R E V I E W
Open Access
On difference equations concerning
Schwarzian equation
Shuang-Ting Lan
1and Zong-Xuan Chen
2**Correspondence: [email protected]
2School of Mathematical Sciences, South China Normal University, Guangzhou, China
Full list of author information is available at the end of the article
Abstract
Consider the difference equation
3f(z)
f(z) –
3 2
2f(z)
f(z)
2k
= P(z,f(z))
Q(z,f(z)),
whereP(z,f) andQ(z,f) are prime polynomials inf(z) with degfP=p, degfQ=q, and d= max{p,q}> 0. We give the supremum ofd, an estimation of the sum of
Nevanlinna exceptional values of meromorphic solutionf(z) of the equation, and study the value distributions of their difference
f(z) and divided difference f(fz()z).
MSC: 30D35; 34A20
Keywords: meromorphic solution; difference; Nevanlinna exceptional value
1 Introduction and main results
In this paper, we use the basic notions of Nevanlinna theory, such asT(r,f),m(r,f),N(r,f),
and so on; see [–]. LetS(r,f) denote any quantity satisfyingS(r,f) =o(T(r,f)) for allr
outside a set of finite logarithmic measure. We callf an admissible solutionof a difference
(or differential) equation if all coefficientsα of the equation satisfyT(r,α) =S(r,f). In
addition, we denote byσ(f) the order of growth of a meromorphic functionf(z) and by
λ(f) andλ(f), respectively, the exponents of convergence of zeros and poles off(z), which
are defined by
σ(f) = lim
r→∞
logT(r,f)
logr , λ(f) =rlim→∞
logN(r,f)
logr , λ
f
=lim
r→∞
logN(r,f)
logr .
Ifλ(f –a) <σ(f), thenais called a Borel exceptional value off.
Fora∈C∪ {∞}, we denote byδ(a,f) the deficiency ofatof(z), which is defined by
δ(a,f) = lim
r→∞
m(r, f–a)
T(r,f) (ifa∈C), δ(∞,f) =rlim→∞
m(r,f) T(r,f).
Obviously,δ(a,f)≥. Ifδ(a,f) > , thenais called a Nevanlinna exceptional value off.
The forward differencesnf(z),n∈N+, are defined in the standard way [] by
f(z) =f(z+ ) –f(z), n+f(z) =nf(z+ ) –nf(z).
Ishizaki [] studied Schwarzian differential equations and obtained the following:
Theorem A Letα,α, . . . ,αsbe distinct constants.If
f f
–
f f
=f
f –
f f
k
= P(z,f) Q(z,f)
possesses an admissible solution,then we have
s
j=
δ(αj,f)≤ – d k,
where d=max{degfP(z,f),degfQ(z,f)}.
Chen and Li [] investigated difference equations and obtained the following theorem, which can be regarded as a difference analogue of Theorem A.
Theorem B Let f(z)be an admissible solution of the difference equation
f(z)
f(z) –
f(z)
f(z)
k
=Rz,f(z)= P(z,f(z))
Q(z,f(z)) (.)
such thatσ(f) < ,where k (≥)is an integer, P(z,f)and Q(z,f)are polynomials with
degfP(z,f) =p,degfQ(z,f) =q,and d=max{p,q}.Letα, . . . ,αsbe s(≥)distinct complex constants.Then
s
j=
δ(αj,f)≤ – q
k. (.)
In particular,if N(r,f) =S(r,f),then
s
j=
δ(αj,f)≤ – d
k. (.)
Lan and Chen [] studied the value distribution of transcendental meromorphic solu-tions and their differences of some difference equation concerning a Schwarzian equation.
From Theorem B we do a rough calculation: iff(z) has two finite Borel exceptional
val-ues, (.) shows thatq≤k; iff(z) has only one finite Borel exceptional value, (.) shows
q≤k((.) shows thatd≤k); iff(z) has no finite Nevanlinna exceptional values, (.)
shows thatq≤k((.) shows thatd≤k). The upper bound ofdseems to bear onk. We
ask the following interesting questions: what is the supremum ofd? whether inequalities
(.) and (.) can be improved? Once we fixdandk, can we get the number of
Nevan-linna exceptional values of meromorphic solutions of (.)? In this paper, we answer these questions and obtain the following theorem.
(i) d≤k;
(ii) sj=δ(αj,f)≤–dk;
(iii) ifN(r,f) =S(r,f),thenq=d≤k,andsj=δ(αj,f)≤ –dk. Example . The functionf(z) =ztanπz satisfies the difference equation
f(z)
f(z) –
f(z)
f(z)
=P(z,f) Q(z,f),
where
P(z,f) = f+ (z+ )f+z+ z+ f+zz+ z+ f
+ zz+ z+ f+ zz+ z+ f+zz– z– f
+ z(z+ ),
Q(z,f) = –f(f+z)f+f+z+z.
We see thatk= andp=q= . So,d=max{p,q}= = k. Obviously,f(z) =ztanπz
has
no Nevanlinna exceptional value, that is, for any distinct constantsαj(j= , . . . ,s), we have
s
j=δ(αj,f) = =– d k.
Example . The functionf(z) =ez+zsatisfies the difference equation
f(z)
f(z) –
f(z)
f(z)
=P(z,f) Q(z,f),
where
P(z,f) = –(e– )f+ (e– )(e– )z+ (e– )z+e– f
– (e– )z– (e– )z– (e– )z– (e– )(e– )z– ,
Q(z,f) = (e– )f+( +e)z+ ( –e)z+ .
We see thatk= andp=q= . So,d=max{p,q}= = k. Obviously,f(z) has no finite
Nevanlinna exceptional value, that is, for any distinct constantsαj(j= , . . . ,s), we have
s
j=δ(αj,f) = = –dk. Remark .
(i) Comparing Theorem . and Theorem B, we see that the result of Theorem . is better than that of Theorem B.
(ii) Under conditions of Theorem .(iii), the result shows thatsj=δ(αj,f)≤ –dk< ,
that is,f(z)has no finite Borel exceptional value.
(iii) Examples . and . show the equalities in Theorem . can be reached. So, the result of Theorem . is precise.
In general,λ(f)≤σ(f), wheref(z) is a meromorphic function. For example, iff(z) =
ez+ andf
f(z) = (e– )ez+ satisfiesλ(f) =σ(f). Even for a meromorphic function of small
lower order,f(z) may have finitely many zeros. To show this, Bergweiler and Langley []
constructed a meromorphic function like this and obtained the following:
Theorem C Let φ(r) be a positive nondecreasing function on [,∞) that satisfies
limr→∞φ(r) =∞.Then there exists a transcendental and meromorphic function f in the
plane with
lim
r→∞
T(r,f)
r <∞, rlim→∞ T(r,f)
φ(r)logr<∞
such that g(z) =f(z)has only one zero.Moreover,the function g satisfies
lim
r→∞
T(r,g)
φ(r)logr<∞.
Many authors tried hard to investigate zeros of difference and divided difference of meromorphic solutions of difference equation and obtained profound results (see [, ,
], etc.). Fixing the connection ofdandk, we get the number of Nevanlinna exceptional
values of a meromorphic solutionf of (.) and discover some very good properties off.
These results are stated as follows.
Theorem . Suppose that f(z)is an admissible meromorphic solution of difference
equa-tion(.)of finite order,where k∈N+and d=max{deg
fP,degfQ}= k.Then
(i) f(z)has no Nevanlinna exceptional value; (ii) λ(f) =λ(f) =σ(f),λ(ff) =λ(f
f
) =σ(f); (iii) T(r,f) = T(r,f) +S(r,f).
Setd= kin Theorem .(iii). By the proof of Theorem .(iii) we easily obtain the
fol-lowing corollary.
Corollary . Suppose that f(z)is an admissible entire solution of difference equation(.) of finite order,where k∈N+and d=max{deg
fP,degfQ}= k.Then
(i) f(z)has no finite Nevanlinna exceptional value; (ii) T(r,f) =T(r,f) +S(r,f).
The following Example . satisfies the conditions and results of Theorem ..
Example . In Example ., we see thatf(z) =ztanπz
is also a transcendental
meromor-phic solution of finite order of the equation
f(z+ )f(z– ) = –z.
By the following Lemma .,f(z) has no Nevanlinna exceptional value, and
λ(f) =λ
f
=σ(f),λ
f f
=λ
f f
By calculation we have
f(z) =ztan
πz +tan
πz +z+
–tanπz .
Thus,
T(r,f) = T
r,tanπz
+S
r,tanπz
= T(r,f) +S(r,f).
2 Lemmas for the proofs of theorems
Lemma .([, ]) Let f(z)be a meromorphic function of finite orderσ,and letηbe a nonzero complex constant.Then,for eachε( <ε< ),we have
m
r,f(z+η) f(z)
+m
r, f(z) f(z+η)
=Orσ–+ε.
Lemma .([]) Let f(z)be a meromorphic function with exponent of convergence of poles
λ(
f) =λ<∞,and letη= be fixed.Then,for eachε( <ε< ), Nr,f(z+η)=Nr,f(z)+Orλ–+ε+O(logr).
Lemma .([]) Let f(z)be a meromorphic function of orderσ=σ(f),σ<∞,and letη
be a fixed nonzero complex number.Then,for eachε> , Tr,f(z+η)=Tr,f(z)+Orσ–+ε+O(logr).
Lemma .([], Lemma .) Suppose that f(z)is a nonconstant meromorphic function in
|z|<R.Let aj(j= , , . . . ,q)be distinct finite complex numbers.Then,for <r<R,we have
m
r, q
j= f –aj
= q
j= m
r, f –aj
+O().
Lemma .([] (Valiron-Mohon’ko)) Let f(z)be a meromorphic function.Then,for all irreducible rational functions in f,
Rz,f(z)= an(z)f(z)
n+· · ·+a (z) bm(z)f(z)m+· · ·+b
(z)
with meromorphic coefficients ai(z)and bj(z)being small with respect to f,the characteristic function of R(z,f(z))satisfies
Tr,Rz,f(z)=max{m,n}T(r,f) +S(r,f).
Lemma .([]) Suppose that h(z)is a nonconstant rational function.If w(z)is a tran-scendental meromorphic solution of finite order of the equation
w(z+ )w(z– ) =h(z).
(i) w(z)has no Nevanlinna exceptional value;
(i) g(r)≤h(r)outside of an exceptional set of finite linear measure,or
(ii) g(r)≤h(r),r∈/H∪(, ],whereH⊂(,∞)is a set of finite logarithmic measure. Then,for anyα> ,there existsr> such thatg(r)≤h(αr)for allr>r. From Lemma . we easily obtain the following:
Lemma . Let g(r)be a nondecreasing function,and E be a set of finite logarithmic mea-sure or finite linear meamea-sure.Then
lim
3 Proofs of theorems
Proof of Theorem. (i) We first prove thatd≤k. Lemma . shows that
Combining this with Lemma ., we have
T(r,R) =m(r,R) +N(r,R) =N(r,R) +S(r,f) =dT(r,f) +S(r,f). (.)
Rewrite (.) in the form
Similarly, ifzis a pole off(z)and not a pole off(z+ ),f(z+ ), orf(z+ ), thenzcannot
Combining this with Lemma ., we obtain
≤T(r,Q) +S(r,f)
Combining this with Lemma ., we obtain
So,
s
j=
δ(αj,f)≤ – d
k.
Proof of Theorem. (i) Using the same method as in Theorem .(i), we still have
(.)-(.). By the factd= kand (.) we have
kN(r,f) + kN
r,
f(z)
= kT(r,f) + kTr,f(z)+S(r,f) = kT(r,f) +S(r,f).
From this it follows that
N(r,f) =T(r,f) +S(r,f) (.)
and
N
r,
f(z)
=Tr,f(z)+S(r,f) = T(r,f) +S(r,f). (.)
By (.) we see that
m(r,f) =T(r,f) –N(r,f) =S(r,f), (.)
Hence,
m(r,f) =oT(r,f), r∈/E,
whereE⊂(,∞) is a set of finite logarithmic measure. Therefore,
δ(∞,f) = lim
r→∞
m(r,f) T(r,f) ≤rlim→∞
r∈/E
m(r,f) T(r,f)=rlim→∞
r∈/E
o(T(r,f)) T(r,f) = .
So,δ(∞,f) = .
We deduce from (.) that
m
r,
f(z)
=Tr,f(z)–N
r,
f(z)
=S(r,f). (.)
For any givena∈C, by (.) and Lemma . we have
m
r, f(z) –a
≤m
r, f(z) f(z) –a
+m
r,
f(z)
=S(r,f),
that is,
m
r, f(z) –a
whereE⊂(,∞) is a set of finite logarithmic measure. Hence,
δ(a,f) = lim
r→∞
m(r,f(z)–a)
T(r,f) ≤rlim→∞ r∈/E
m(r,f(z)–a)
T(r,f) =rlim→∞ r∈/E
o(T(r,f)) T(r,f) = .
So,δ(a,f) = .
Thus, for anya∈C∪ {∞}, we haveδ(a,f) = . So,f(z) has no Nevanlinna exceptional
value.
(ii) By (.) we easily obtain thatλ(f) =σ(f) =σ(f). By (.) and Lemma . we see
m(r,f)≤m
r,f f
+m(r,f) =S(r,f).
Together with (.), we have
N(r,f) =T(r,f) –m(r,f) =T(r,f) +S(r,f) = T(r,f) +S(r,f) or
N(r,f) = +o()T(r,f), r∈/E,
whereE⊂(,∞) is a set of finite logarithmic measure.
By Lemmas . and . we see
λ
f
= lim
r→∞
logN(r,f)
logr =rlim→∞ r∈/E
logN(r,f)
logr
= lim
r→∞ r∈/E
log( +o())T(r,f)
logr =rlim→∞
logT(r,f)
logr =σ(f). So,λ(f) =σ(f).
Finally, we prove thatλ(ff) =λ(f f
) =σ(f). Setg(z) = ff(z(z)). Then
⎧ ⎪ ⎨ ⎪ ⎩
f(z+ ) = (g(z) + )f(z),
f(z+ ) = (g(z+ ) + )f(z+ ) = (g(z+ ) + )(g(z) + )f(z),
f(z+ ) = (g(z+ ) + )f(z+ ) = (g(z+ ) + )(g(z+ ) + )(g(z) + )f(z).
Substituting this into (.), we have
R(z,f) =
(g(z) + )((g(z+ ) + )(g(z+ ) + ) – )
g(z) –
g(z+ )(g(z) + )
g(z)
k .
Combining this with (.) and Lemma ., we obtain
dT(r,f) =Tr,R(z,f)+S(r,f)
≤MTr,g(z)+Tr,g(z+ )+Tr,g(z+ )+S(r,f)
= MTr,g(z)+S(r,g) +S(r,f), (.)
By the definition ofS(r,f) andS(r,g), (.) can be rewritten as
d+o()T(r,f)≤M+o()T(r,g), r∈/E,
whereE⊂(,∞) is a set of finite logarithmic measure.
Applying Lemmas . and . to the last inequality, we have
σ(f) = lim
from which it follows that
N
By the definition ofS(r,ff), the last equality shows that
By (.), (.), and (.) we see that
Applying Lemmas . and . to (.), we obtain
λ
The authors declare that they have no competing interests. Authors’ contributions
Both authors contributed equally to the manuscript and jointly worked on the results. Both authors typed, read, and approved the final manuscript.
Author details
1Department of Mathematics, Guangzhou Civil Aviation College, Guangzhou, China.2School of Mathematical Sciences, South China Normal University, Guangzhou, China.
Acknowledgements
The authors would like to thank the referee for valuable suggestions to the present paper. This research was supported by the Natural Science Foundation of Guangdong Province in China Nos. 2016A030310106, 2014A030313422.
Received: 5 July 2016 Accepted: 28 November 2016 References
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