On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives ()
1. Introduction and Results
Let
denote the complex plane and f be a nonconstant meromorphic function on
. We assume the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as
(see, e.g., [2] [3] ), and
denotes any quantity that satisfies the condition
as
outside of a possible exceptional set of finite linear measure. A meromorphic function a is called a small function with respect to f, provided that
.
Let f and g be two nonconstant meromorphic functions. Let a be a small function of f and
We say that f, g share a counting multiplicities (CM) if
have the same zeros with the same multiplicities and we say that f, g share a ignoring multiplicities (IM) if we do not consider the multiplicities. In addition, we say that f
and g share ¥ CM, if
share 0 CM, and we say that f and g share ¥ IM, if
share 0 IM. Suppose that f and g share a IM. Throughout this paper, we denote by
the reduced counting function of those common a-points of f and g in
, where the multiplicity f each a-point of f is greater than that of the corresponding a-point of g, and denote by
the counting function for common simple 1-point of both f and g, and
the counting function of those 1-points of f and g where
. In the same way, we can define
and
If f and g share 1 IM, it is easy to see that
![]()
In addition, we need the following definitions:
Definition 1.1. Let f be a non-constant meromorphic function, and let p be a positive integer and
Then by
we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater that p, by
we denote the corresponding reduced counting function (ignoring multiplicities). By
we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not less than
by
we denote the corresponding reduced counting function (ignoring multiplicities,) where and what follows,
mean
respectively, if
.
Definition 1.2. Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer. We define
![]()
where
![]()
Remark 1.1. From the above inequalities, we have
![]()
Definition 1.3. Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer. We define
![]()
Remark 1.2. From the above inequality, we have
![]()
Definition 1.4. (see [4] ). Let k be a nonnegative integer or infinity. For
we denote by
the set of all a-points of f, where an a-point of multiplicity m is counted m times if
and
times if
. If
, we say that f, g share the value a with weight k.
We write f, g share
to mean that f, g share the value a with weight k; clearly if f, g share
, then f, g share
for all integers p with
. Also, we note that f, g share a value a IM or CM if and only if they share
or
, respectively.
R. Bruck [5] first considered the uniqueness problems of an entire function sharing one value with its derivative and proved the following result.
Theorem A. Let f be a non-constant entire function satisfying
. If f and
share the value 1 CM, then
for some nonzero constant c.
Bruck [5] further posed the following conjecture.
Conjecture 1.1. Let f be a non-constant entire function
be the first iterated order of f. If
is not a positive integer or infinite, f and
share the value 1 CM, then
for some nonzero constant.
Yang [6] proved that the conjecture is true if f is an entire function of finite order. Yu [7] considered the problem of an entire or meromorphic function sharing one small function with its derivative and proved the following two theorems.
Theorem B. Let f be a non-constant entire function and
be a meromorphic small function. If
and
share 0 CM and
, then
.
Theorem C. Let f be a non-constant non-entire meromorphic function and
be a meromorphic small function. If
1) f and a have no common poles.
2)
and
share 0 CM.
3) ![]()
then
where k is a positive integer.
In the same paper, Yu [7] posed the following open questions.
1) Can a CM shared be replaced by an IM share value?
2) Can the condition
of theorem B be further relaxed?
3) Can the condition 3) in theorem C be further relaxed?
4) Can in general the condition 1) of theorem C be dropped?
In 2004, Liu and Gu [8] improved theorem B and obtained the following results.
Theorem D. Let f be a non-constant entire function
be a meromorphic small function. If
and
share 0 CM and
then
.
Lahiri and Sarkar [9] gave some affirmative answers to the first three questions improving some restrictions on the zeros and poles of a. They obtained the following results.
Theorem E. Let f be a non-constant meromorphic function, k be a positive integer, and
be a meromorphic small function. If
1) a has no zero (pole) which is also a zero (pole) of f or
with the same multiplicity.
2)
and
share ![]()
3)
then
.
In 2005, Zhang [10] improved the above results and proved the following theorems.
Theorem F. Let f be a non-constant meromorphic function,
be integers. Also let
be a meromorphic small function. Suppose that
and
share
. If
and
(1)
or
and
(2)
or
and
(3)
then ![]()
In 2015, Jin-Dong Li and Guang-Xiu Huang proved the following Theorem.
Theorem G. Let f be a non-constant meromorphic function,
be integers. Also let
be a meromorphic small function. Suppose that
and
share
. If
and
(4)
and
(5)
or
and
(6)
then ![]()
In this paper, we pay our attention to the uniqueness of more generalized form of a function namely
and
sharing a small function.
Theorem 1.1. Let f be a non-constant meromorphic function,
be integers. Also let
be a meromorphic small function. Suppose that
and
share
. If
and
(7)
and
(8)
or
and
(9)
then ![]()
Corollary 1.2. Let f be a non-constant meromorphic function,
be integers. Also let
be a meromorphic small function. Suppose that
and
share
. If
and ![]()
or
and ![]()
or
and ![]()
then ![]()
2. Lemmas
Lemma 2.1 (see [1] ). Let f be a non-constant meromorphic function,
be two positive integers, then
![]()
clearly ![]()
Lemma 2.2 (see [1] ). Let
(10)
where F and G are two non constant meromorphic functions. If F and G share 1 IM and
, then
![]()
Lemma 2.3 (see [11] ). Let f be a non-constant meromorphic function and let
![]()
be an irreducible rational function in f with constant coefficients
and
where
and
. Then
![]()
where ![]()
3. Proof of the Theorem
Proof of Theorem 1.1. Let
and
Then F and G share
, except the zeros and poles of
. Let H be defined by (10).
Case 1. Let ![]()
By our assumptions, H have poles only at zeros of
and
and poles of F and G, and those 1-points of F and G whose multiplicities are distinct from the multiplicities of corresponding 1-points of G and F respectively. Thus, we deduce from (10) that
(11)
here
is the counting function which only counts those points such that
but
.
Because F and G share 1 IM, it is easy to see that
(12)
By the second fundamental theorem, we see that
(13)
Using Lemma 2.2 and (11), (12) and (13), we get
(14)
We discuss the following three sub cases.
Sub case 1.1.
. Obviously.
(15)
Combining (14) and (15), we get
(16)
that is
![]()
By Lemma 2.1 for
, we get
![]()
So
![]()
which contradicts with (7).
Sub case 1.2.
. It is easy to see that
(17)
and
(18)
Combining (14) and (17) and (18), we get
(19)
that is
![]()
By Lemma 2.1 for
, we get
![]()
So
![]()
which contradicts with (8).
Sub case 1.3.
. It is easy to see that
(20)
(21)
Similarly we have
(22)
Combining (14) and (20)-(22), we get
(23)
that is
![]()
By Lemma 2.1 for
and for
respectively, we get
![]()
So
![]()
which contradicts with (9).
Case 2. Let ![]()
on integration we get from (10)
(24)
where C, D are constants and
. we will prove that
.
Sub case 2.1. Suppose
. If
be a pole of f with multiplicity p such that
then it is a pole of G with multiplicity
respectively. This contradicts (24). It follows that
and hence
Also it is clear that
From (7)-(9) we know respectively
(25)
(26)
and
(27)
Since
, from (24) we get
![]()
Suppose
.
Using the second fundamental theorem for F we get
![]()
i.e.,
![]()
So, we have
and so
which contradicts (25)-(27).
If
then
(28)
and from which we know
and hence,
If ![]()
We know from (28) that
![]()
So from Lemma 2.1 and the second fundamental theorem we get
![]()
![]()
which is absurd. So
and we get from (28) that
which implies ![]()
In view of the first fundamental theorem, we get from above
![]()
which is impossible.
Sub case 2.2.
and so from (24) we get
![]()
If
then
![]()
and ![]()
By the second fundamental theorem and Lemma 2.1 for
and Lemma 2.3 we have
![]()
Hence
![]()
So, it follows that
![]()
![]()
and
![]()
This contradicts (7)-(9). Hence
and so
that is
This completes the proof of the theorem.