Entire functions share two half small functions
The paper generalizes a result by P. Li and C. C. Yang [Illinois J. Math. – 2000. – 44. – P. 349 – 362] and extends the previous work of G. Qiu [Kodai Math. J. – 2000. – 23. – P. 1 – 11].
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| author | Al-Khaladi, A. H. H. Аль-Халаді, А. Х. Х. |
| author_facet | Al-Khaladi, A. H. H. Аль-Халаді, А. Х. Х. |
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| description | The paper generalizes a result by P. Li and C. C. Yang [Illinois J. Math. – 2000. – 44. – P. 349 – 362] and extends the
previous work of G. Qiu [Kodai Math. J. – 2000. – 23. – P. 1 – 11]. |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.5
A. H. H. Al-Khaladi (College Comput. Sci. and Math., Tikrit Univ., Iraq)
ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS
ЦIЛI ФУНКЦIЇ ПОДIЛЯЮТЬ ДВI НАПIВМАЛI ФУНКЦIЇ
The paper generalizes a result by P. Li and C. C. Yang [Illinois J. Math. – 2000. – 44. – P. 349 – 362] and extends the
previous work of G. Qiu [Kodai Math. J. – 2000. – 23. – P. 1 – 11].
У роботi узагальнено результат П. Лi та Ц. Ц. Янга [Illinois J. Math. – 2000. – 44. – P. 349 – 362] та розширено
результати попередньої роботи Г. Кiу [Kodai Math. J. – 2000. – 23. – P. 1 – 11].
1. Introduction and main results. Throughout f denotes an entire function, i.e., a function that is
analytic in the whole complex plane, and f \prime denotes its derivative. We use the same signs as given
in Nevanlinna theory (see [3, 4]). In particular S(r, f) denotes any quantity satisfying S(r, f) =
= o(T (r, f)) as r \rightarrow \infty , except possibly on a set of finite linear measure. A meromorphic function
\alpha is said to be a small function of f if T (r, \alpha ) = S(r, f). We say that two nonconstant meromorphic
functions f and g share the value or small function \alpha IM (ignoring multiplicities), if f - \alpha and
g - \alpha have the same zeros. Let k be a positive integer, we denote by Nk)
\biggl(
r,
1
f - \alpha
\biggr)
the counting
function of zeros of f - \alpha with multiplicity \leq k and by N(k
\biggl(
r,
1
f - \alpha
\biggr)
the counting function of
zeros of f - \alpha with multiplicity > k. We denote by N=k
\biggl(
r,
1
f - \alpha
\biggr)
counting function of zeros of
f - \alpha which have the multiplicity k . In the same manner we define
\=Nk)
\biggl(
r,
1
f - \alpha
\biggr)
, \=N(k
\biggl(
r,
1
f - \alpha
\biggr)
and \=N=k
\biggl(
r,
1
f - \alpha
\biggr)
,
where in counting the zeros of f - \alpha we ignore the multiplicities.
If g(z) - \alpha (z) = 0 whenever f(z) - \alpha (z) = 0, then we write f = \alpha \Rightarrow g = \alpha (some times we
say f and g share half \alpha IM). Thus f and g share \alpha IM if and only if f = \alpha \leftrightarrow g = \alpha , where
f = \alpha \leftrightarrow g = \alpha means f = \alpha \Rightarrow g = \alpha and g = \alpha \Rightarrow f = \alpha .
On the problems of uniqueness of an entire and its first derivative that share some values. E. Mues
and N. Steinmets (see [5]) proved the following:
Theorem A. If a nonconstant entire function f and its derivative f \prime share two distinct finite
values IM, then f \equiv f \prime .
Li and Yang (see [1]) extended this result as follows:
Theorem B. Let f be a nonconstant entire function and a, b be two distinct complex numbers.
If f = a \Rightarrow f \prime = a and f = b \Rightarrow f \prime = b, then only one of the following cases holds:
(I) f \equiv f \prime ;
(II) if ab \not = 0, then f(z) = a+ ce
b
b - a
z or f(z) = b+ ce
a
a - b
z;
c\bigcirc A. H. H. AL-KHALADI, 2018
978 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS 979
(III) if ab = 0, then f(z) = (a+ b)(ce
1
4
z - 1)2, where c is a nonzero constant.
On the other hand G. Qiu (see [2]) generalized Theorem A to the following:
Theorem C. Let f be a nonconstant entire function, \alpha and \beta be two distinct small functions of
f with \alpha \not \equiv \infty and \beta \not \equiv \infty . If f and f \prime share \alpha and \beta IM, then f \equiv f \prime .
In this paper, we will generalize and extends the above results to obtain the following results:
Theorem 1. Let f be a nonconstant entire function, \alpha and \beta be two distinct small functions of
f with \alpha \not \equiv \infty and \beta \not \equiv \infty . If f = \alpha \Rightarrow f \prime = \alpha and f = \beta \Rightarrow f \prime = \beta , then exactly one of the
following four cases must occur:
(i) f \equiv f \prime ;
(ii) if \alpha \not \equiv \alpha \prime and \beta \not \equiv \beta \prime , then f(z) = \beta + c(\beta - \alpha )e
\int z
0 (\alpha - \alpha \prime
\alpha - \beta
)(t)dt or f(z) = \alpha + c(\beta - \alpha )\times
\times e
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
;
(iii) if \alpha \equiv \alpha \prime and \beta \not \equiv \beta \prime , then f(z) = \alpha + (\beta - \alpha )
\biggl(
1 + ce
1
4
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
\biggr) 2
or if \alpha \not \equiv \alpha \prime and
\beta \equiv \beta \prime , then f(z) = \beta + (\alpha - \beta )
\biggl(
1 + ce
1
4
\int z
0 (\alpha - \alpha \prime
\alpha - \beta
)(t)dt
\biggr) 2
, where c is a nonzero constant;
(iv) if \alpha \equiv \alpha \prime and \beta \equiv \beta \prime , then T (r, f) = N=2
\biggl(
r,
1
f - \alpha
\biggr)
+ S(r, f) = N1)
\biggl(
r,
1
f \prime - \alpha
\biggr)
+
+ S(r, f) = T (r, f \prime ) + S(r, f) = N1)
\biggl(
r,
1
f \prime - \beta
\biggr)
+ S(r, f) = N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r, f).
From Theorem 1, we deduce the following corollaries:
Corollary 1. Let f be a nonconstant entire function, \alpha and \beta be two distinct small functions of
f with \alpha \not \equiv \infty and \beta \not \equiv \infty . If f = \alpha \Rightarrow f \prime = \alpha and f = \beta \Rightarrow f \prime = \beta , and if \alpha \not \equiv \alpha \prime or \beta \not \equiv \beta \prime ,
then f as in Theorem 1 (i) – (iii).
Corollary 2. Let f be a nonconstant entire function, \alpha and \beta be two distinct small functions of
f with \alpha \not \equiv \infty and \beta \not \equiv \infty . If f = \alpha \Rightarrow f \prime = \alpha and f = \beta \Rightarrow f \prime = \beta , and if \=N
\biggl(
r,
1
f - \alpha
\biggr)
=
= \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
+ S(r, f), then f as in Theorem 1 (i) – (iii).
Corollary 3. Let f be a nonconstant entire function, \alpha and \beta be two distinct small functions
of f with \alpha \not \equiv \infty and \beta \not \equiv \infty . If f = \alpha \leftrightarrow f \prime = \alpha and f = \beta \Rightarrow f \prime = \beta , then only one of the
following cases holds:
(i) f \equiv f \prime ;
(ii) if \alpha \not \equiv \alpha \prime and \beta \not \equiv \beta \prime , then f(z) = \beta + c(\beta - \alpha )e
\int z
0 (\alpha - \alpha \prime
\alpha - \beta
)(t)dt
;
(iii) if \alpha \equiv \alpha \prime and \beta \not \equiv \beta \prime , then \alpha \equiv \beta \prime and f(z) = \alpha + (\beta - \beta \prime )(1 + ce
1
4
z)2 or if \alpha \not \equiv \alpha \prime and
\beta \equiv \beta \prime , then f(z) = \beta + (\alpha - \beta )(1 + ce
1
4
\int z
0 (\alpha - \alpha \prime
\alpha - \beta
)(t)dt
)2, where c is a nonzero constant.
Remark 1. If \alpha \equiv a and \beta \equiv b are constants, then Corollary 1 becomes Theorem B. Therefore,
Corollary 1 is generalization Theorem B.
2. If f and f \prime share \alpha IM, then \=N
\biggl(
r,
1
f - \alpha
\biggr)
= \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
and hence the case (iv) in
Theorem 1 is impossible. Therefore Corollary 2 is extension of Theorem C.
3. Also in Theorem 1, if \alpha (z) \equiv z and \beta (z) \equiv 1, then f(z) = 1+(z - 1)ez - 1. That is Theorem
1 is strictly an extension and generalization of Theorems B and C.
4. It is obvious that Corollary 3 is an extension of Theorem C.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
980 A. H. H. AL-KHALADI
2. Some lemmas. For the proof of our results we need the following lemmas:
Lemma 1 [2]. Let f be a nonconstant entire function, \alpha 1 and \alpha 2 be two distinct small
functions of f with \alpha 1 \not \equiv \infty and \alpha 2 \not \equiv \infty . Set
\Delta (f) =
\bigm| \bigm| \bigm| \bigm| f - \alpha 1 \alpha 1 - \alpha 2
f \prime - \alpha \prime
1 \alpha \prime
1 - \alpha \prime
2
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| f - \alpha 2 \alpha 1 - \alpha 2
f \prime - \alpha \prime
2 \alpha \prime
1 - \alpha \prime
2
\bigm| \bigm| \bigm| \bigm| . (2.1)
Then
\Delta (f) \not \equiv 0, (2.2)
m
\biggl(
r,
\Delta (f)f
(f - \alpha 1)(f - \alpha 2)
\biggr)
= S(r, f), (2.3)
m
\biggl(
r,
\Delta (f)
f - \alpha i
\biggr)
= S(r, f), i = 1, 2, (2.4)
2\sum
j=1
N
\biggl(
r,
1
f - \alpha j
\biggr)
- N
\biggl(
r,
1
\Delta (f)
\biggr)
\leq
2\sum
j=1
\=N
\biggl(
r,
1
f - \alpha j
\biggr)
+ S(r, f). (2.5)
Lemma 2 [1]. Let f be a nonconstant meromorphic function and \alpha , \beta , \gamma be small functions
of f with \alpha \not \equiv 0 or \gamma \not \equiv 0. Furthermore, let g = \alpha f2 + \beta f + \gamma . If \=N(r, f) + \=N
\biggl(
r,
1
f
\biggr)
= S(r, f)
and N1)
\biggl(
r,
1
g
\biggr)
= S(r, f), then \beta 2 - 4\alpha \gamma \equiv 0.
Lemma 3 [3, p. 47]. Let f be a nonconstant meromorphic function and a1, a2, a3 be distinct
small functions of f, then
T (r, f) \leq
3\sum
j=1
\=N
\biggl(
r,
1
f - aj
\biggr)
+ S(r, f).
3. Proof of Theorem 1. Suppose that f \not \equiv f \prime and that the auxiliary function
\omega =
\Delta (f)(f - f \prime )
(f - \alpha )(f - \beta )
, (3.1)
where \Delta (f) is defined by (2.1), \alpha 1 = \alpha and \alpha 2 = \beta . From (2.2) we know that \Delta (f) \not \equiv 0. Therefore
it follows that \omega \not \equiv 0. It is easy to see from (2.5) that N(r, \omega ) = S(r, f). By (2.3) we obtain
m(r, \omega ) \leq m
\biggl(
r,
\Delta (f)f
(f - \alpha )(f - \beta )
\biggr)
+m
\biggl(
r, 1 - f \prime
f
\biggr)
= S(r, f).
Thus
T (r, \omega ) = S(r, f). (3.2)
From the fact f = \alpha \Rightarrow f \prime = \alpha , f = \beta \Rightarrow f \prime = \beta and Lemma 3 we know that
T (r, f) \leq \=N
\biggl(
r,
1
f - \alpha
\biggr)
+ \=N
\biggl(
r,
1
f - \beta
\biggr)
+ \=N(r, f) + S(r, f) \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS 981
\leq \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
+ \=N
\biggl(
r,
1
f \prime - \beta
\biggr)
+ S(r, f) \leq
\leq 2T (r, f \prime ) + S(r, f) \leq 2T (r, f) + S(r, f).
It follows that every S(r, f) is also an S(r, f \prime ) and vice versa. From now on we will write S(r) for
the common error term. Let F =
f - \alpha
\beta - \alpha
. Then from (2.1) we get
\Delta (f) = (\beta - \alpha )2F \prime . (3.3)
Substituting F and \Delta (f) into (3.1), \omega is expressed to
\omega =
F \prime [\alpha - \alpha \prime + (\beta - \beta \prime - \alpha + \alpha \prime )F - (\beta - \alpha )F \prime ]
F (F - 1)
, (3.4)
which may also be written F 2 = a1F + a2F
\prime + a3FF \prime + a4F
\prime 2, where T (r, aj) = S(r), j = 1, 2,
3, 4. From the definition of F and the last formula we see that
2T (r, f) + S(r) = 2T (r, F ) = 2m(r, F ) + 2N(r, F ) =
= 2m(r, F ) + S(r) \leq
\leq m(r, F ) +m(r, F \prime ) + 2m
\biggl(
r,
F \prime
F
\biggr)
+ S(r) \leq
\leq m(r, F ) +m(r, F \prime ) + S(r) \leq
\leq 2m(r, F ) + S(r).
That is T (r, F \prime ) = T (r, F ) + S(r) = T (r, f) + S(r). We rewrite (3.4) in the form\biggl[
F - 1
2
- \beta - \beta \prime - \alpha + \alpha \prime
2\omega
F \prime
\biggr] 2
=
=
\Biggl[ \biggl(
\beta - \beta \prime - \alpha + \alpha \prime
2\omega
\biggr) 2
- \beta - \alpha
\omega
\Biggr]
F \prime 2 +
\beta - \beta \prime + \alpha - \alpha \prime
2\omega
F \prime +
1
4
. (3.5)
In the following we shall treat three cases:
Case 1: \alpha \not \equiv \alpha \prime and \beta \not \equiv \beta \prime . Since f = \alpha \Rightarrow f \prime = \alpha and f = \beta \Rightarrow f \prime = \beta , so the zeros of
f - \alpha and f - \beta with multiplicities longer than one are zeros of \alpha - \alpha \prime and \beta - \beta \prime respectively. It
follows that
\=N(2
\biggl(
r,
1
f - \alpha
\biggr)
+ \=N(2
\biggl(
r,
1
f - \beta
\biggr)
= S(r).
From this, (3.1) and (3.2) we deduce that
\=N
\biggl(
r,
1
\Delta (f)
\biggr)
\leq N
\biggl(
r,
1
\omega
\biggr)
+ \=N(2
\biggl(
r,
1
f - \alpha
\biggr)
+ \=N(2
\biggl(
r,
1
f - \beta
\biggr)
= S(r),
and so from (3.3),
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
982 A. H. H. AL-KHALADI
\=N
\biggl(
r,
1
F \prime
\biggr)
+ \=N
\bigl(
r, F \prime \bigr) = S(r).
Appling Lemma 2 to equation (3.5) we find that\biggl(
\beta - \beta \prime + \alpha - \alpha \prime
2\omega
\biggr) 2
-
\Biggl[ \biggl(
\beta - \beta \prime - \alpha + \alpha \prime
2\omega
\biggr) 2
- \beta - \alpha
\omega
\Biggr]
\equiv 0.
That is \omega \equiv (\alpha - \alpha \prime )(\beta - \beta \prime )
\alpha - \beta
. Substituting this into (3.5) gives
\biggl[
F - 1 -
\biggl(
\alpha - \beta
\alpha - \alpha \prime
\biggr)
F \prime
\biggr] \biggl[
F +
\biggl(
\alpha - \beta
\beta - \beta \prime
\biggr)
F \prime
\biggr]
\equiv 0,
which implies that
f(z) = \beta + c(\beta - \alpha )e
\int z
0 (\alpha - \alpha \prime
\alpha - \beta
)(t)dt or f(z) = \alpha + c(\beta - \alpha )e
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
,
where c is a nonzero constant.
Case 2: \alpha \equiv \alpha \prime and \beta \not \equiv \beta \prime or \alpha \not \equiv \alpha \prime and \beta \equiv \beta \prime . Without loss of generality, we can assume
that \alpha \equiv \alpha \prime and \beta \not \equiv \beta \prime . According to the discussion in Case 1 we know that \=N(2
\biggl(
r,
1
f - \beta
\biggr)
=
= S(r), and so from the definition of F we obtain \=N(2
\biggl(
r,
1
F - 1
\biggr)
= S(r). Since the zeros of
f - \alpha are all the zeros of f \prime - \alpha = f \prime - \alpha \prime , it follows that N1)
\biggl(
r,
1
f - \alpha
\biggr)
= S(r). Further, we
can conclude from (3.1) that the zeros of f - \alpha which multiplicity p (\geq 3) are the zeros of \omega . Thus,
from (3.2) we get
\=N(3
\biggl(
r,
1
f - \alpha
\biggr)
\leq N
\biggl(
r,
1
\omega
\biggr)
+ S(r) = S(r).
Thus
\=N
\biggl(
r,
1
f - \alpha
\biggr)
= \=N=2
\biggl(
r,
1
f - \alpha
\biggr)
+ S(r).
From this and the definition of F we get
\=N
\biggl(
r,
1
F
\biggr)
= \=N=2
\biggl(
r,
1
F
\biggr)
+ S(r).
From (3.4) we easily see that the zero of F \prime must be the zero of F with multiplicity 2 if it is not
zero of \omega . Let h =
F
F \prime 2 . Then we have
\=N
\biggl(
r,
1
h
\biggr)
+ \=N(r, h) = S(r). (3.6)
Equation (3.4) can be written as
(\beta - \alpha )(F \prime - \delta F )2 = F
\bigl[ \bigl(
(\beta - \alpha )\delta 2 - \omega
\bigr)
F + \omega
\bigr]
, (3.7)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS 983
where \delta =
\beta - \beta \prime
2(\beta - \alpha )
. If (\beta - \alpha )\delta 2 - \omega \not \equiv 0, then from (3.7), we find that
F (z0) =
\biggl(
- \omega
(\beta - \alpha )\delta 2 - \omega
\biggr)
(z0) \Rightarrow F \prime (z0) =
\biggl(
- \omega \delta
(\beta - \alpha )\delta 2 - \omega
\biggr)
(z0),
and thus h(z0) =
\biggl(
\omega - (\beta - \alpha )\delta 2
\omega \delta 2
\biggr)
(z0). Noting that F (z1) = 1 implies that F \prime (z1) =
=
\biggl(
\beta - \beta \prime
\beta - \alpha
\biggr)
(z1) and thus h(z1) =
\biggl(
\beta - \alpha
\beta - \beta \prime
\biggr) 2
(z1), by Lemma 3, we get
T (r, F ) \leq \=N
\biggl(
r,
1
F - 1
\biggr)
+ \=N
\Biggl(
r,
1
F + \omega
(\beta - \alpha )\delta 2 - \omega
\Biggr)
+ S(r) \leq
\leq \=N
\Biggl(
r,
1
h - ( \beta - \alpha
\beta - \beta \prime )2
\Biggr)
+ \=N
\Biggl(
r,
1
h - \omega - (\beta - \alpha )\delta 2
\omega \delta 2
\Biggr)
+ S(r) \leq
\leq 2T (r, h) + S(r).
Therefore \omega , \alpha and \beta are small functions of h. From the definition of h and equation (3.7), we
obtain \biggl(
hF \prime - \beta - \beta \prime
2\omega
\biggr) 2
= h+
(\beta - \beta \prime )2
4\omega 2
- \beta - \alpha
\omega
.
Therefore h +
(\beta - \beta \prime )2
4\omega 2
- \beta - \alpha
\omega
has no simple zero. Hence by Lemma 2, we get
(\beta - \beta \prime )2
4\omega 2
-
- \beta - \alpha
\omega
\equiv 0. That is \omega \equiv (\beta - \beta \prime )2
4(\beta - \alpha )
. Thus (3.7) becomes
\biggl(
1
\delta
F \prime - F
\biggr) 2
= F. (3.8)
Let G =
1
\delta
F \prime - F. We get F = G2 and thus F \prime = 2GG\prime . From (3.8) we have
\biggl(
2
\delta
G\prime - G
\biggr) 2
\equiv 1.
Hence either
2
\delta
G\prime - G \equiv 1 or
2
\delta
G\prime - G \equiv - 1. If
2
\delta
G\prime - G \equiv - 1, then we find that f = \alpha - (\beta - \alpha )(1+
+ce
1
2
s), where s =
\int z
0 \delta (t)dt and c is a nonzero constant. From this and f = \beta \Rightarrow f \prime = \beta we arrive
at a contradiction. Therefore
2
\delta
G\prime - G \equiv 1. From this it is easy to see that f = \alpha +(\beta - \alpha )(1+ce
1
2
s)2,
where s =
\int z
0 \delta (t)dt and c is a nonzero constant.
Case 3: \alpha \equiv \alpha \prime and \beta \equiv \beta \prime . By the discussion in Case 2 we know that
N1)
\biggl(
r,
1
f - \alpha
\biggr)
+N1)
\biggl(
r,
1
f - \beta
\biggr)
= S(r) (3.9)
and
N(3
\biggl(
r,
1
f - \alpha
\biggr)
+N(3
\biggl(
r,
1
f - \beta
\biggr)
\leq 3N
\biggl(
r,
1
\omega
\biggr)
\leq 3T (r, \omega ) +O(1) = S(r). (3.10)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
984 A. H. H. AL-KHALADI
From \alpha \equiv \alpha \prime , \beta \equiv \beta \prime and (3.4) we deduce that
m
\biggl(
r,
1
F \prime
\biggr)
\leq m
\biggl(
r,
1
\omega
\biggr)
+m
\biggl(
r,
F \prime
F (F - 1)
\biggr)
+ S(r) \leq
\leq T (r, \omega ) +m
\biggl(
r,
F \prime
F
\biggr)
+m
\biggl(
r,
F \prime
F - 1
\biggr)
+ S(r) = S(r),
so that
m
\biggl(
r,
1
f - \alpha
\biggr)
+m
\biggl(
r,
1
f - \beta
\biggr)
= m
\biggl(
r,
1
F
\biggr)
+m
\biggl(
r,
1
F - 1
\biggr)
+ S(r) =
= m
\biggl(
r,
1
F (F - 1)
\biggr)
+ S(r) \leq
\leq m
\biggl(
r,
1
F \prime
\biggr)
+ S(r) = S(r).
Combining this, (3.9) and (3.10) we obtain
T (r, f) = N=2
\biggl(
r,
1
f - \alpha
\biggr)
+ S(r) = N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r). (3.11)
Set
\Gamma = 2
f \prime \prime - \beta
f \prime - \beta
- f \prime - \beta
f - \beta
. (3.12)
Since \beta \equiv \beta \prime , m(r,\Gamma ) = S(r). It follows from (3.12) that if z\beta is a zero of f - \beta with multiplicity
2, then \Gamma (z\beta ) = O(1). Thus, from (3.11) we get
N(r,\Gamma ) \leq \=N
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r). (3.13)
Also, if z\alpha is a zero of f - \alpha with multiplicity 2, then\bigl[
2(f \prime \prime - \beta ) - (\Gamma + 1)(\alpha - \beta )
\bigr]
(z\alpha ) = 0. (3.14)
On the other hand, differentiating (3.1) twice and then using f(z\alpha ) = \alpha , we arrive at [2(f \prime \prime - \alpha ) -
- \omega ](z\alpha ) = 0. If we now eliminate f \prime \prime (z\alpha ) between this and (3.14) we obtain
[\omega - (\Gamma - 1)(\alpha - \beta )] (z\alpha ) = 0. (3.15)
Set
\Omega = 2
f \prime \prime - \alpha
f \prime - \alpha
- f \prime - \alpha
f - \alpha
.
Similarly as the above, we have m(r,\Omega ) = S(r),
N(r,\Omega ) \leq \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
- \=N=2
\biggl(
r,
1
f - \alpha
\biggr)
+ S(r)
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ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS 985
and
[\omega - (\Omega - 1)(\alpha - \beta )] (z\beta ) = 0.
We discuss the following four subcases:
Subcase 3.1: \omega - (\Gamma - 1)(\alpha - \beta ) \equiv 0 and \omega - (\Omega - 1)(\alpha - \beta ) \equiv 0. Then \Gamma \equiv \Omega . Hence\biggl(
f \prime - \beta
f \prime - \alpha
\biggr) 2
= c
\biggl(
f - \beta
f - \alpha
\biggr)
, where c is a nonzero constant. Therefore 2T (r, f \prime ) + S(r) = T (r, f).
This is impossible because f is an entire function.
Subcase 3.2: \omega - (\Gamma - 1)(\alpha - \beta ) \not \equiv 0 and \omega - (\Omega - 1)(\alpha - \beta ) \not \equiv 0. Then from (3.15), (3.1)
and (3.13) we deduce that
\=N=2
\biggl(
r,
1
f - \alpha
\biggr)
\leq N
\biggl(
r,
1
\omega - (\Gamma - 1)(\alpha - \beta )
\biggr)
+ S(r) \leq
\leq T (r, \omega ) + T (r,\Gamma ) + S(r) =
= N(r,\Gamma ) + S(r) \leq
\leq \=N
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r).
Together with (3.11) we have
T (r, f) \leq \=N
\biggl(
r,
1
f \prime - \beta
\biggr)
+ S(r) \leq T (r, f \prime ) + S(r) \leq T (r, f) + S(r). (3.16)
Consequently,
N(2
\biggl(
r,
1
f \prime - \beta
\biggr)
+m
\biggl(
r,
1
f \prime - \beta
\biggr)
= S(r). (3.17)
Similarly, from \omega - (\Omega - 1)(\alpha - \beta ) \not \equiv 0 we get
N(2
\biggl(
r,
1
f \prime - \alpha
\biggr)
+m
\biggl(
r,
1
f \prime - \alpha
\biggr)
= S(r).
From this, (3.11), (3.16) and (3.17) we arrive at the conclusion (iv).
Subcase 3.3: \omega - (\Gamma - 1)(\alpha - \beta ) \not \equiv 0 and \omega - (\Omega - 1)(\alpha - \beta ) \equiv 0. Since \omega - (\Gamma - 1)(\alpha - \beta ) \not \equiv 0,
by the discussion in Subcase 3.2 we have (3.16) and (3.17). From \omega - (\Omega - 1)(\alpha - \beta ) \equiv 0, we find
that \=N(2
\biggl(
r,
1
f \prime - \alpha
\biggr)
= S(r) and
f \prime = \alpha \Rightarrow f = \alpha . (3.18)
Otherwise f \prime (z0) = \alpha \Rightarrow f(z0) \not = \alpha holds for a sequence z0 whose counting function is an S(r).
We set
\nu =
\Delta (f \prime )(f - f \prime )
(f \prime - \alpha )(f \prime - \beta )
. (3.19)
From (2.1) we conclude that \nu \not \equiv 0. Again by (2.4), (3.11), (3.17) and (3.16) we see that
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
986 A. H. H. AL-KHALADI
m(r, \nu ) = m
\left( r,
\Delta (f \prime )
\Bigl(
f - \beta
f \prime - \beta - 1
\Bigr)
f \prime - \alpha
\right) \leq m
\biggl(
r,
\Delta (f \prime )
f \prime - \alpha
\biggr)
+m
\biggl(
r,
f - \beta
f \prime - \beta
\biggr)
+ S(r) \leq
\leq m
\biggl(
r,
f - \beta
f \prime - \beta
\biggr)
+ S(r) = N
\biggl(
r,
1
f - \beta
\biggr)
- N
\biggl(
r,
1
f \prime - \beta
\biggr)
+ S(r) =
= T (r, f) - T (r, f \prime ) + S(r) = S(r). (3.20)
From (3.19), (2.5), (3.18), (3.11), (3.17) we deduce that
N(r, \nu ) \leq \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
- \=N
\biggl(
r,
1
f - \alpha
\biggr)
+ \=N
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N
\biggl(
r,
1
f - \beta
\biggr)
=
= N1)
\biggl(
r,
1
f \prime - \alpha
\biggr)
- \=N=2
\biggl(
r,
1
f - \alpha
\biggr)
+N1)
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N=2
\biggl(
r,
1
f - \beta
\biggr)
+
+S(r) = N1)
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r). (3.21)
Let z1 be a common zero of f - \alpha (or f - \beta ) and f \prime - \alpha (or f \prime - \beta ) with multiplicities 2 and 1
respectively. From (3.1) and (3.19) it follows that (\omega - 2\nu )(z1) = 0. If \omega - 2\nu \not \equiv 0, then from (3.2),
(3.20), (3.21), (3.17), (3.11) and (3.16) we conclude that
\=N=2
\biggl(
r,
1
f - \alpha
\biggr)
+ \=N=2
\biggl(
r,
1
f - \beta
\biggr)
\leq N
\biggl(
r,
1
\omega - 2\nu
\biggr)
\leq T (r, \omega ) + T (r, \nu ) +O(1) =
= m(r, \nu ) +N(r, \nu ) + S(r) =
= N1)
\biggl(
r,
1
f \prime - \beta
\biggr)
- \=N=2
\biggl(
r,
1
f - \beta
\biggr)
+ S(r) \leq
\leq T (r, f \prime ) - 1
2
T (r, f) + S(r) =
=
1
2
T (r, f) + S(r).
That is 2T (r, f) \leq T (r, f) + S(r), a contradiction. Therefore we have \omega - 2\nu \equiv 0 . From this it is
easy to arrive at the contradiction.
Subcase 3.4: \omega - (\Gamma - 1)(\alpha - \beta ) \equiv 0 and \omega - (\Omega - 1)(\alpha - \beta ) \not \equiv 0. Similarly as the Subcase
3.3, we will arrive at the same contradiction.
Theorem 1 is proved.
4. Proof of corollaries. We proof only Corollary 3; proofs of the remaining corollaries are easy.
If \alpha \not \equiv \alpha \prime and \beta \not \equiv \beta \prime , then f(z) = \alpha + c(\beta - \alpha )e
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
. By differentiating both sides of this
last function with respect to z, we obtain
f \prime (z) - \alpha = c(\beta - \alpha \prime )
\Biggl(
e
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
+
\alpha \prime - \alpha
c(\beta - \alpha \prime )
\Biggr)
.
But this is a contradiction to our assumption that f = \alpha \leftrightarrow f \prime = \alpha . If \alpha \equiv \alpha \prime and \beta \not \equiv \beta \prime , then
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
ENTIRE FUNCTIONS SHARE TWO HALF SMALL FUNCTIONS 987
f(z) = \alpha + (\beta - \alpha )
\biggl(
1 + ce
1
4
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
\biggr) 2
.
Differentiating once gives
f \prime (z) - \alpha =
\biggl(
1 + ce
1
4
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
\biggr) \biggl(
\beta \prime - \alpha + c[\beta \prime - \alpha + 1/2(\beta - \beta \prime )]e
1
4
\int z
0 (\beta - \beta \prime
\beta - \alpha
)(t)dt
\biggr)
.
Since f = \alpha \leftrightarrow f \prime = \alpha , we have either \beta \prime - \alpha + 1/2(\beta - \beta \prime ) \equiv 0 or \beta \prime - \alpha \equiv 0. If \beta \prime - \alpha +
+ 1/2(\beta - \beta \prime ) \equiv 0, then we can write
f(z) - \alpha = (\beta - \alpha )
\biggl(
1 +
c\surd
\beta - \alpha
\biggr) 2
.
This is impossible. Therefore \beta \prime - \alpha \equiv 0, in this case f(z) = \alpha + (\beta - \beta \prime )
\Bigl(
1 + ce
1
4
z
\Bigr) 2
. Finally,
if f = \alpha \leftrightarrow f \prime = \alpha , then it is clear that \=N
\biggl(
r,
1
f - \alpha
\biggr)
= \=N
\biggl(
r,
1
f \prime - \alpha
\biggr)
. Thus the case (iv) in
Theorem 1 does not appear. Now complete the proof of Corollary 3.
References
1. Li P., Yang C. C. When an entire function and its linear differential polynomial share two values // Illinois J. Math. –
2000. – 44. – P. 349 – 362.
2. Qiu G. Uniqueness of entire functions that share some small functions // Kodai Math. J. – 2000. – 23. – P. 1 – 11.
3. Hayman W. K. Meromorphic functions. – Oxford: Clarendon Press, 1964.
4. Yang C. C., Yi H. X. Uniqueness theory of meromorphic functions. – Kluwer Acad. Publ., 2004.
5. Mues E., Steinmets N. Meromorphe funktionen die mit ihrer ableitung werte teiken // Manuscripta Math. – 1979. –
29. – P. 195 – 206.
Received 21.10.13
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
|
| id | umjimathkievua-article-1610 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:05Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2b/21f1a1eb0c72e691055420eb0b8e132b.pdf |
| spelling | umjimathkievua-article-16102019-12-05T09:20:38Z Entire functions share two half small functions Цiлi функцiї подiляють двi напiвмалi функцiї Al-Khaladi, A. H. H. Аль-Халаді, А. Х. Х. The paper generalizes a result by P. Li and C. C. Yang [Illinois J. Math. – 2000. – 44. – P. 349 – 362] and extends the previous work of G. Qiu [Kodai Math. J. – 2000. – 23. – P. 1 – 11]. У роботi узагальнено результат П. Лi та Ц. Ц. Янга [Illinois J. Math. – 2000. – 44. – P. 349 – 362] та розширено результати попередньої роботи Г. Кiу [Kodai Math. J. – 2000. – 23. – P. 1 – 11]. Institute of Mathematics, NAS of Ukraine 2018-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1610 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 7 (2018); 978-987 Український математичний журнал; Том 70 № 7 (2018); 978-987 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1610/592 Copyright (c) 2018 Al-Khaladi A. H. H. |
| spellingShingle | Al-Khaladi, A. H. H. Аль-Халаді, А. Х. Х. Entire functions share two half small functions |
| title | Entire functions share two half small functions |
| title_alt | Цiлi функцiї подiляють двi напiвмалi функцiї |
| title_full | Entire functions share two half small functions |
| title_fullStr | Entire functions share two half small functions |
| title_full_unstemmed | Entire functions share two half small functions |
| title_short | Entire functions share two half small functions |
| title_sort | entire functions share two half small functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1610 |
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