Abstract
The fast growing solutions of the following linear differential equation \((*)\) is investigated by using a more general scale \({[p,q]_{,\varphi }}\)-order,
where \(A_i(z)\) are entire functions in the complex plane, \(i=0,1,\ldots ,k-1\). The growth relationships between entire coefficients and solutions of the equation \((*)\) is found by using the concepts of \({[p,q]_{,\varphi }}\)-order and \({[p,q]_{,\varphi }}\)-type, which extend and improve some previous results.
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1 Introduction and Main Results
We assume that the reader is familiar with the fundamental results and the standard notation of Nevanlinna theory in the complex plane \({\mathbb {C}}\), see [8, 15] for more details. Considering the linear differential equation
where \(A_0(z),\ldots ,A_{k-1}(z)\) are entire functions in \({\mathbb {C}}\) and \(k(\ge 2)\) is integer. Usually, order and hyper order are used to study the growth of solutions of Eq. (1.1), for example, see [7, 10, 14, 15, 18, 19, 20, 21] and therein references. For the fast growing entire function, the iterated order is defined to measure their growing. It is well-known that Kinnunen firstly used the idea of iterated order to study the fast growing of solutions of Eq. (1.1) in [13]. Since then, the iterated order of solutions of Eq. (1.1) is very interesting topic, many results concerning iterated order of solutions of Eq. (1.1) have been obtained, for example [3, 9] and therein references. To estimate precisely the fast growing of entire functions, the concept of [p, q]-order is defined in [12]. From then, many results concerning [p, q]-order of solutions of Eq. (1.1) have been found by different researchers, for example [16, 17] and theirin references.
In [4], Chyzhykov and Semochko have pointed out that the definition of [p, q]-order have weaknesses is that it do not cover arbitrary growth, and given Examples 1.4 and 1.7 in [4] to show the case. And the same time, they given more general growth scale of meromorphic function as follows.
Definition 1
([4]) Let \(\varphi\) be an increasing unbounded function on \([1,+\infty )\), and f be a meromorphic function. The \(\varphi\)-orders of f are defined by
If f is an entire function, then the \(\varphi\)-orders are defined by
Remark 1
([4]) Let \(\varphi \in \Phi\) and f be an entire function. Then
The properties of \(\Phi\) and \(\varphi\) will be shown in the following Sect. 2. Furthermore, Chyzhykov and Semochko studied the growth of solutions of Eq. (1.1) by using the concept of \(\varphi\)-order.
Theorem 1.1
([4]) Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions. Then all nontrivial solutions f of Eq. (1.1) satisfy
Theorem 1.2
([4]) Let \(\varphi \in \Phi\), and \(l=\max \left\{ j|\rho _\varphi ^{0}(A_j)\ge \beta , j=0,\ldots ,k-1\right\} .\) Then Eq. (1.1) possesses at most l entire linearly independent solutions f with \(\rho _\varphi ^{1}(f)<\beta .\)
Theorem 1.3
([4]) Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions such that \(\rho _\varphi ^{0}(A_0)>\max \left\{ \rho _\varphi ^{0}(A_j),j=1,\ldots ,k-1\right\}\). Then all nontrivial solutions f of Eq. (1.1) satisfy \(\rho _\varphi ^{1}(f)=\rho _\varphi ^{0}(A_0).\)
Recently, Belaïdi defined the concept of \(\varphi\)-type of meromorphic functions which is used to study the growth of solutions of Eq. (1.1), and the following Theorem 1.4 is obtained.
Definition 2
([2]) Let \(\varphi\) be an increasing unbounded function on \([1,+\infty )\), and f be a meromorphic function with \(\rho _\varphi ^{i}(f) \in (0,+\infty ),i=0,1\). The \(\varphi\)-types of f are defined by
If f is an entire function, then the \(\varphi\)-types of f are defined by
Theorem 1.4
([2]) Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions. Assume that
and
Then all nontrivial solutions f of Eq. (1.1) satisfy \({\tilde{\rho }}_\varphi ^{1}(f)={\tilde{\rho }}_\varphi ^{0}(A_0).\)
Motivated to the [p, q]-order of meromorphic function. We introduce the concepts of \([p,q]_{,\varphi }\)-order and \([p,q]_{,\varphi }\)-type, where \(p\ge q \ge 1\). For all \(r\in (0,+\infty )\), \(\exp _1r=e^r\), \(\exp _{n+1}r=\exp (\exp _nr)\) and \(\log _1r=\log r\) and \(\log _{n+1}r=\log (\log _nr)\), \(n\in N\). We also denote \(\exp _0r=r=\log _0r\), \(\exp _{-1}r=\log _1r\). The \([p,q]_{,\varphi }\)-order and \([p,q]_{,\varphi }\)-type are defined as follows, respectively.
Definition 3
Let \(\varphi\) be an increasing unbounded function on \([1,+\infty )\), and f be a meromorphic function. The \([p,q]_{,\varphi }\)-orders of f are defined by
If f is an entire function, then the \([p,q]_{,\varphi }\)-orders of f are defined by
Definition 4
Let \(\varphi\) be an increasing unbounded function on \([1,+\infty )\), and f be a meromorphic function with \(\rho _{[p,q],\varphi }^{i}(f) \in (0,+\infty ),i=0,1\). The \([p,q]_{,\varphi }\)-types of f are defined by
If f is an entire function with \({\tilde{\rho }}_{[p,q],\varphi }^{i}(f) \in (0,+\infty ), i=0,1\), then the \([p,q]_{,\varphi }\)-types of f are defined by
The following two examples show that \([p,q]_{,\varphi }\)-order is indeed superior to \(\varphi\)-order when studying the same fast growth functions.
Example 1
It follows from [5] that \(\exp _4(\alpha (\log r)^\beta )\) is convex in \(\log r\). Then there exists an entire function f that satisfies
where \(\alpha ,\beta >0\).
For \(\varphi (r)=(\log _2 r)^{\frac{1}{\beta }}\), we can get that
however,
Example 2
It follows from [5] that \(\exp _2(\alpha (\log r)^\beta )\) is convex in \(\log r\). Then there exists an entire function f that satisfies
where \(\alpha ,\beta >0\).
For \(\varphi (r)=(\log _2 r)^{\frac{1}{\beta }}\), we can get that
however,
Here, we study the growth of solutions of Eq. (1.1) by using the concepts of \([p,q]_{,\varphi }\)-order and \([p,q]_{,\varphi }\)-type, Theorems 1.5–1.8 are obtained which are generalization of previous results from Chyzhykov-Semochko [4] and Belaïdi [2].
Theorem 1.5
Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions. Then all nontrivial solutions f of Eq. (1.1) satisfy
Theorem 1.6
Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions, \(m=\max \left\{ j|\rho ^{0}_{[p,q],\varphi }(A_j)\ge \lambda ,j=0,\ldots ,k-1\right\} .\) Then Eq. (1.1) possesses at most m entire linearly independent solutions f with \(\rho ^{1}_{[p,q],\varphi }(f)<\lambda .\)
Theorem 1.7
Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions such that \(\rho ^{0}_{[p,q],\varphi }(A_0)>\max \left\{ \rho ^{0}_{[p,q],\varphi }(A_j),j=1,\ldots ,k-1\right\}\). Then all nontrivial solutions f of Eq. (1.1) satisfy \(\rho ^{1}_{[p,q],\varphi }(f)=\rho ^{0}_{[p,q],\varphi }(A_0).\)
Theorem 1.8
Let \(\varphi \in \Phi\), \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions. Assume that
and
Then all nontrivial solutions f of Eq. (1.1) satisfy \({\tilde{\rho }}^{1}_{[p,q],\varphi }(f)={\tilde{\rho }}^{0}_{[p,q],\varphi }(A_0).\)
2 Properties of \([p,q]_{,\varphi }\)-order
In [4], Chyzhykov and Semochko defined the class of positive unbounded increasing function on \([1,+\infty )\) by \(\Phi\) such that \(\varphi (e^{t})\) is slowly growing, i. e.,
First, we recall properties of functions from the class \(\Phi\).
Proposition 2.1
([4]) If \(\varphi \in \Phi\), then
Remark 2
If \(\varphi\) is non-decreasing, then (2.2) is equivalent to the definition of the class \(\Phi\).
Next, we obtain some basic properties of \([p,q]_{,\varphi }\)-order by using standard method.
Proposition 2.2
Let \(\varphi \in \Phi\), and f be an entire function. Then
Proof
First, we prove that this is true when \(j=1\), and it can be proved for the case of \(j=0\) by using similar reason as the case of \(j=1\).
According to the monotonicity of function \(\varphi\) and the following inequality
we get that
Next, by (2.3) and choose \(R=kr,\,k>1\), we have
In fact, by the properties of function \(\varphi\),
Hence,
It is implies that
Therefore, this is completely proved. \(\square\)
Proposition 2.3
Let \(\varphi \in \Phi\), and let \(f,f_1,f_2\) be three meromorphic functions. Then the following statements hold.
-
(i)
\(\rho _{[p,q],\varphi }^{j}(f_1+f_2)\le \max \left\{ \rho _{[p,q],\varphi }^{j}(f_1), \rho _{[p,q],\varphi }^{j}(f_2)\right\} ,j=0,1.\)
-
(ii)
\(\rho _{[p,q],\varphi }^{j}(f_1f_2)\le \max \left\{ \rho _{[p,q],\varphi }^{j}(f_1), \rho _{[p,q],\varphi }^{j}(f_2)\right\} ,j=0,1.\)
-
(iii)
\(\rho _{[p,q],\varphi }^{j}(\frac{1}{f})= \rho _{[p,q],\varphi }^{j}(f)~ for~ f\ne 0, j=0,1.\)
-
(iv)
\(for\,\,a \in {\mathbb {C}}{\setminus }\left\{ 0\right\} , we\,\,have\,\, \rho _{[p,q],\varphi }^{j}(af)=\rho _{[p,q],\varphi }^{j}(f), \tau _{[p,q],\varphi }^{j}(af)=\tau _{[p,q],\varphi }^{j}(f),\) j=0,1.
Proof
(i) We prove that this is true when \(j=1\), and similarly it can be proved for the case of \(j=0\). Let \(a=\rho _{[p,q],\varphi }^{1}(f_1)\), \(b=\rho _{[p,q],\varphi }^{1}(f_2)\). Without loss of generality, suppose that \(a\le b <+\infty\). Now by the definition of \(\rho _{[p,q],\varphi }^1\)-order, for any \(\varepsilon >0\) and sufficiently large r,
It follows from the properties of Nevanlinna characteristic functions that
Hence,
It is implies that
The properties (ii), (iii) and (iv) can be proved by using similar way as in the proof of the case (i). \(\square\)
Proposition 2.4
Let \(\varphi \in \Phi\), and \(f_1,f_2\) be two meromorphic functions. If \(\rho _{[p,q],\varphi }^{j}(f_1)<\rho _{[p,q],\varphi }^{j}(f_2), j=0,1\), then
Proof
Obviously, we can easily conclude that this is true by Proposition 2.3. \(\square\)
Proposition 2.5
Let \(\varphi \in \Phi\), and \(f_1,f_2\) be two meromorphic functions. Then the following statements hold.
-
(i)
If \(\,\,0<\rho _{[p,q],\varphi }^{j}(f_1)< \rho _{[p,q],\varphi }^{j}(f_2)<+\infty , 0<\tau _{[p,q],\varphi }^{j}(f_1)<\tau _{[p,q],\varphi }^{j}(f_2),j=0,1,\,then\)
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1+f_2)=\tau _{[p,q],\varphi }^{j}(f_1f_2)=\tau _{[p,q],\varphi }^{j}(f_2). \end{aligned}$$ -
(ii)
If \(\,\,0<\rho _{[p,q],\varphi }^{j}(f_1)= \rho _{[p,q],\varphi }^{j}(f_2)=\rho _{[p,q],\varphi }^{j}(f_1+f_2),j=0,1, \,then\)
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1+f_2) \le \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$Moreover, if \(\,\, \tau _{[p,q],\varphi }^{j}(f_1)\ne \tau _{[p,q],\varphi }^{j}(f_2),\,\) then
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1+f_2) = \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$ -
(iii)
If \(\,\,0<\rho _{[p,q],\varphi }^{j}(f_1)= \rho _{[p,q],\varphi }^{j}(f_2)=\rho _{[p,q],\varphi }^{j}(f_1f_2),j=0,1,\,then\)
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1f_2) \le \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$Moreover, if \(\,\, \tau _{[p,q],\varphi }^{j}(f_1)\ne \tau _{[p,q],\varphi }^{j}(f_2), \,\) then
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1f_2) = \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$
Proof
We just prove the case of \(j=1\), and the case of \(j=0\) is very similar.
(i) By the definition of the \(\tau _{[p,q],\varphi }^{1}\)-type, for any given \(\varepsilon >0\), there exists a sequence \(\left\{ r_n\right\}\) which tending to infinity and \(N_1\in Z^{+}\), such that for \(n>N_1\),
On the other hand, there exists \(N_2\in Z^{+}\), such that for \(n>N_2\),
Obviously,
Set \(N=\max \left\{ N_1,N_2\right\}\). By the properties of \(\varphi\) and \(n>N\), we have
It follows from Proposition 2.4 that \(\rho _{[p,q],\varphi }^{1}(f_1+f_2)=\rho _{[p,q],\varphi }^{1}(f_2)\). By the monotonicity of \(\varphi\), we have
And then
Since \(\rho _{[p,q],\varphi }^{1}(f_1+f_2)=\rho _{[p,q],\varphi }^{1}(f_2)>\rho _{[p,q],\varphi }^{1}(f_1)=\rho _{[p,q],\varphi }^{1}(-f_1)\), then
Thus \(\tau _{[p,q],\varphi }^{1}(f_2)=\tau _{[p,q],\varphi }^{1}(f_1+f_2)\).
Next we prove that \(\tau _{[p,q],\varphi }^{1}(f_1f_2)=\tau _{[p,q],\varphi }^{1}(f_2)\). Obviously, \(T(r,f_1f_2) \ge T(r,f_2)-T(r,f_1)-\log 2\). By using similar discussion as in the proof above, we obtain easily that
Since \(\rho _{[p,q],\varphi }^{1}(f_1f_2)=\rho _{[p,q],\varphi }^{1}(f_2)>\rho _{[p,q],\varphi }^{1}(f_1)=\rho _{[p,q],\varphi }^{1}(\frac{1}{f_{1}})\), then
So, \(\tau _{[p,q],\varphi }^{1}(f_2)=\tau _{[p,q],\varphi }^{1}(f_1f_2)\).
(ii) By (2.5), we have
Hence, by the monotonicity of \(\varphi\),
Without loss of generality, suppose \(\tau _{[p,q],\varphi }^{1}(f_1)<\tau _{[p,q],\varphi }^{1}(f_2)\). Then, by (2.6) and \(\rho _{[p,q],\varphi }^{1}(f_1+f_2)=\rho _{[p,q],\varphi }^{1}(f_1)=\rho _{[p,q],\varphi }^{1}(-f_1)\), we get
By (2.6) and (2.7), \(\tau _{[p,q],\varphi }^{1}(f_{1}+f_2)\)=\(\max \left\{ \tau _{[p,q],\varphi }^{1}(f_1), \tau _{[p,q],\varphi }^{1}(f_2)\right\}\).
(iii) is proved by using similar reason as in the proof of (i) and (ii). \(\square\)
The following Corollary can be obtain from (i) and (ii) of Proposition 2.5.
Corollary 2.6
Let \(\varphi \in \Phi\), and let \(f_1,f_2\) be two meromorphic functions.
-
(i)
If \(0<\rho _{[p,q],\varphi }^{j}(f_1)= \rho _{[p,q],\varphi }^{j}(f_2)=\rho _{[p,q],\varphi }^{j}(f_1+f_2), j=0,1,\) then
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1) \le \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1+f_2), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$ -
(ii)
If \(0<\rho _{[p,q],\varphi }^{j}(f_1)= \rho _{[p,q],\varphi }^{j}(f_2)=\rho _{[p,q],\varphi }^{j}(f_1f_2), j=0,1,\) then
$$\begin{aligned} \tau _{[p,q],\varphi }^{j}(f_1) \le \max \left\{ \tau _{[p,q],\varphi }^{j}(f_1f_2), \tau _{[p,q],\varphi }^{j}(f_2)\right\} . \end{aligned}$$
Proposition 2.7
Let \(\varphi \in \Phi\), and f be a meromorphic function. Then
Proof
Set \(\rho _{[p,q],\varphi }^{1}(f)=\alpha\). From the definition of \(\rho _{[p,q],\varphi }^{1}\)-order, for any \(\varepsilon >0\), there exists \(r_0 >1\), such that for all \(r \ge r_0\),
Obviously, \(T(r,f^{'})\le 2T(r,f)+m(r,\frac{f^{'}}{f})\). By the Lemma of logarithmic derivative (p.34 in [8]), we have
where \(E\subset [0,+\infty )\) is of finite linear measure. By Lemma 3.2 in Sect. 3 and for all sufficiently large r,
It is implies that \(\rho _{[p,q],\varphi }^{1}(f)\ge \rho _{[p,q],\varphi }^{1}(f^{'})\).
On the other hand, we prove the inequality \(\rho _{[p,q],\varphi }^{1}(f) \le \rho _{[p,q],\varphi }^{1}(f^{'})\). The definition of \(\rho _{[p,q],\varphi }^{1}(f^{'})=\beta\) implies that for any given above \(\varepsilon >0\), there exists \(r_{1}>1\), such that for all \(r>r_1\),
By the properties of \(\varphi\) and
we can get that
By the monotonicity of \(\varphi\), we get
It is implies that \(\rho _{[p,q],\varphi }^{1}(f) \le \rho _{[p,q],\varphi }^{1}(f^{'})\). \(\square\)
3 Auxiliary Results
In the proof of Theorems 1.5 and 1.6, the classical reduced order method is adopted for Eq. (1.1), which aims to find the estimation of \(m(r,A_j)(j=0,\ldots ,k-1)\) by using the estimation of \(m(r,\frac{f^{(k)}}{f})(k\ge 1)\). The following lemma is an estimation of \(m(r,\frac{f^{(k)}}{f})\).
Lemma 3.1
Let f be a meromorphic function of order \(\rho _{[p,q],\varphi }^{1}(f)=\rho\), \(k\in {\mathbb {N}}\), and \(\varphi \in \Phi\). Then for any \(\varepsilon >0\),
outside, possibly, an exceptional set E of finite linear measure.
Proof
Let \(k=1\). The definition of \(\rho _{[p,q],\varphi }^{1}\)-order implies that for any \(\varepsilon >0\), there exists \(r_0>1\), such that for all \(r>r_0\),
It follows from (3.1) and the lemma of logarithmic derivative that
where \(E\subset (0, +\infty )\) is of finite linear measure.
Now, we assume that for some \(k \in {\mathbb {N}}\),
Since \(N(r,f^{(k)})\le (k+1)N(r,f)\), we deduce
It follows from (3.2) and (3.3) that \(m\left( r,\frac{f^{(k+1)}}{f^{(k)}}\right) =O(\exp _{p-2}[\varphi ^{-1}\log _q r^{\rho +\varepsilon }])\), \(r\notin E\). Thus,
\(\square\)
The following lemma is needed to prove Theorems 1.5 and 1.6.
Lemma 3.2
([1]) Let \(g: [0,+\infty ) \rightarrow {\mathbb {R}}\) and \(h: [0,+\infty ) \rightarrow {\mathbb {R}}\) be monotone nondecreasing functions such that \(g(r) \le h(r)\) outside an exceptional set E of finite linear measure. Then for any \(\alpha >1\), there exists \(r_0>0\) such that \(g(r) \le h(\alpha r)\) for all \(r>r_0\).
Wiman-Valiron theory is needed in proving our results, which can be found [15]. Let \(f(z)=\sum \limits _{n=0}^{+\infty } a_{n}z^{n}\) be an entire function. Then
are called the maximal term and the central index of f, respectively.
Lemma 3.3
([15, p. 51]) Let f be a transcendental entire function, let \(0<\delta < \frac{1}{4}\) and z such that \(\vert z\vert =r\) and \(\vert f(z)\vert >M(r,f)\nu (r,f)^{-\frac{1}{4}+\delta }\). Then there exists a set \(E\subset \mathbb {R_+}\) of finite logarithmic measure such that
holds for integer \(m\ge 0\) and \(r \notin E\).
The following estimation of the radius r of the polynomial P(z) is used in the proof of Theorem 1.5.
Lemma 3.4
([15, p.10]) Let \(P(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{0}\) be a polynomial, where \(a_{n}\ne 0\). Then all zero of P(z) lie in the discs D(0, r) of radius
We need the following two lemmas to get estimations of T(r, f) and m(r, f), which is used in proving Theorems 1.6 and 1.8.
Lemma 3.5
Let f be a meromorphic function with \(\rho _{[p,q],\varphi }^{0}(f)= \rho _0\in (0,+\infty ).\) Then, for all \(\mu (<\rho _0)\), there exists a set \(E\in [1,+\infty )\) of infinite logarithmic measure, such that \(\varphi (e^{\log _{p-1}T(r,f)})>\mu \log _q r\) holds for all \(r \in E\).
Proof
The definition of \(\rho _{[p,q],\varphi }^{0}\)-order implies that there exists a sequence \((R_j)^{+\infty }_{j=1}\) satisfying
From the equality above, for any \(\varepsilon \in (0,\rho _0-\mu )\), there exists an integer \(j_1\) such that for \(j \ge j_1\),
Since \(\mu <\rho _0-\varepsilon\), there exists an integer \(j_2\) such that for \(j \ge j_2\),
It follows from this inequality and (3.4) that for \(j \ge j_3=\max \left\{ j_1,j_2\right\}\) and for any \(r\in [R_j, (1+\frac{1}{j})R_j]\),
Set \(E=\bigcup \limits _{j=j_3}^{+\infty }[R_j,(1+\frac{1}{j})R_j]\). It is easy to show that E is of infinite logarithmic measure,
\(\square\)
We can also prove the following result by using similar reason as in the proof of Lemma 3.5.
Lemma 3.6
Let \(\varphi \in \Phi\), and f be an entire function with \({\tilde{\rho }}_{[p,q],\varphi }^{0}(f)= \rho _0 \in (0,+\infty )\) and \({\tilde{\tau }}_{[p,q],\varphi }^{0}(f)\in (0,+\infty )\). Then for any given \(\beta <{\tilde{\tau }}_{[p,q],\varphi }^{0}(f)\), there exists a set \(E\in [1,+\infty )\) of infinite logarithmic measure such that for all \(r \in E\),
The following lemma is used to prove Theorem 1.7 for the case of \(q=1\).
Lemma 3.7
([9]) Let f be a solution of Eq. (1.1), and let \(1 \le \gamma <+\infty\). Then for all \(0<r<R\), where \(0<R<+\infty\),
where \(C>0\) is a constant which depends on \(\gamma\) and the initial value of f in a point \(z_0\), where \(A_j \ne 0\) for some \(j=0,\ldots ,k-1\), and where
The following logarithmic derivative estimation was found in [6] from Gundersen.
Lemma 3.8
([6]) Let f be a transcendental meromorphic function, and let \(\alpha >1\) be a given constant. Then there exists a set \(E\subset [1,+\infty )\) with finite logarithmic measure and a constant \(B>0\) that depends only on \(\alpha\), and i, j, \(0 \le i < j \le k-1\), such that for all z satisfying \(\vert z \vert =r \notin [0,1]\bigcup E\),
Lemma 3.9
Let \(\varphi \in \Phi\) and \(A_0(z),\ldots ,A_{k-1}(z)\) be entire functions. Then, every nontrivial solution f of Eq. (1.1) satisfies
Proof
Set
By the definition of \({\tilde{\rho }}_{[p,q],\varphi }^{0}(A_j)\), for any \(\varepsilon >0\) and for sufficiently large r,
By Lemma 3.7 for \(\gamma =1\), we have
It follows from (3.5), (3.6) and Proposition 2.2 that
\(\square\)
4 Proofs of Theorems 1.5 and 1.6
The classical way of reducing the order is adopted for Eq. (1.1) in proofs of Theorems 1.5 and 1.6, and \(T(r,A_{j})(j=0,1,\ldots ,k-1)\) is estimated by \(T(r,\frac{f^{(k)}}{f})(k\ge 1)\) in reducing the order.
To state our proving concisely, let E represents the finite logarithmic measure, I represents the infinite logarithmic measure and F represents the finite linear measure in the proofs of Theorems 1.5–1.8. Next we start prove our results by using the similar way as in the proofs of Theorems 1.1–1.4.
Proof of Theorem 1.5
Set \(\gamma _{[p,q],\varphi }=\sup \left\{ \rho _{[p,q],\varphi }^{1}(f)|L(f)=0\right\}\), and
\(\alpha _{[p,q],\varphi }=\sup \left\{ \rho _{[p,q],\varphi }^{0}(A_j)|j=0,1,\ldots ,k-1\right\}\).
First, we prove that \(\alpha _{[p,q],\varphi }\le \gamma _{[p,q],\varphi }\). If \(\gamma _{[p,q],\varphi }=+\infty\), it is trivial. Hence we just consider the case of \(\gamma _{[p,q],\varphi }<+\infty\). Let \(f_1,\ldots , f_k\) be a solution base of Eq. (1.1) with \(\rho _{[p,q],\varphi }^{1}(f_j)< +\infty , j=1,\ldots ,k\). It is clear that \(W=W(f_1,\ldots , f_k)\ne 0\) by the properties of the Wronsky determinant.
It follows from Propositions 2.3 and 2.7 that \(\rho _{[p,q],\varphi }^{1}(W)<\infty\). By properties of the Wronsky determinant ([15, p.55]),
where
In view of Proposition 2.3 we can conclude that \(\rho _{[p,q],\varphi }^{1}(A_i)< \infty , i=0,1,\ldots ,k-1\).
By Lemma 3.1 to \(f_i, i=1,\ldots ,k,\)
We now apply the standard order reduction procedure ( [15, p.53–57]). Denote
\(A_k=1\), and \(\nu _1^{(-1)}:=\frac{f}{f_1}\), i.e., \((\nu _1^{(-1)})^{'}:=\nu _1\). Hence,
Substituting (4.1) into (1.1) and using the fact that \(f_1\) solves (1.1), we obtain
where
By \(\gamma _{[p,q],\varphi }<+\infty\) and Proposition 2.7, the meromorphic functions
are solutions of (4.2) of finite \(\rho _{[p,q],\varphi }^1\)-order.
Next, we claim that
when
In fact, we prove it by induction on i following [15]. By equality (4.2) for \(j=k-2\), we have \(A_{1,k-2}=A_{k-1}+k \frac{f'}{f}\). By Lemma 3.1 and (4.4),
We assume that
Since
by Lemma 3.1, (4.4) and (4.6), we have
We may now proceed as above the order reduction procedure for (4.2). In each reduction step, we obtain a solution base of meromorphic functions of finite \(\rho _{[p,q],\varphi }^{1}\)-order according to (4.3), and the implication (4.4) and (4.5) remains valid. Hence, we finally obtain an equation of the form \(w^{'}+B(z)w=0\), and w is any solution of the equation with \(\rho _{[p,q],\varphi }^{1}(w)<\infty\). Then
Observing the reasoning corresponding to (4.4) and (4.5) in the subsequent reduction steps,
It implies that
By Lemma 3.2 and Proposition 2.1, for sufficiently large r, \(j=0,\ldots ,k-1,\)
Hence, \(\frac{\varphi (e^{\log _{p-1}T(r,A_j)})}{\log _q r}\le \gamma _{[p,q],\varphi }+2\varepsilon\). This implies that \(\alpha _{[p,q],\varphi }\le \gamma _{[p,q],\varphi }\).
We next prove the converse inequality under the assumption that \(\alpha _{[p,q],\varphi }<+\infty\).
By Lemma 3.3, there exists a set \(E\subset \mathbb {R_+}\) of finite logarithmic measure, such that for all z satisfies \(|f(z)|=M(r,f)\) and \(|z|=r \not \in E\),
Substituting (4.8) into (1.1),
The definition of \({\tilde{\rho }}_{[p,q],\varphi }^{0}\)-order and Proposition 2.2 yields that for any \(\varepsilon >0\) there exists \(r_0 >1\), such that for all \(r\ge r_0\),
By Lemma 3.4 and Proposition 2.1,
It follows from [11, p.36–37] that
This implies that \(\gamma _{[p,q],\varphi }\le \alpha _{[p,q],\varphi }\).
\(\square\)
Proof of Theorem 1.6
By the assumption there exist two numbers \(\lambda _1\) and \(\lambda\) such that \(\rho _{[p,q],\varphi }^{0}(A_m)\ge \lambda\) and \(\rho _{[p,q],\varphi }^{0}(A_l)\le \lambda _1<\lambda\) for \(l=m+1,\ldots ,k-1\).
Let \(f_1,\ldots ,f_{m+1}\) be linearly independent solutions of (1.1) such that \(\rho _{[p,q],\varphi }^{1}(f_i)\) \(<\lambda\), \(i=1,\ldots ,m+1\). If \(m=k-1\), then all \(f_1,\ldots ,f_{k}\) are of \(\rho _{[p,q],\varphi }^{1}(f_i)<\lambda\), this contradict with Theorem 1.5. Hence, \(m<k-1\). Applying the order reduction procedure as in the proof of Theorem 1.5. We use the notation \(\nu _0\) instead of f, and \(A_{0,0},\ldots ,A_{0,k-1}\) instead of \(A_{0},\ldots ,A_{k-1}\). On the general reduction step, we obtain an equation of the form
where
and the functions
determine at each reduction step a solution base of (4.9) in terms of the preceding solution base. We may express (1.1) and the mth reduction steps by the following Table. The rows correspond to (4.9) for \(\nu _0,\ldots ,\nu _m\), i.e., the first row corresponds to (1.1), and columns from k to 0 give the coefficients of these equations, while the last column lists those solutions with \(\rho _{[p,q],\varphi }^{1}(f)<\lambda\).
k | k-1 | . | \(\textbf{m}\) | m-1 | . | 0 | \(\rho _{[p,q],\varphi }^{1}(f)<\lambda\) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\nu _0\) | 1 | \(A_{0,k-1}\) | . | \(\mathbf {A_{0,m}}\) | \(A_{0,m-1}\) | . | \(A_{0,0}\) | \(\nu _{0,1},\ldots ,\nu _{0,m+1}\) | |||||
\(\nu _1\) | 1 | . | \(A_{1,m}\) | \(\mathbf {A_{1,m-1}}\) | . | \(A_{1,0}\) | \(\nu _{1,1},\ldots ,\nu _{1,m}\) | ||||||
. | . | . | . | . | . | ||||||||
. | . | . | . | . | . | ||||||||
. | . | . | . | . | . | ||||||||
\(\nu _{m-1}\) | \(A_{m-1,m}\) | \(A_{m-1,m-1}\) | . | \(A_{m,0}\) | \(\nu _{m-1,1}\),\(\nu _{m-1,1}\) | ||||||||
\(\nu _m\) | \(A_{m,m}\) | \(A_{m,m-1}\) | . | \(\mathbf {A_{m,0}}\) | \(\nu _{m,1}\) |
By Lemma 3.1 and (4.10), we see that in the second row, corresponding to the first reduction step, \(m(r,A_{1,l})=O\left\{ \exp _{p-2}[\varphi ^{-1}(\log _q r^{\lambda _1+\varepsilon })]\right\}\), \(r \notin F\), \(l=m,\ldots ,k-2\), while \(\lambda _1+\varepsilon <\lambda\) and \(m(r,A_{1,m-1}) \ne O\left\{ \exp _{p-2}[\varphi ^{-1}(\log _q r^{\lambda _1+\varepsilon })]\right\}\), \(r \notin F\).
Similarly, in each reduction step (4.10) implies that
when \(l=m+1-j,\ldots ,k-(j+1)\), i.e., for all coefficients to the left from the boldface coefficient \(A_{j, m-j}\), while for \(j=1,\ldots ,m\),
In particular,
Applying Lemma 3.5 to the coefficient \(A_{m,0}\) with the constant \(\lambda\), and obtain that
On the other hand, after the mth reduction step, by (4.10), (4.11) and Lemma 3.1, we have
That implies that
Since \(\rho _{[p,q],\varphi }^{0}(\nu _{m,1})<\lambda _1\), in view of Propositions 2.3 and 2.7,
Therefore,
By Lemma 3.2, for sufficiently large r,
By (4.12) and (4.13), we obtain the contradiction with our assumption. Hence, there exists at most m linearly independent solutions Eq. (1.1) with \(\rho _{[p,q],\varphi }^{1}(f)<\lambda\). \(\square\)
5 Proofs of Theorems 1.7 and 1.8
Proof of Theorem 1.7
Let f be a nontrivial solution of Eq. (1.1). We denote \(\rho _{[p,q],\varphi }^{1}(f)=\rho _1\) and \(\rho _{[p,q],\varphi }^{0}(A_0)=\rho _0\). The inequality \(\rho _0 \le \rho _1\) follows from Theorem 1.6 when \(m=0\) and \(\lambda =\rho _0\).
To prove the conserve inequality, by Lemma 3.7 for \(\gamma =1\), Proposition 2.1 and the definition of \(\rho _{[p,q],\varphi }^{0}\)-order, for any \(\varepsilon >0\),
Therefore,
It is implies that \(\rho _1\le \rho _0\), and then Theorem 1.7 is proved. \(\square\)
Proof of Theorem 1.8
Suppose that f is a nontrivial solution of Eq. (1.1). From (1.1), we can write
If \(\max \left\{ {\tilde{\rho }}_{[p,q],\varphi }^{0}(A_j),j=1,\ldots ,k-1\right\}< {\tilde{\rho }}_{[p,q],\varphi }^{0}(A_0)=\rho _0<+\infty\), and by Theorem 1.7, then
Suppose that
and
First, we prove that \(\rho _1={\tilde{\rho }}_{[p,q],\varphi }^{1}(f) \ge {\tilde{\rho }}_{[p,q],\varphi }^{0}(A_0)=\rho _0\). By assumption there exists a set \(K\subseteq \left\{ 1,2,\ldots ,k-1\right\}\) such that
and
Thus, we choose \(\lambda _1\) and \(\lambda _2\) satisfying
For sufficiently large r,
and
where \(0<\alpha <\rho _0\). By Lemma 3.6, there exists a set \(I \subset [1,+\infty )\) with infinite logarithmic measure, such that for all \(r \in I\),
By Lemma 3.8, there exists a constant \(B>0\) and a set \(E\subset [1,+\infty )\) having finite logarithmic measure, such that for all z satisfying \(\vert z\vert =r \notin E\bigcup [0,1]\),
Set \(\rho _1={\tilde{\rho }}_{[p,q],\varphi }^{1}(f)\). By Proposition 2.2, for any given \(\varepsilon \in (0,\max \left\{ \frac{\lambda _2 -\lambda _1}{2},\rho _0 -\rho _1\right\} )\) and sufficiently large \(\vert z\vert =r \notin E \bigcup [0,1]\),
Hence, substituting (5.2),(5.3), (5.4) and (5.5) into (5.1), for sufficiently large \(\vert z \vert =r \in I \setminus (E\cup [0,1])\),
Obviously, \(I \setminus (E\cup [0,1])\) is of infinite logarithmic measure. By (5.6), there exists a sequence of points \(\left\{ \vert z_n \vert \right\} =\left\{ r_n\right\} \subset I {\setminus }( E\cup [0,1])\) tending to \(+\infty\), such that
By the monotonicity of the function \(\varphi ^{-1}\), we obtain that \(\lambda _1 \ge \lambda _2\). This contradiction implies
On the other hand, by Lemma 3.9, we have
Hence, every nontrivial solution f of Eq. (1.1) satisfies \({\tilde{\rho }}_{[p,q],\varphi }^{1}(f) ={\tilde{\rho }}_{[p,q],\varphi }^{0}(A_0).\) \(\square\)
6 Conclusions
We define new measure \({[p,q]_{,\varphi }}\)-order to describe the growing of meromorphic function, and the new measure is used to study the growth of solutions of complex differential equations.
Availability of data and material
All data generated or analysed during this study are included in this published article.
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Funding
The National Natural Science Foundation of China (Grant No. 12261023, 11861023) and Postgraduate Research Funding in Guizhou Province (Grant No. QianJiaoHeYJSKYJJ[2021]087).
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J. R. Long and H. Y. Qin carried out the research, study, methodology and writing L. Tao participated in the validity confirmation and advisor role. All authors readed and approved the final manuscript.
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Long, J., Qin, H. & Tao, L. On \([p,q]_{,\varphi }\)-Order and Complex Differential Equations. J Nonlinear Math Phys 30, 932–955 (2023). https://doi.org/10.1007/s44198-023-00107-7
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DOI: https://doi.org/10.1007/s44198-023-00107-7
Keywords
- Linear differential equations
- Entire functions
- \([p, q]_{,\varphi }\)-order
- \([p, q]_{,\varphi }\)-type