# Order of growth of linear complex differential equations when the
coefficients have same order

Naveen Mehra and S. K. Chanyal  Naveen Mehra, Department of Mathematics,
Kumaun University, D.S.B. Campus, Nainital-263001, Uttarakhand, India
<EMAIL_ADDRESS>S. K. Chanyal, Department of Mathematics, Kumaun
University, D.S.B. Campus, Nainital-263001, Uttarakhand, India
<EMAIL_ADDRESS>
###### Abstract.

This article contains the theorems which shows that when
$A(z)=h_{1}(z)e^{P_{1}(z)}$ and $B(z)=h_{0}(z)e^{P_{0}(z)}$ are of same
order,then all the non-trivial solutions of equation
$f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0$ are of infinite order. Moreover we
extend these results to higher order differential equations.

###### Key words and phrases:

entire function, order of growth and complex differential equation.

###### 2020 Mathematics Subject Classification:

34M10, 30D35

The research work of the first author is supported by research fellowship from
Council of Scientific and Industrial Research (CSIR), New Delhi.

##  1\. Introduction and statement of main results

Consider the differential equation,

$f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0$ (1)

where $A(z)$ and $B(z)(\not\equiv 0)$ are entire functions. Then, it is well
known that all solutions of equation (1) are entire functions(see [9]).

######  Definition 1.

The order of growth $\rho(f)$ and lower order of growth $\mu(f)$ of an entire
function $f$ are defined by

$\displaystyle{\rho(f)=\limsup\limits_{r\to\infty}\displaystyle{\frac{\log^{+}\log^{+}{M(r,f)}}{\log{r}}}}$
$\displaystyle{\mu(f)=\liminf\limits_{r\to\infty}\displaystyle{\frac{\log^{+}\log^{+}{M(r,f)}}{\log{r}}}}$

respectively, where $M(r,f)=max_{|z|=r}|f(z)|$.

######  Definition 2.

Exponent of convergence of zeros of function $f$ is defined by

$\displaystyle{\lambda(f)=inf\\{\alpha>0\displaystyle{|}\sum\limits_{n=1}^{\infty}|a_{n}|^{-\alpha}<\infty\\}},$

where $(a_{n})_{n\in\mathbb{N}}$ is a sequence of non-zero complex
numbers(not-necessarily distinct) with $\lim_{n\to\infty}a_{n}=\infty$.

If $A(z)$ and $B(z)$ are polynomials, then all solutions of equation (1) are
of finite order(see [14]). Moreover, if either of $A(z)$ and $B(z)$ or both
are entire transcendental function, then equation (1) have atleast one
solution of infinite order(see [4]). Following classical result is summary of
[6] and [8].

######  Theorem 1.

Suppose that $A(z)$ and $B(z)$ are entire functions satisfying one of the
following conditions:

  1. (i)

[6] $\rho(A)<\rho(B)$;

  2. (ii)

[6] $A(z)$ is a polynomial and $B(z)$ is a transcendental entire function;

  3. (iii)

[8] $\rho(B)<\rho(A)\leq\frac{1}{2}$.

Then every non-trivial solution of equation (1) are of infinite order.

Following are the examples of differential equations having finite order
solutions when $\rho(B)\leq\rho(A)$.

######  Example 1.

In the differential equation $f^{\prime\prime}+(e^{z}-1)f^{\prime}+e^{z}f=0$,
$f(z)=e^{z}$ is a non-trivial solution of order $1$ and order of both
coefficients are equal.

######  Example 2.

The differential equation $f^{\prime\prime}+(ze^{z}+z+1)f^{\prime}+zf=0$ has a
solution $f=e^{-z}+1$.

Above examples shows the differential equation (1) has finite order solution
when $\rho(B)\leq\rho(A)$. Consider $A(z)=h_{1}(z)e^{P_{1}(z)}$ and
$B(z)=h_{0}(z)e^{P_{0}(z)}$, where $h_{1}(z)$ and $h_{0}(z)$ are
transcendental entire functions; $P_{1}(z)$ and $P_{0}(z)$ are polynomials
satisfying $\rho(h_{1})<\rho(h_{0})$. Gundersen [7] proved that all non-
trivial solutions are of infinite order when $\deg P_{1}\neq\deg P_{0}$. What
if $\deg P_{1}=\deg P_{0}$?

######  Example 3.

$f^{\prime\prime}(z)-z^{2}e^{z}+ze^{z}f=0$ has finite order non-trivial
solution $f(z)=2z$.

In example 3, we have $\rho(A)=\rho(B)=\deg(z)=1$, where $\lambda(A)<\rho(A)$
and $\lambda(B)<\rho(B)$. Let

$P_{1}(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots+a_{0}$ (2)

and

$P_{0}(z)=b_{n}z^{n}+b_{n-1}z^{n-1}+\cdots+b_{0}$ (3)

where $a_{n}\neq 0$ and $b_{n}\neq 0$. Kwon[12] considered this case with some
restrictions and gives condition when order of solution is infinite. He proved
the following:

######  Theorem 2.

[12] Consider the equation

$f^{\prime\prime}+h_{1}(z)e^{P_{1}(z)}f^{\prime}+h_{0}(z)e^{P_{0}(z)}f=0,$ (4)

where $P_{1}(z)$ and $P_{0}(z)$ are non-constant polynomials as given in (2)
and (3) with $a_{n}\neq 0$, $b_{n}\neq 0$ and $h_{1}(z)$, $h_{2}(z)\neq 0$ are
entire functions satisfying $\rho(h_{1})<degP_{1}$ , $\rho(h_{0})<degP_{0}$.
Then all non-trivial solutions are of infinite order if any of the following
holds:

  1. (i)

$\arg a_{n}\neq\arg b_{n}$ or $a_{n}=cb_{n}$ with $0<c<1$,

  2. (ii)

$a_{n}=b_{n}$ and $\deg(P_{1}-P_{0})=m\geq 1$, $\rho(h_{1})<m$ and
$h_{0}(z)<m$,

  3. (iii)

$a_{n}-cb_{n}$ with $c>1$ and $\deg(P_{1}-cP_{0})=m\geq 1$, $\rho(h_{1})<m$
and $0<\rho(h_{0})<\frac{1}{2}$ or

  4. (iv)

$a_{n}=cb_{n}$ with $c\geq 1$, $P_{1}(z)-cP_{0}(z)$ be a constant and
$\rho(h_{1})<\rho(h_{0})<\frac{1}{2}$.

In this article, we shall like to expand Kwon’s result when
$P_{1}(z)-cP_{0}(z)=constant$ and $c\geq 1$. His result has a limitation on
the $\rho(h_{0})<\frac{1}{2}$. In our first result we consider
$\rho(h_{1})<\mu(h_{0})<\frac{1}{2}$. The statement of our first result is
following:

######  Theorem 3.

Let $P_{1}(z)$ and $P_{0}(z)$ be non-constant polynomials as given in (2) and
(3) with $a_{n}\neq 0$, $b_{n}\neq 0$ and $h_{1}(z)$, $h_{0}(z)(\not\equiv 0)$
be an entire functions satisfying $\rho(h_{1})<degP_{1}$ and
$\rho(h_{0})<degP_{0}$. Let $P_{1}(z)-cP_{0}(z)$ be a constant with $c\geq 1$
and $\rho(h_{1})<\mu(h_{0})\leq\frac{1}{2}$. Then all non-trivial solutions of
(4) are of infinite order.

In Theorem 3, we restrict $\mu(h_{0})$ to be less than half. If we consider
$h_{1}(z)$ to have Fabry gaps or multiply connected Fatou component, we
consider $\mu(h_{0})$ cannot be less than half.

######  Definition 3.

An entire function $f=\sum_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ is
said to have Fabry gaps if $\frac{\lambda_{n}}{n}\to\infty$.

There are many research articles in which authors used coefficient having a
Fabry gaps. Long[13], Kumar and Saini[11] also used Fabrys gap in their
articles.

######  Theorem 4.

Let $P_{1}(z)$ and $P_{0}(z)$ be non-constant polynomials as given in (2) and
(3) with $a_{n}\neq 0$, $b_{n}\neq 0$ and $h_{1}(z)$, $h_{0}(z)(\not\equiv 0)$
be an entire functions satisfying $\rho(h_{1})<degP_{1}$ and
$\rho(h_{0})<degP_{0}$. Let $P_{1}(z)-cP_{0}(z)$ be a constant with $c\geq 1$
and let $h_{1}(z)$ has Fabry gaps with $\mu(h_{0})>\rho(h_{1})$. Then every
non-constant solution $f$ of (4) has infinite order.

######  Definition 4.

For the entire function, the set of family of iterates
$\\{f^{n}:n\in\mathbb{N}\\}$ is well defined and forms a normal family; is
called Fatou set.

Interested reader may visit [3] for detailed analysis about Fatou set. The
complement of Fatou set is known as Julia set. G. Zhang and L. Yang[15]
considered entire function having multiply connected Fatou component in his
paper.

######  Theorem 5.

Let $P_{1}(z)$ and $P_{0}(z)$ be non-constant polynomials as given in (2) and
(3) with $a_{n}\neq 0$, $b_{n}\neq 0$ and $h_{1}(z)$, $h_{0}(z)(\not\equiv 0)$
be an entire functions satisfying $\rho(h_{1})<degP_{1}$ and
$\rho(h_{0})<degP_{0}$. Let $P_{1}(z)-cP_{0}(z)$ be a constant with $c\geq 1$
and $h_{1}(z)$ has multiply connected Fatou component with
$\rho(h_{1})\leq\mu(h_{0})$. Then all non-trivial solutions of (4) are of
infinite order.

##  2\. Preliminary Lemma

This section is devoted to preliminaries results which are used to prove our
main theorems when a transcendental entire function $f$ is of finite order
Gundersen[6] gave a well known result. This result is used extensively.

######  Lemma 1.

[6] Let $f$ be a transcendental entire function with $\rho<\infty$, let
$\Gamma=\\{(k_{1},j_{1}),\\\ (k_{2},j_{2}),\ldots,(k_{n},j_{n})\\}$ denotes
finite set of distinct pairs of integers that satisfy $k_{i}>j_{i}\geq 0$, for
$i=1,2,\ldots,n$, and let $\epsilon>0$ be a given constant. Then the following
three statements holds:

  1. (i)

there exists a set $E_{1}\subset[0,2\pi)$ that has linear measure $0$, such
that if $\psi_{0}\in[0,2\pi)-E_{1}$, then there is a constant
$R_{0}=R_{0}(\psi_{0})>0$ so that for all z satisfying $\arg z=\psi_{0}$ and
$|z|\geq R_{0}$ and for all $(k,j)\in\Gamma$, we have

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq|z|^{(k-j)(\rho-1+\epsilon))}.$
(5)

  2. (ii)

there exists a set $E_{2}\subset(1,\infty)$ that has finite logarithmic
measure, such that for all z with $|z|\notin E_{2}\cup[0,1]$ and for all
$(k,j)\in\Gamma$, the inequality (5) holds.

  3. (iii)

there exists a set $E_{3}\subset[0,\infty)$ that has finite linear measure,
such that for all z with $|z|\notin E_{3}$ and for all $(k,j)\in\Gamma$, we
have

$\left|\frac{f^{(k)}(z)}{f^{(j)}(z)}\right|\leq|z|^{(k-j)(\rho+\epsilon)}.$
(6)

Besicovitch’s $\cos\pi\rho$ theorem provides the condition for a function $f$
of order less than 1. It says that $\log m(r,f)>(cos\pi\rho^{\prime})\log
M(r,f)$ when $\rho<\rho^{\prime}<1$, $|z|=r$ belongs to a set of upper density
greater than $1-\frac{\rho}{\rho^{\prime}}$, where
$m(r,f)=\min\\{|f|:|z|=r\\}$ and $M(r,f)=\max\\{|f|:|z|=r\\}$. Barry[2] proved
this result for $\mu<1$.

######  Lemma 2.

[2] Assume that $f$ is an entire function with $0\leq\mu(f)<1$. Then, for
every $\beta\in(\mu(f),1)$, $m(r,f)>M(r,f)\cos\pi\alpha$ for $|z|=r\in
F\subset[1,\infty)$, where $\overline{\log dens}F\geq 1-\frac{\mu(f)}{\beta}$.

Combining Theorem 1 from [5] and Lemma 2.2 from [10], we get the following
lemma. It is proved by Long[13]. It is the property of finite order entire
functions having Fabry gaps.

######  Lemma 3.

[13] Assume $f(z)=\sum\limits_{n=0}^{\infty}a_{\lambda_{n}}z^{\lambda_{n}}$ be
an entire function of finite order of growth with Fabry gaps, and let $u(z)$
be an entire function with $\rho(u)\in(0,\infty)$. Then for any given
$\epsilon\in(0,\varsigma)$, where $\varsigma=\min(1,\rho(u))$, there exists a
set $G\subset(1,\infty)$ satisfying $\overline{\log dense}G\geq\xi$, where
$\xi\in(0,1)$ is a constant such that for all $|z|=r\in G$,

$\log M(r,u)>r^{\rho(u)-\epsilon},\ \log m(r,g)>(1-\xi)\log M(r,g),$

where $M(r,u)=\max\\{|u(z)|:|z|=r\\},\ m(r,g)=\min\\{|g(z)|:|z|=r\\}$ and
$M(r,g)=\max\\{|g(z)|:|z|=r\\}$.

Every multiply connected Fatou component of a transcendental meromorphic
function must be Baker wandering domain(proved by Baker[1]) and if a
transcendental meromorphic function has Baker wandering domain, then
$\mathcal{J}$ has only bounded components(see [16] Remark on page $25$). So,
following lemma was proved by Zheng[16](see Corollary 1), also holds for
transcendental meromorphic function having multiply connected Fatou component.

######  Lemma 4.

[16] Assume that f is a transcendental meromorphic function with at most
finitely many poles. If $\mathcal{J}(f)$ has only bounded components, then for
any complex number $z$, there exist a constant $0<\gamma<1$ and two sequences
$\\{r_{n}\\}$ and $\\{R_{n}\\}$ of positive numbers with $r_{n}\to\infty$ and
$R_{n}/r_{n}\to\infty(n\to\infty)$ such that

$M(r,f)^{\gamma}\leq L(r,f),r\in H,$

where $H=\cup_{n=1}^{\infty}\\{r:r_{n}<r<R_{n}\\}.$

##  3\. Proof of the main theorems

### Proof of Theorem 3.

###### Proof.

Let $\rho(f)<\infty$. Then by Lemma 1, there exist a set $E\subset[0,\infty)$
having finite linear measure such that for $r\not\in E$

$\left|\frac{f^{(k)}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|\leq
r^{k\rho(f)}.$ (7)

Since $\mu(h_{0})<1/2$, by Lemma 2 for $r\in F$, where $\overline{\log
dens}F\geq 1-\frac{\mu(f)}{\beta}$, we have

$M(r,h_{0})\cos\pi\alpha\ <|h_{0}(re^{\iota\theta})|.$ (8)

By definition of order, we have for $\epsilon>0$

$|h_{1}(re^{r\iota\theta})|\leq\exp{r^{\rho(h_{1})+\epsilon}},$ (9)

holds for all $r\geq r_{0}$, where $r_{0}>0$. From (1),we have

$|h_{0}(re^{\iota\theta})e^{(1-c)P_{0}(re^{\iota\theta})}|\leq|e^{-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|+|h_{1}(re^{\iota\theta})e^{P_{1}(re^{\iota\theta})-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|.$
(10)

Using (7), (8) and (9) in (10), for very large $r\in F\setminus E$ on a curve
$C$ be a tending to $\infty$ such that $ReP_{0}(z)=0$, we get

$M(r,h_{0})\cos\pi\beta<(1+\exp{r^{\rho(h_{1})+\epsilon}})r^{2\rho(f)},$

which is a contradiction, since $\rho(h_{1})<\mu(h_{0})$. ∎

### Proof of Theorem 4.

###### Proof.

Assume that solution $f$ of equation (1) to be of finite order. Using Lemma 3
on transcendental entire function $h_{0}(z)$ having Fabry gaps, for $r\in G$,
where $\overline{\log dens}G\geq\xi$, for $\xi\in(0,1)$ we have

$M(r,h_{0})^{1-\xi}<|h_{0}(re^{\iota\theta})|.$ (11)

From (1), we have

$|h_{0}(re^{\iota\theta})e^{(1-c)P_{0}(re^{\iota\theta})}|\leq|e^{-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|+|h_{1}(re^{\iota\theta})e^{P(re^{\iota\theta})-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|.$
(12)

Using (7), (9) and (11) in (12), for very large $r\in G\setminus E$ on a curve
$C$ be a tending to $\infty$ such that $ReP_{0}(z)=0$, we get

$M(r,h_{0})^{1-\xi}<(1+\exp{r^{\rho(h_{1})+\epsilon}})r^{2\rho(f)},$

which is a contradiction. ∎

### Proof of Theorem 5.

###### Proof.

Suppose that $\rho(f)<\infty$. Considering $h_{0}$ to be an entire function
having multiply connected Fatou component, by Lemma 4 for $r\in H$, where
$H=\cup_{n=1}^{\infty}\\{r:r_{n}<r<R_{n}\\}$ , we have

$M(r,h_{0})^{\gamma}<|h_{0}(re^{\iota\theta})|$ (13)

From (1), we have

$|h_{0}(re^{\iota\theta})e^{(1-c)P_{0}(re^{\iota\theta})}|\leq|e^{-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|+|h_{1}(re^{\iota\theta})e^{P_{1}(re^{\iota\theta})-cP_{0}(re^{\iota\theta})}|\left|\frac{f^{\prime}(re^{\iota\theta})}{f(re^{\iota\theta})}\right|.$
(14)

Using (7), (9) and (13) in (14), for very large $r\in H\setminus E$ on a curve
$C$ be a tending to $\infty$ such that $ReP_{0}(z)=0$, we get

$M(r,h_{0})^{\gamma}<(1+\exp{r^{\rho(h_{1})+\epsilon}})r^{2\rho(f)},$

which is not possible. ∎

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