Unicity of meromorphic functions and their linear differential polynomials

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R E S E A R C H

Open Access

Unicity of meromorphic functions and their

linear differential polynomials

Chenguang Wang

1*

_{and Qi Han}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Jining University, Qufu, Shandong 273155, P.R. China

Full list of author information is available at the end of the article

Abstract

This paper studies the unicity of meromorphic functions that share an arbitrary nonzero small function with their general linear diﬀerential polynomials. The result derived here extends one by Brück in 1996 and others.

MSC: 30D35; 30D30

Keywords: Nevanlinna theory; meromorphic functions; small functions; entire functions; diﬀerential polynomials

1 Introduction

In , R Nevanlinna [–] proved his celebratedfive-valueandfour-valuetheorems, which gave birth to the study of the unicity of meromorphic functions in the open com-plex planeC; in , by considering the unicity of an entire functionf and its first order derivativef, Rubel and Yang [] showed thatf≡fwhen they share two distinct, finite val-uesCM, which was generalized to meromorphic functions independently by Gundersen [] as well as by Mues and Steinmetz [] in . Since these results, there have been many papers on several related problems aboutf and its derivatives, and we refer the reader to [, –] and the references therein for more details.

Before we proceed, we spare the reader for a moment and assume the familiarity with the basics of Nevanlinna’s value distribution theory of meromorphic functions inCsuch as theﬁrstandsecondmain theorems, and the common notations such as the charac-teristic function T(r,f), theproximity function m(r,f) and thecounting functions N(r,f) (with multiplicities) andN¯(r,f) (without multiplicities);S(r,f) denotes any quantity satis-fyingS(r,f) =o(T(r,f)) asr→ ∞except possibly on a set of ﬁnite Lebesgue measure, not necessarily the same at each occurrence.

Leta,f,g be some meromorphic functions onC.ais said to be a small function tof, providedT(r,a) =S(r,f). Givena, a small function to bothf andgor some value inC∪ {∞}, one says thatf andgsharea CMprovided the two meromorphic functionsf–aand

g–ahave the same zeros with counting multiplicities.

Now, we continue our discussion and observe a question that arises naturally: When onlyf andfshare one valueCM, then what can one have accordingly? Towards this di-rection, in , Brück [] ﬁrst proved the following result.

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Theorem A Let f be a nonconstant entire function with N(r,_f) =S(r,f).When f and f

share a ﬁnite,nonzero value a CM,then there is a constant c= such that f–a

f–a=c. ()

This interesting result was extended by Zhang [] in , when he proved, among some related results, the following one.

Theorem B Let f be a nonconstant meromorphic function such that

¯

N(r,f) +N

r, 

f

<λ+o()Tr,f forλ∈

, 

. ()

When f and fshare a ﬁnite,nonzero value a CM,then we have().

Quite recently, via replacing the ﬁnite, nonzero valueaby a small functiona(z) related tof, Al-Khaladi [, ] obtained the following striking result.

Theorem C Let f be a nonconstant meromorphic function such that

¯

N(r,f) +N¯

r, 

f

<λ+o()Tr,f forλ∈

, 

. ()

When f and fshare a small function a(z) (≡,∞)CM,then we have_ff–_–a_a=  –_ac,where c

is a constant such that c_a= and N(r,  –c_a) =S(r,f).

Remark D There are reasons why the condition on the zeros offis posed by Brück []: To control the multiple values off for each ﬁnite value, inspired by the exponential func-tion; yet, it is not natural to generalize this condition by the zeros off(k) _when_k_≥_{ as}

there is no need to control the multiple values off.

Next, we deﬁne a linear diﬀerential polynomial of orderk≥, such as

L(f) :=af+af+· · ·+akf(k) (ak≡), ()

associated withf, whereajare small functions off forj= , , . . . ,k. Then we can prove

the following theorems, which are the main results of this paper.

Theorem  Let f be a nonconstant meromorphic function satisfying()withλ∈(,_k_+ ).

When f andL(f)share a small function a(z) (≡,∞)CM,then we have_Lf₍–_f_)–a_a=  –_ac for a constant c with c_a= and N(r,  –c_a) =S(r,f).

Theorem  Under the assumptions of Theorem,we further suppose that f,a(z) (≡,∞)

and aj,j= , , . . . ,k,are all entire,then we have either f =L(f),or_Lf₍–_f_)–a_a=c for a constant

c= and a(z)must also be a constant as well.

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was not suitably cited in the key reference of [] (not the paper [] itself ), despite the conclusions as well as the methods being very similar.

It is worth to mention that the machinery used in this paper is standard Nevanlinna’s value distribution theory while the methods involved are combinations of those already applied in Al-Khaladi [, ] as well as in Han and Yi [].

2 Proofs of the main results

This section is devoted to the detailed proofs of Theorems  and . We ﬁrst deﬁne the following auxiliary functionαas

α:= f–a

L–a. ()

Here and hereafter, we useLforL(f) for brevity. Our hypotheses implyαis such a mero-morphic function thatT(r,α) =O(T(r,f)) yetN(r,α) =S(r,f).

Case I.ais a ﬁnite, nonzero constant.

Rewrite () asf–a=αL–aαand diﬀerentiate it to yield

f= (αL)–aα=αβf–γ, () where we introduced two meromorphic functionsβ:=(α_αL_f) andγ :=aα

α.

Notice thatα≡. We get easily

_

α=β–γ

 f,

–_αα =β+γ

f (f) –γ_f.

()

Substitute _αfrom the top line to the bottom line in () to derive

γ 

f–β

α

α =β

₊_γ f

(f) –γ 

f. ()

That is, for the meromorphic functionδ:=γ(α

α +

γ

γ –

f

f), one has

δ

f=β

₊_βα

α. ()

Now, whenγ ≡, then we seeαis a constant, so that_Lf₍_f–_)–a_a=cfor a constantc=α= . As a consequence, we assume in the following thatγ ≡.

Next, ifδ≡, thenα

α +

γ

γ –

f

f ≡; usingαγ=aα, one derives

αγ =aα=cf. ()

Here,c=  is a constant. Hence, it follows thatN(r,f) =O(N(r,α)) =S(r,f), which

fur-ther yieldsN(r,_α)≤kN¯(r,f) +S(r,f) =S(r,f). On the other hand, combining () and () provides us with the following identity:

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Ifc= –, thenαL=cfor a constantc= ; yet, since nowf= –aα, we haveL= –aL(α)

via an easy computation, so that –aαL(α) =c. Rewrite it as

c

α = –a

L(α)

α . ()

This then implies thatm(r,_α)≤_m(r,L_α(α)) +S(r,f) =S(r,f). Consequently, using the ﬁrst main theorem, one hasT(r,α) =S(r,f) and thusT(r,f) =S(r,f) from (). This is a con-tradiction. Ifc= –, then from () one observes

 +c

α = (αL)

αf =β. ()

An easy calculation saysm(r,_α)≤m(r,β) +S(r,f) =S(r,f), so thatT(r,α) =S(r,f) and thus

T(r,f) =S(r,f) by (). This is a contradiction again.

All these foregoing discussions imply thatδ≡. So, from (), we have

m

r, 

f

≤m

r,α

α

+m(r,β) +m

r,β

β

+m

r, δ

=m

r, δ

+S(r,f) =T(r,δ) –N

r, δ

+S(r,f)

=m(r,δ) +N(r,δ) –N

r, δ

+S(r,f).

Keeping in mindm(r,γ) =S(r,f), we havem(r,δ) =S(r,f) and thus

m

r, 

f

+N

r, δ

=N(r,δ) +S(r,f). ()

Resetδ=a(α α)

₊_a₍α α)

_–_aα α

f

f and note that the zeros ofαcome from the poles off.

Whenf(z) =  with multiplicityp≥k+ , thenα(z)=  and L(f)(z) = (L(f))(z) = 

with multiplicities at leastp–k+  andp–k, respectively; using (), one observesα(z) = 

with multiplicity at least p–k sincef– (αL)=aα; thus, recallingγ =aα

α, we know γ(z) =  with multiplicity at leastp–k. Thereby,δ(z) =  with multiplicity at leastp–

k– . As a result, it follows that

N(k+

r, 

f

≤N

r, δ

+ (k+ )N¯(k+

r, 

f

+S(r,f). ()

Whenδ(z) =∞, then eitherα(z) =  (fromf(z) =∞in fact) orf(z) =∞with

multi-plicity  ofδ(z) =∞, orf(z) =  with multiplicity  ofδ(z) =∞. In view of () for the

case wheref(z) = , we can thus conclude that

N(r,δ)≤N¯(r,f) +N¯k)

r, 

f

+S(r,f). ()

Here,Nk)(r,_f),N¯k)(r,_f), andN(k+(r,_f),N¯(k+(r,_f) are the counting functions of the

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Altogether with (), (), and (), we obtain

Tr,f=m

r, 

f

+Nk)

r, 

f

+N(k+

r, 

f

=N(r,δ) +Nk)

r, 

f

+ (k+ )N¯(k+

r, 

f

+S(r,f) ≤N¯(r,f) +N¯k)

r, 

f

+Nk)

r, 

f

+ (k+ )N¯(k+

r, 

f

+S(r,f),

which further implies that, ask≥ andNk)(r,_f)≤kN¯k)(r,_f) +S(r,f),

Tr,f≤(k+ )

¯

N(r,f) +N¯

r, 

f

+S(r,f). ()

This obviously contradicts our assumption. As a result, whenais a ﬁnite, nonzero con-stant, it follows that_Lf₍–_f_)–a_a=cwith a constantc= .

Case II.ais a nonconstant, small function.

Like before, rewrite () asf –a=αL–aαand diﬀerentiate it to yield

f–a= (αL)– (αa). () When (f–a)(z) = , then we have{(αL)– (αa)}(z) = . Recall that the zeros ofα

come from the poles off. As a consequence, one deduces that

H(z) :=

(αL)

αf –

(aα) αa

(z) = . ()

IfH≡, then we have, noticing thatm(r,H) =S(r,f), ¯

N

r, 

f–a

≤N

r, 

H

+S(r,f)

=T(r,H) +S(r,f) =N(r,H) +S(r,f).

Note whenH(z) =∞, then we have eitherα(z) =  (fromf(z) =∞actually) orf(z) =∞

- with multiplicitykofH(z) =∞, orf(z) =  - with multiplicityk–  ofH(z) =∞. Thus,

we can derive that, similar to (),

N(r,H)≤k

¯

N(r,f) +N¯

r, 

f

+S(r,f). ()

Apply the second main theorem for three small functions tofto yield

Tr,f≤ ¯N

r, 

f

+N¯r,f+N¯

r, 

f–a

+S(r,f), ()

which together with () yields () again. This gives a contradiction. Thus,H≡. That is, (αL)≡_af(aα). Using (), one derives

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so thata≡(aα); in other words,α=  –_ac for a constantcwith _ac = . We thus conclude that _Lf₍–_f_)–a_a=  –_ac andN(r,  –_ac) =N(r,α) =S(r,f).

Finally, let us prove the associated Theorem . Whenf,a, andaj(j= , , . . . ,k) are all

entire functions, we haveα(z) =eω(z)_{for some entire function}_ω₍_z_{). If}_a_{is constant, then}

we have the same result as in Case I: _Lf₍–_f_)–a_a =cwith a constantc= ; ifais nonconstant, then_ac =  –eω(z)_{admits many zeros, so that}_a_{cannot be entire unless}_c_{= , which provides}

us with the casef=L(f).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Jining University, Qufu, Shandong 273155, P.R. China.}2_{Department of Mathematics,}

University of Houston, Houston, Texas 77204, USA.

Acknowledgements

Both authors are greatly indebted to the anonymous referees.

Received: 29 March 2014 Accepted: 1 October 2014 Published: 13 October 2014 References

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doi:10.1186/1029-242X-2014-388

Cite this article as:Wang and Han:Unicity of meromorphic functions and their linear differential polynomials.