uses the Phragmén-Lindelöf theorem. The authors’ final result also relates to Q1. They provide a lower bound on the iterated order of such infinite order solutions.

(Reviewer’s notes: 1) Sectors are often identified with the corresponding angles in this paper. 2) One can easily adjust the inequality in (3.4) when \(\beta< 1\), as well as the integrals in (3.2) and (4.7). 3) As the case when \(j=0\), \(j=s-1\) may also happen in (i), so that the estimate \(0\leq j\leq s-1\) is sharp in this sense. For example, \(f=e^{-z}+z\) solves \(f^{(4)}+f^{\prime\prime\prime}+(z+1)e^zf^{\prime\prime}-f'-f=0\). Take \(\alpha=5/12\), \(\beta=1\), \(\delta=1/5\) and \(S(0)=\{0\leq \arg z \leq \pi/3\}\). Then for any \(\varepsilon\) small enough, we have \(j=1\) in this case.)

Reviewer: Kazuya Tohge (Kanazawa)