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On the Exponent of Convergence for the Zeros of the Solutions of ![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_IEq1_HTML.gif)
Journal of Inequalities and Applications volume 2010, Article number: 428936 (2010)
Abstract
Let and
be entire functions of order less than 1 with
and
transcendental. We prove that every solution
of the equation
,
,
being has zeros with infinite exponent of convergence.
1. Introduction
It is assumed that the reader of this paper is familiar with the basic concepts of Nevanlinna theory [1, 2] such as , and
. Suppose that
is a meromorphic function, then the order of growth of the function
and the exponent of convergence of the zeros of
are defined, respectively, as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ1_HTML.gif)
Let be a measurable subset of
. The lower logarithmic density and the upper logarithmic density of
are defined, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ2_HTML.gif)
where is the characteristic function of
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ3_HTML.gif)
Now let us recall some of the previous results on the linear differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ4_HTML.gif)
where is an entire function of finite order, When
is polynomial, many authors [3–6] have studied the properties of the solutions of (1.4). If
is a transcendental entire function with
, Gundersen [7] proved that every nontrivial solution of (1.4) has infinite order of growth. In [8], Wang and Laine considered the nonhomogeneous equation of type
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ5_HTML.gif)
where are entire functions of order less than one and
are complex numbers. In fact, they have proved the following theorem.
Theorem 1.1.
Suppose that are entire functions of order less than one, and suppose that
with
and
. Then every nontrivial solution of (1.5) is of infinite order.
Corollary 1.2.
Suppose that , where
is a nonvanishing entire function with
and
with
. Then every nontrivial solution of (1.4) is of infinite order.
2. Results
We observe that all the above results concern the order of growth only. In this paper, we are going to prove the following theorem which concerns the exponent of convergence.
Theorem 2.1.
Let and
be entire functions of order less than 1 with
and
being transcendental. Then every solution
of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ6_HTML.gif)
has zeros with infinite exponent of convergence.
The hypothesis that is transcendental is not redundant since Frei [4] has shown that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ7_HTML.gif)
has solutions of finite order, for certain constants .
We notice that Theorem 2.1 fails for . For any entire function
the function
solves (2.1) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ8_HTML.gif)
3. Some Lemmas
Throughout this paper we need the following lemmas. In 1965, Hayman [9] proved the following lemma.
Lemma 3.1.
Let the function be meromorphic of finite order
in the plane and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ9_HTML.gif)
for all outside a set
of upper logarithmic density
, where the positive constant
depends only on
and
.
In 1962, Edrei and Fuchs [10] proved the following lemma.
Lemma 3.2.
Let be a meromorphic function in the complex plane and let
have measure
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ10_HTML.gif)
In 2007, Bergweiler and Langley [11] proved the following lemma.
Lemma 3.3.
Let be a transcendental entire function of order
. For large
define
to be the length of the longest arc of the circle
on which
, with
if the minimum modulus
satisfies
. Then at least one of the following is true:
(i)there exists a set of positive upper logarithmic density such that
for
;
(ii)for each the set
has lower logarithmic density at least
We deduce the following.
Lemma 3.4.
Let , let
be a positive integer, and let
have logarithmic density
. Let
be a transcendental entire function such that
on a path
tending to infinity and for all
with
and
. Then
has order at least
.
Proof.
Assume that and choose a polynomial
of degree at most
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ11_HTML.gif)
is transcendental entire. Then we have for all
and for all
with
and
. With the notation of Lemma 3.3, we see that
for all large
, and so we must have case (ii), as well as
for
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ12_HTML.gif)
Since has logarithmic density
this gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ13_HTML.gif)
4. Proof of Theorem 2.1
Let ,
and
be as in the hypotheses. We can assume that
. Suppose that
is a solution of (2.1) having zeros with finite exponent of convergence. Then we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ14_HTML.gif)
where and
are entire functions with
. We can assume that
, since if
is constant we can replace
by
and
by
. Using (2.1) and (4.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ15_HTML.gif)
Lemma 4.1.
One has .
Proof.
Suppose that . Dividing (4.2) by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ16_HTML.gif)
Hence, provided lies outside a set of finite measure,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ17_HTML.gif)
and so, using the fact that and
have order less than
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ18_HTML.gif)
This holds outside a set of finite measure and so for all large
, since we may take
with
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ19_HTML.gif)
Lemma 4.1 is proved.
Let denote large positive constants. Choose
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ20_HTML.gif)
There exists an -set
[2, page 84] such that for all large
outside
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ21_HTML.gif)
and using the Poisson-Jensen formula,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ22_HTML.gif)
Moreover, there exists a set of logarithmic density
such that for
the circle
does not meet the
-set
.
Lemma 4.2.
The functions and
are both transcendental.
Proof.
Let be small and positive and suppose that
or
is a polynomial. Let
be large with
and
. Since
is small it follows from (4.2) and (4.8) that
. Choose
with
such that the intersection of
with the ray
given by
is bounded. Applying Lemma 3.4 to the function
, with
a subpath of
, gives
, but
may be chosen arbitrarily small, and this contradicts (4.7).
The next step is to estimate in the right half-plane.
Lemma 4.3.
Let be a large positive integer and let
. Then for large
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ23_HTML.gif)
one has, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ24_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ25_HTML.gif)
Proof.
Let be large and satisfy (4.10), and assume that (4.11) does not hold. Then (4.8) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ26_HTML.gif)
Also, (4.7), and (4.9) give
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ27_HTML.gif)
Here denote positive constants which may depend on
but not on
. Using (4.8), (4.12) and (4.14) we get, from (4.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ28_HTML.gif)
Now divide (4.2) by . We obtain, using (4.15),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ29_HTML.gif)
which gives and (4.12). This proves Lemma 4.3.
Lemma 4.4.
Let and
be as in Lemma 4.3. Choose
such that the ray
has bounded intersection with the
-set
. Let
be the union of the ray
and the arcs
, where
is the set chosen following (4.9). Then one of the following holds:
(i)one has (4.11) for all large in
;
(ii)one has (4.12) for all large in
.
Proof.
This follows simply from continuity. For each large in
we have (4.11) or (4.12), but we cannot have both because of (4.14). This proves Lemma 4.4.
Lemma 4.5.
Let . Then for large
with
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ30_HTML.gif)
Proof.
Let be as in the hypotheses. Since
we only need to prove (4.17) for
. Assume that
. Then dividing (4.2) by
gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ31_HTML.gif)
by (4.8), and so (4.17) follows using (4.7). This proves Lemma 4.5.
Lemma 4.6.
If conclusion (i) of Lemma 4.4 holds then , while if conclusion (ii) of Lemma 4.4 holds then
.
Proof.
Suppose that conclusion (i) of Lemma 4.4 holds. Choose such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ32_HTML.gif)
and let be small compared to
. Assume that
in Lemma 4.4 is small compared to
, in particular so small that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ33_HTML.gif)
where is the positive constant from Lemma 3.1. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ34_HTML.gif)
and let be the exceptional set of Lemma 3.1, with
. Then for large
we have, using (4.20) and Lemmas 3.1, 3.2, and 4.5,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ35_HTML.gif)
We then have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ36_HTML.gif)
for large . Now take any large
. Since
has logarithmic density
, while
has upper logarithmic density at most
, and since
is small, there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F428936/MediaObjects/13660_2010_Article_2146_Equ37_HTML.gif)
which proves Lemma 4.6 in this case. The alternative case, in which we have conclusion (ii) in Lemma 4.4, is proved the same way, using in place of
.
To finish the proof suppose first that conclusion (ii) of Lemma 4.4 holds. Then Lemma 3.4 implies that has order at least
. Since
may be chosen arbitrarily small, this contradicts Lemma 4.6. The same contradiction, with
replaced by
, arises if conclusion (i) of Lemma 4.4 holds, and the proof of the theorem is complete.
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Acknowledgment
The author thanks Professor J. K. Langley for the invaluable discussions on the results of this paper during his visit in summer 2008 and summer 2010 to the University of Nottingham in the U.K.
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Alotaibi, A. On the Exponent of Convergence for the Zeros of the Solutions of .
J Inequal Appl 2010, 428936 (2010). https://doi.org/10.1155/2010/428936
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DOI: https://doi.org/10.1155/2010/428936