# On the Deficiencies of Some Differential-Difference Polynomials

Authors: Xiu-Min Zheng; Hong Yan Xu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The characteristic functions of differential-difference polynomials are investigated, and the result can be viewed as a differential-difference analogue of the classic Valiron-Mokhon’ko Theorem in some sense and applied to investigate the deficiencies of some homogeneous or nonhomogeneous differential-difference polynomials. Some special differential-difference polynomials are also investigated and these results on the value distribution can be viewed as differential-difference analogues of some classic results of Hayman and Yang. Examples are given to illustrate our results at the end of this paper.


---

## Body

## 1. Introduction

Throughout this paper, we use standard notations in the Nevanlinna theory (see, e.g., [1–3]). Let f(z) be a meromorphic function. Here and in the following the word “meromorphic’’ means being meromorphic in the whole complex plane. We use normal notations m(r,f), T(r,f), N(r,f), N(r,1/f), σ(f), λ(f), and λ(1/f). And we also use σ2(f) to denote the hyperorder of f(z) and δ(α,f) to denote the Nevanlinna deficiency of α with respect to f(z). Moreover, we denote by S(r,f) any real quantity satisfying S(r,f)=o(T(r,f)) as r→∞ outside of a possible exceptional set of finite logarithmic measure.Recently, with some establishments of difference analogues of the classic Nevanlinna theory (two typical and most important ones can be seen in [4–6]), there has been a renewed interest in the properties of complex difference expressions and meromorphic solutions of complex difference equations (see, e.g., [4–17]). By combining complex differentiates and complex differences, we proceed in this way in this paper.It is well known that the following Valiron-Mokhon’ko Theorem, due to Valiron [18] and A. Z. Mokhon’ko and V. D. Mokhon’ko [19], is of essential importance in the theory of complex differential equations and functional equations.Theorem A (see [2, 3]).
Letf(z) be a meromorphic function. Then for all irreducible rational functions in f,
(1)R(z,f(z))=∑i=0m‍ai(z)f(z)i∑j=0n‍bj(z)f(z)j,
with meromorphic coefficients ai(z), bj(z), the characteristic function of R(z,f(z)) satisfies
(2)T(r,R(z,f(z)))=dT(r,f)+O(Ψ(r)),
where d=max⁡{m,n} and Ψ(r)=max⁡i,j{T(r,ai),T(r,bj)}.Noting that the difference analogue of Theorem A may not hold, we have obtained a result of this type in [16] by adding some additional assumptions as follows.Theorem B (see [16]).
Suppose thatP(z,f) is a difference polynomial of the form(3)P(z,f)=∑λ∈I‍aλ(z)f(z)i0f(z+c1)i1⋯f(z+cn)in,
containing just one monomial of degree d(P), and f(z) is a transcendental meromorphic function of finite order. If f(z) also satisfies N(r,f)+N(r,1/f)=S(r,f), then we have
(4)T(r,P(z,f))=d(P)T(r,f)+S(r,f).In this paper, we consider removing the assumption “P(z,f) contains just one monomial of degree d(P)’’ in Theorem B and obtain a weaker result, which is also generalized into differential-difference case. The concrete result can be seen in Section 2.Next, we recall a classic result concerning Picard’s values of meromorphic functions and its derivatives, due to Hayman [20].Theorem C (see [20]).
Letf(z) be a transcendental entire function. Then(a)
forn≥3 and a≠0, Ψ(z)=f′(z)-a(f(z))n assumes all finite values infinitely often;(b)
forn≥2, Φ(z)=f′(z)(f(z))n assumes all finite values except possibly zero infinitely often.Corresponding difference analogues of Theorem C can be seen in [12, 17].Theorem D (see [12, 17]).
Letf(z) be a transcendental entire function of finite order, and let c be a nonzero complex constant. Then(a)
forn≥3 and a≠0, Ψ1(z)=f(z+c)-af(z)n assumes all finite complex values infinitely often;(b)
forn≥2, Φ1(z)=f(z+c)f(z)n assumes all finite complex values except possibly zero infinitely often.After Theorem C, many results have been obtained on the value distribution of differential polynomials. A typical one is as follows.Theorem E (see [21, 22]).
Letf be a transcendental meromorphic function with N(r,f)+N(r,1/f)=S(r,f), and let Ψ be a differential polynomial in f of the form
(5)Ψ(z)=∑‍a(z)f(z)l0f′(z)l1⋯f(k)(z)lk
with no constant term. Furthermore, assume the degree, n, of Ψ is greater than one and l0<n, 0≤li≤n, for all i≠0. Then δ(a,Ψ)<1 for all a≠0,∞. Moreover, if all the terms of Ψ have different degrees at least two, that is, Ψ is nonhomogeneous, then δ(a,Ψ)≤1-(1/2n) for all a≠∞.We also consider deficiencies of difference polynomials of meromorphic functions of finite order in [16], which can be viewed as difference analogues of Theorem E, as well as generalizations of Theorem D.In this paper, we proceed to investigate deficiencies of differential-difference polynomials of meromorphic functions. The concrete results can be seen in Section3.Examples are given in Section4 to illustrate our results.


## 2. A Differential-Difference Analogue of Valiron-Mokhon’ko Theorem

In what follows, we will consider differential-difference polynomials. A differential-difference polynomial is a polynomial inf(z), its shifts, its derivatives, and derivatives of its shifts (see [14]), that is, an expression of the form
(6)P(z,f)=∑λ∈I‍aλ(z)f(z)λ0,0f′(z)λ0,1⋯f(m)(z)λ0,m×f(z+c1)λ1,0f′(z+c1)λ1,1⋯f(m)(z+c1)λ1,m⋯f(z+cn)λn,0f′(z+cn)λn,1⋯f(m)(z+cn)λn,m=∑λ∈I‍aλ(z)∏i=0n∏j=0mf(j)(z+ci)λi,j,
where I is a finite set of multi-indices λ=(λ0,0,…,λ0,m,λ1,0,…,λ1,m,…,λn,0,…,λn,m), and c0(=0) and c1,…,cn are distinct complex constants. And we assume that the meromorphic coefficients aλ(z), λ∈I of P(z,f) are of growth S(r,f). We denote the degree of the monomial ∏i=0n∏j=0mf(j)(z+ci)λi,j of P(z,f) by d(λ)=∑i=0n∑j=0mλi,j. Then we denote the degree and the lower degree of P(z,f) by
(7)d(P)=max⁡λ∈I{d(λ)},d*(P)=min⁡λ∈I{d(λ)},
respectively. In particular, we call P(z,f) a homogeneous differential-difference polynomial if d(P)=d*(P). Otherwise, P(z,f) is nonhomogeneous.In the following, we assumed(P)≥1 and P(z,f)≢P(z,0).We prove a weaker differential-difference version of the classic Valiron-Mokhon’ko Theorem as follows.Theorem 1.
Suppose thatf(z) is a transcendental meromorphic function, and P(z,f) is a differential-difference polynomial of the form (6). If f(z) also satisfies σ2(f)<1 and
(8)N(r,f)+N(r,1f)=S(r,f),
then one has
(9)d*(P)T(r,f)+S(r,f)≤T(r,P(z,f))≤d(P)T(r,f)+S(r,f).Remark 2.
IfP(z,f) is a homogeneous differential-difference polynomial in addition, then
(10)T(r,P(z,f))=d(P)T(r,f)+S(r,f).Remark 3.
Especially, assumption (8) can be replaced by the assumption “max⁡{λ(f),λ(1/f)}<σ(f)’’. In fact, if f(z) satisfies max⁡{λ(f),λ(1/f)}<σ(f), then f(z) is of regular growth, and (8) holds consequently.To prove Theorem1, we need the following lemmas.Lemma 4 (see [6]).
Letf(z) be a nonconstant meromorphic function, ε>0, and c∈ℂ. If ζ=σ2(f)<1, then
(11)m(r,f(z+c)f(z))=o(T(r,f)r1-ζ-ε)
for all r outside of a set of finite logarithmic measure.Lemma 5 (see [6]).
LetT:[0,+∞)→[0,+∞) be a nondecreasing continuous function and let s∈(0,+∞). If the hyperorder of T is strictly less than one, that is, lim⁡r→∞¯(log⁡2T(r)/log⁡r)=ζ<1 and δ∈(0,1-ζ), then
(12)T(r+s)=T(r)+o(T(r)rδ),
where r runs to infinity outside of a set of finite logarithmic measure.It is shown in [23, p.66] and [7, Lemma 1] that the inequality
(13)(1+o(1))T(r-|c|,f)≤T(r,f(z+c))≤(1+o(1))T(r+|c|,f)
holds for c≠0 and r→∞. And from the proof, the above relation is also true for counting function. By combing Lemma 5 and these inequalities, we immediately deduce the following lemma.Lemma 6.
Letf(z) be a nonconstant meromorphic function of σ2(f)<1, and let c be a nonzero complex constant. Then one has
(14)T(r,f(z+c))=T(r,f)+S(r,f),N(r,f(z+c))=N(r,f)+S(r,f),N(r,1f(z+c))=N(r,1f)+S(r,f).Lemma 7.
Letf(z) be a transcendental meromorphic function of σ2(f)<1, and let P(z,f) be a differential-difference polynomial of the form (6); then we one has
(15)m(r,P(z,f))≤d(P)m(r,f)+S(r,f).
Furthermore, if f(z) also satisfies
(16)N(r,f)=S(r,f),
then one has
(17)T(r,P(z,f))≤d(P)T(r,f)+S(r,f).Proof.
Fori=0,1,…,n,  j=0,1,…,m, we define gi,j(z)=f(j)(z+ci)/f(z). We also define
(18)gi,j*(z)={gi,j(z),if|gi,j(z)|>1;1,if|gi,j(z)|≤1,f*(z)={f(z),if|f(z)|>1,1,if|f(z)|≤1.
Thus,
(19)|P(z,f)|≤∑λ∈I‍(|aλ(z)||f(z)|d(λ)∏i=0n∏j=0m|gi,j(z)|λi,j)≤(∑λ∈I‍|aλ(z)|∏i=0n∏j=0m|gi,j*(z)|λi,j)|f*(z)|d(P)≤(∑λ∈I‍|aλ(z)|∏i=0n∏j=0m|gi,j*(z)|d(λ))|f*(z)|d(P)≤(∑λ∈I‍|aλ(z)|)(∏i=0n∏j=0m|gi,j*(z)||f*(z)|)d(P).
By the definitions of f*(z) and gi,j*(z), i=0,1,…,n, j=0,1,…,m, we have
(20)m(r,f*)=m(r,f),m(r,gi,j*)=m(r,gi,j),i=0,…,n,j=0,…,m.
It follows by (19) and (20) that
(21)m(r,P(z,f))≤d(P)m(r,f*)+d(P)∑i=0n‍∑j=0m‍m(r,gi,j*)+S(r,f)=d(P)m(r,f)+d(P)∑i=0n‍∑j=0m‍m(r,gi,j)+S(r,f).
Lemmas 4 and 6 and the logarithmic derivative lemma imply that, for i=0,1,…,n and j=0,1,…,m,
(22)m(r,gi,j)=m(r,f(j)(z+ci)f(z))≤m(r,f(j)(z+ci)f(z+ci))+m(r,f(z+ci)f(z))=S(r,f(z+ci))+S(r,f)=S(r,f).
Then (15) follows by (21) and (22).
It is easy to find that(23)N(r,P(z,f))=O(N(r,f)+∑i=1n‍N(r,f(z+ci)))+S(r,f).
Then (16), (23), and Lemma 6 yield that
(24)N(r,P(z,f))=S(r,f).
Thus, (17) follows by (15) and (24).Lemma 8.
Letf(z) be a transcendental meromorphic function of σ2(f)<1, and let P(z,f) be a differential-difference polynomial of the form (6); then one has
(25)m(r,P(z,f)fd(P))≤(d(P)-d*(P))m(r,1f)+S(r,f).Proof.
Similar to (19), we have
(26)|P(z,f)f(z)d(P)|≤∑λ∈I‍(|aλ(z)|∏i=0n∏j=0m|gi,j(z)|λi,j|g(z)|d(P)-d(λ))≤(∑λ∈I‍|aλ(z)||g*(z)|d(P)-d(λ))×∏i=0n∏j=0m|gi,j*(z)|d(P)≤(∑λ∈I‍|aλ(z)|)×∏i=0n∏j=0m|gi,j*(z)|d(P)|g*(z)|d(P)-d*(P),
where g(z)=1/f(z) and
(27)g*(z)={g(z),if|g(z)|>1,1,if|g(z)|≤1.
By the definition of g*(z), we have m(r,g*)=m(r,g)=m(r,1/f). Thus, (20), (22), and (26) yield that
(28)m(r,P(z,f)fd(P))≤(d(P)-d*(P))m(r,g*)+d(P)∑i=0n‍∑j=0m‍m(r,gi,j*)+S(r,f)≤(d(P)-d*(P))m(r,1f)+S(r,f),
that is, (25).Now, we can finish the proof of Theorem1 in the end.Proof of Theorem1.
We deduce from (8), (24), and Lemma 8 that
(29)d(P)T(r,f)=T(r,fd(P))≤m(r,P(z,f)fd(P))+N(r,P(z,f)fd(P))+T(r,P(z,f))+O(1)≤(d(P)-d*(P))m(r,1f)+N(r,P(z,f))+d(P)N(r,1f)+T(r,P(z,f))+O(1)≤(d(P)-d*(P))T(r,f)+T(r,P(z,f))+S(r,f),
that is,
(30)d*(P)T(r,f)+S(r,f)≤T(r,P(z,f)).
Then, (9) follows by (17) and (30).


## 3. Deficiencies of Some Differential-Difference Polynomials

In the following, we assume thatα(z)(≢0) is a meromorphic function of growth S(r,f).In this section, we will apply Theorem1 to consider the deficiencies of general homogeneous or nonhomogeneous differential-difference polynomials.Theorem 9.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (8), and P(z,f) is a differential-difference polynomial of the form (6).(a)
IfP(z,f) is a homogeneous differential-difference polynomial, then one has
(31)lim⁡r→∞¯N¯(r,1/(P(z,f)-α))T(r,P(z,f))=1,δ(α,P(z,f))=0.(b)
IfP(z,f) is a nonhomogeneous differential-difference polynomial with 2d*(P)>d(P), then one has
(32)lim⁡r→∞¯N¯(r,1/(P(z,f)-α))T(r,P(z,f))≥2d*(P)-d(P)d*(P),δ(α,P(z,f))≤1-2d*(P)-d(P)d*(P)<1.Thus, P(z,f)-α(z) has infinitely many zeros, whether P(z,f) is homogeneous or nonhomogeneous.Furthermore, one considers some differential-difference polynomials of special forms, which are generalizations of both differential cases and difference cases, that is, Theorems C–E.Theorem 10.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (16), P(z,f) is a differential-difference polynomial of the form (6), and F(f)=(fv+av-1(z)fv-1+⋯+a1(z)f+a0(z))u, u,v∈ℕ, is a polynomial of f(z) with meromorphic coefficients ai(z), i=0,…,v-1 of growth S(r,f). If uv>d(P), u≠1, then
(33)Q1(z,f)=F(f)P(z,f)
satisfies
(34)lim⁡r→∞¯N¯(r,1/(Q1(z,f)-α))T(r,Q1(z,f))≥(u-1)(uv-d(P))u(uv+d(P)),δ(α,Q1(z,f))≤1-(u-1)(uv-d(P))u(uv+d(P))<1.
Thus, Q1(z,f)-α(z) has infinitely many zeros.WhenF(f) is of a special form fv, we can deduce the following result from Theorem 9.Theorem 11.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (16), and P(z,f) is a differential-difference polynomial of the form (6). If v∈ℕ∖{1} and v+2d*(P)>d(P), then
(35)Q2(z,f)=fvP(z,f)
satisfies δ(β,Q2(z,f))<1, where β∈ℂ∖{0}. Thus, Q2(z,f)-β has infinitely many zeros.Remark 12.
On the one hand, we can also apply Theorem9 to Q1(z,f) with the assumption “2(d*(P)+d*(F))>d(P)+uv’’ and obtain the same result as Theorem 10. But our present assumption “uv>d(P)’’ has no concern with d*(P) and d*(F), so we think Theorem 10 is better to some extent. On the other hand, we can also apply Theorem 10 to Q2(z,f) with the assumption “v>d(P),’’ which is stronger than “v+2d*(P)>d(P)’’ in Theorem 11, showing Theorem 11 is better to some extent.Theorem 13.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (16), P(z,f) is a differential-difference polynomial of the form (6), and F(f)=(fv+av-1(z)fv-1+⋯+a1(z)f+a0(z))u, u,v∈ℕ, is a polynomial of f(z) with meromorphic coefficients ai(z), i=0,…,v-1 of growth S(r,f). If (u-1)uv/(2u-1)>d(P), u≠1, then
(36)Q3(z,f)=F(f)+P(z,f)
satisfies
(37)lim⁡r→∞¯N¯(r,1/(Q3(z,f)-α))T(r,Q3(z,f))≥1-1u-2u-1u2vd(P),δ(α,Q3(z,f))≤1u+2u-1u2vd(P)<1.
Thus,Q3(z,f)-α(z) has infinitely many zeros.Whenu=1, one can consider some special cases as follows.Theorem 14.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (16), and P(z,f) is a differential-difference polynomial of the form (6).(a)
Ifv>d(P)+2≥3, then
(38)Q4(z,f)=fv+P(z,f)satisfies
(39)lim⁡r→∞¯N¯(r,1/(Q4(z,f)-α))T(r,Q4(z,f))≥1-d(P)+2v,δ(α,Q4(z,f))≤d(P)+2v<1.(b)
If(v-1)v/(2v-1)>d(P), v≥3, then Q4(z,f) satisfies
(40)lim⁡r→∞¯N¯(r,1/(Q4(z,f)-α))T(r,Q4(z,f))≥1-1v-2v-1v2d(P),δ(α,Q4(z,f))≤1v+2v-1v2d(P)<1.Especially, it holds for v=d(P)+2=3.(c)
Ifv≥d(P)+2≥3 and f also satisfies N(r,1/f)=S(r,f), then Q4(z,f) satisfies δ(α,Q4(z,f))<1. Especially, it holds for v=d(P)+2>3.Thus, Q4(z,f)-α(z) has infinitely many zeros.If we assume thatN(r,1/f)=S(r,f) in addition, the following result follows immediately by Theorem 9.Theorem 15.
Suppose thatf(z) is a transcendental meromorphic function satisfying σ2(f)<1 and (8), and P(z,f) is a differential-difference polynomial of the form (6). If 2min⁡{d*(P),v}>max⁡{d(P),v}, then Q4(z,f) satisfies δ(α,Q4(z,f))<1. Thus, Q4(z,f)-α(z) has infinitely many zeros.Remark 16.
Noting that, whenv>3, (v-1)v/(2v-1)≤v-2  hold, we see that the assumption “v>d(P)+2’’ in Theorem 14(a) is weaker than the assumption “(v-1)v/(2v-1)>d(P)’’ in Theorem 14(b). And these assumptions in Theorem 14 have no concern with d*(P)); thus they are different from the assumption “2min⁡{d*(P),v}>max⁡{d(P),v}’’ in Theorem 15.Remark 17.
From the proofs behind, it is easy to find that(41)λ(P(z,f)-α)=σ(P(z,f))=σ(f),λ(Qi(z,f)-α)=σ(Qi(z,f))=σ(f),i=1,3,4,
hold, respectively, in Theorems 9, 10, 13, 14(a) and (b), and 15.
Now, we give the proofs of Theorems9–15.Proof of Theorem9.
It follows by Theorem1 that
(42)S(r,f)=S(r,P(z,f)).
We deduce from (8), (24), (25), and (42) that
(43)N¯(r,1P(z,f))≤N¯(r,1fd(P))+N¯(r,fd(p)P(z,f))≤N¯(r,1f)+m(r,P(z,f)fd(P))+N(r,P(z,f)fd(P))+O(1)≤(d(P)-d*(P))m(r,1f)+S(r,f)≤(d(P)-d*(P))m(r,1f)+S(r,P(z,f)).
Thus, an application of the second main theorem and (24), (42), and (43) imply that
(44)T(r,P(z,f))≤N¯(r,P(z,f))+N¯(r,1P(z,f))+N¯(r,1P(z,f)-α)+S(r,P(z,f))≤(d(P)-d*(P))m(r,1f)+N¯(r,1P(z,f)-α)+S(r,P(z,f)).(a)
Ifd(P)=d*(P), then it follows by (44) that
(45)T(r,P(z,f))≤N¯(r,1P(z,f)-α)+S(r,P(z,f)),by which (31) holds.(b)
If2d*(P)>d(P), then we deduce from (30) and (44) that(46)T(r,P(z,f))≤(d(P)-d*(P))T(r,f)+N¯(r,1P(z,f)-α)+S(r,P(z,f))≤d(P)-d*(P)d*(P)T(r,P(z,f))+N¯(r,1P(z,f)-α)+S(r,P(z,f));that is,
(47)2d*(P)-d(P)d*(P)T(r,P(z,f))≤N¯(r,1P(z,f)-α)+S(r,P(z,f)).
Since2d*(P)-d(P)>0, (32) follows immediately by (47).Proof of Theorem10.
We deduce from (16), (17), and (24) that
(48)T(r,Q1(z,f))≤(uv+d(P))T(r,f)+S(r,f),(49)N(r,Q1(z,f))=O(N(r,f))+N(r,P(z,f))+S(r,f)=S(r,f)
hold. Next, we consider N¯(r,1/Q1(z,f)). Let z0 be a zero of Q1(z,f) and distinguish three cases.(i)
z0 is not a zero of F(f); then z0 must be a zero of P(z,f) and
(50)u≤ω(1Q1(z,f),z0)+(u-1)ω(1P(z,f),z0),where ω(f,z0) denotes the order of multiplicity of z0 or zero according as z0 is a pole of f(z) or not.(ii)
z0 is a zero of F(f) but not a pole of P(z,f). Then
(51)u≤ω(1Q1(z,f),z0).(iii)
z0 is a zero of F(f) and a pole of P(z,f). Then
(52)u≤ω(1F(f),z0)≤ω(1Q1(z,f),z0)+ω(P(z,f),z0).(24) and (50)–(52) yield that
(53)uN¯(r,1Q1(z,f))≤N(r,1Q1(z,f))+(u-1)N(r,1P(z,f))+S(r,f).
Then (48), (49), (53), and an application of the second main theorem to Q1(z,f) imply that(54)T(r,Q1(z,f))≤N¯(r,Q1(z,f))+N¯(r,1Q1(z,f))+N¯(r,1Q1(z,f)-α)+S(r,Q1(z,f))≤1uN(r,1Q1(z,f))+u-1uN(r,1P(z,f))+N¯(r,1Q1(z,f)-α)+S(r,f);
consequently,
(55)T(r,Q1(z,f))≤N(r,1P(z,f))+uu-1N¯(r,1Q1(z,f)-α)+S(r,f).
Moreover, byfd(P)F(f)=fd(P)Q1(z,f)/P(z,f), (16), (24), (25), and Theorem A, we have(56)(d(P)+uv)m(r,f)=m(r,fd(P)Q1(z,f)P(z,f))+S(r,f)≤m(r,Q1(z,f))+m(r,P(z,f)fd(P))+N(r,P(z,f)fd(P))-N(r,fd(P)P(z,f))+S(r,f)≤m(r,Q1(z,f))+d(P)m(r,1f)+d(P)(N(r,1f)-N(r,f))+N(r,P(z,f))-N(r,1P(z,f))+S(r,f)=m(r,Q1(z,f))+d(P)m(r,f)-N(r,1P(z,f))+S(r,f);
consequently,
(57)uvm(r,f)≤m(r,Q1(z,f))-N(r,1P(z,f))+S(r,f).
On the other hand, the evident relation uvω(f,z0)≤ω(F(f),z0)+uv∑j=0v-1ω(aj,z0), where the definition of ω(f,z0) is given after (50), results in
(58)uvN(r,f)≤N(r,F(f))+S(r,f)≤N(r,Q1(z,f))+N(r,1P(z,f))+S(r,f).
We deduce from (57) and (58) that
(59)uvT(r,f)≤T(r,Q1(z,f))+S(r,f).
Then (17), (55), and (59) yield that(60)uvT(r,f)≤N(r,1P(z,f))+uu-1N¯(r,1Q1(z,f)-α)+S(r,f)≤d(P)T(r,f)+uu-1N¯(r,1Q1(z,f)-α)+S(r,f);that is,(61)(u-1)(uv-d(P))uT(r,f)≤N¯(r,1Q1(z,f)-α)+S(r,f).
From (48) and (61), we deduce that
(62)lim⁡r→∞¯N¯(r,1/(Q1(z,f)-α))T(r,Q1(z,f))≥(u-1)(uv-d(P))u(uv+d(P)),δ(α,Q1(z,f))≤1-(u-1)(uv-d(P))u(uv+d(P))<1.Proof of Theorem11.
Assume to the contrary thatδ(β,Q2(z,f))=1. Denoting
(63)Q2(z,f)-β=fv(z)P(z,f)-β=G(z),
we deduce from (16) and (17) that
(64)N(r,1G)=N(r,1Q2(z,f)-β)=S(r,Q2(z,f))=S(r,f).
On the other hand, (16) and (24) yield that
(65)N(r,G)=N(r,Q2(z,f)-β)=S(r,f).
Differentiating both sides of (63), we obtain
(66)fv-1(z)R(z,f)=G′(z),
where R(z,f)=vf′(z)P(z,f)+f(z)P′(z,f). Clearly, we deduce from (16) and (24) that
(67)N(r,R(z,f))=S(r,f).
Moreover, (64) and (65) yield that
(68)N(r,1G′)≤N(r,GG′)+N(r,1G)≤T(r,G′G)+N(r,1G)+O(1)≤m(r,G′G)+N¯(r,G)+2N(r,1G)+O(1)=S(r,G)+S(r,f)=S(r,f).
It follows by (66)–(68) that
(69)N(r,1f)=1v-1N(r,R(z,f)G′)=S(r,f).
Then (16), (69), and the fact 2(d*(P)+v)>d(P)+v imply that the assumptions of Theorem 9(b) are satisfied. Thus, Theorem 9(b) yields that δ(β,Q2(z,f))<1, a contradiction. Therefore, we have δ(β,Q2(z,f))<1.Proof of Theorem13.
We deduce from (16), (17), and (24) that
(70)T(r,Q3(z,f))≤max⁡{uv,d(P)}T(r,f)+S(r,f)=uvT(r,f)+S(r,f).
Denote
(71)H(z)=-P(z,f)+α(z)F(f).
Now, we estimate the poles, the zeros, and 1-points ofH(z) accurately. On the one hand, we see by (71) that the poles of H(z) occur at zeros of F(f) and poles of -P(z,f)+α(z) which are not simultaneously 1-points of H(z), and those poles of H(z) which are zeros of F(f) but not simultaneously zeros of -P(z,f)+α(z) also have multiplicities at least u. On the other hand, we also see by (71) that the zeros of H(z) occur at zeros of -P(z,f)+α(z) and poles of F(f) which are not simultaneously 1-points of H(z). Moreover, 1-points of H(z) occur at zeros of Q3(z,f)-α(z) and occur at the common poles, zeros of F(f) and -P(z,f)+α(z) with the same multiplicities. Thus, it follows by (16) and (24) that
(72)N¯(r,H)+N¯(r,1H)+N¯(r,1H-1)≤1uN(r,H)+N¯(r,1P(z,f)-α)+N¯(r,1Q3(z,f)-α)+S(r,f).
Then (17), (72), and the second main theorem result in(73)T(r,H)≤N¯(r,H)+N¯(r,1H)+N¯(r,1H-1)+S(r,H)≤1uT(r,H)+N¯(r,1P(z,f)-α)+N¯(r,1Q3(z,f)-α)+S(r,f)≤1uT(r,H)+d(P)T(r,f)+N¯(r,1Q3(z,f)-α)+S(r,f);
that is,
(74)(1-1u)T(r,H)≤d(P)T(r,f)+N¯(r,1Q3(z,f)-α)+S(r,f).
Moreover, Theorem A and (17) imply that(75)uvT(r,f)+S(r,f)=T(r,F(f))=T(r,-P(z,f)+αH)≤d(P)T(r,f)+T(r,H)+S(r,f);
that is,
(76)(uv-d(P))T(r,f)≤T(r,H)+S(r,f).
Then (74) and (76) yield that
(77)((u-1)v-2u-1ud(P))T(r,f)≤N¯(r,1Q3(z,f)-α)+S(r,f).
From (70) and (77), we deduce that
(78)lim⁡r→∞¯N¯(r,1/(Q3(z,f)-α))T(r,Q3(z,f))≥1-1u-2u-1u2vd(P),δ(α,Q3(z,f))≤1u+2u-1u2vd(P)<1.To prove Theorem14(c), we also need the following lemma of one of Tumura-Clunie type theorems.Lemma 18 (see [24]).
Letf(z) be a meromorphic function, and suppose that Ψ=anfn+⋯+a0 has small meromorphic coefficients aj(z), an(z)≢0, in the sense of T(r,aj)=S(r,f). Moreover, assume that N¯(r,1/Ψ)+N¯(r,f)=S(r,f). Then Ψ=an(f+(an-1/nan))n.Proof of Theorem14.
(a) We deduce from (16), (17), and (24) that
(79)T(r,Q4(z,f))≤vT(r,f)+S(r,f).
Denote
(80)K(z)=Q4(z,f)-α(z)=fv(z)+P(z,f)-α(z).
Differentiating both sides of (80), we obtain(81)vfv-1(z)f′(z)+P′(z,f)-α′(z)=K′(z)=(fv(z)+P(z,f)-α(z))K′(z)K(z);
that is,
(82)fv-1(z)((vf′(z)f(z)-K′(z)K(z))f(z))=(P(z,f)-α(z))K′(z)K(z)-(P′(z,f)-α′(z))=(P(z,f)-α(z))(K′(z)K(z)-P′(z,f)-α′(z)P(z,f)-α(z)).
It follows by (15)–(17), (24), (79), and (82) that(83)m(r,fv-1)≤m(r,P(z,f)-α)+m(r,K′K)+m(r,P′(z,f)-α′P(z,f)-α)+m(r,1(v(f′/f)-(K′/K))f)≤d(P)m(r,f)+m(r,(vf′f-K′K)f)+N(r,(vf′f-K′K)f)+S(r,K)+S(r,f)≤(d(P)+1)m(r,f)+N(r,K′K)+S(r,f)≤(d(P)+1)m(r,f)+N¯(r,1K)+S(r,f);
that is,
(84)(v-d(P)-2)T(r,f)≤N¯(r,1Q4(z,f)-α)+S(r,f).
From (79) and (84), we deduce that
(85)lim⁡r→∞¯N¯(r,1/(Q4(z,f)-α))T(r,Q4(z,f))≥1-d(P)+2v,δ(α,Q4(z,f))≤d(P)+2v<1.
(b) It suffices to note that we may seefv as (f1)v; then Theorem 14(b) follows immediately by Theorem 13.
(c) By using a similar reasoning as [13, Theorem 1], we can rearrange the expression for the differential-difference polynomial P(z,f) by collecting together all terms having the same total degree and then writing P(z,f) in the form P(z,f)=∑k=0d(P)bk(z)fk(z). Now each of the coefficients bk(z) is a finite sum of products of functions of the form (f(j)(z+ci)/f(z))λi,j=(f(j)(z+cj)/f(z+ci))λi,j(f(z+ci)/f(z))λi,j, with each such product being multiplied by one of the original coefficients aλ(z). We deduce from the logarithmic derivative lemma and Lemmas 4 and 6 that m(r,bk)=S(r,f). Clearly, N(r,bk)=S(r,f) holds by (8) and Lemma 6. Thus, T(r,bk)=S(r,f). Denote
(86)L(z)=Q4(z,f)-α(z)=fv(z)+∑k=0d(P)‍bk(z)fk(z)-α(z).
Assume to the contrary that δ(α,Q4(z,f))=1. Thus, Theorem A yields that
(87)N(r,1L)=N(r,1Q4(z,f)-α)=S(r,Q4(z,f))=S(r,f).
Then (8), (86), (87), Lemma 18, and the assumption that v≥d(P)+2 imply that L(z)≡f(z)v; that is,
(88)P(z,f)=∑k=0d(P)‍bk(z)fk(z)≡α(z).
Noting the fact that T(r,bk)=S(r,f) and T(r,α)=S(r,f), we deduce from Theorem A that (88) is a contradiction. Therefore, we have δ(α,Q4(z,f))<1.


## 4. Examples

Example 1.
We consider nonhomogeneous differential-difference polynomials(89)P1(z,f)=f(z)f2(z+log⁡4)-4f′′(z)f(z+log⁡2)×f′(z+log⁡2)+f′′2(z+log⁡3),P2(z,f)=3f3(z)f′2(z+log⁡4)-2f′(z)f(z+log⁡3)f′′3(z+log⁡2)+f4(z)-f′′3(z),P3(z,f)=f(z)f′(z+log⁡2)f′′(z+log⁡3)-6f′′2(z)
and a homogeneous differential-difference polynomial
(90)P4(z,f)=f′′3(z+log⁡2)-f′(z)f(z+log⁡2)×f′(z+log⁡3)-f(z)f′(z)f′′(z),
where d(P1)=3>2=d*(P1), d(P2)=5>3=d*(P2), d(P3)=3>2=d*(P3), and d(P4)=3=d*(P4). Clearly, the function f(z)=ez satisfies (8) and σ2(f)=0<1. Then we have
(91)d*(P1)T(r,ez)+O(1)=T(r,P1(z,ez))=2rπ+O(1)<d(P1)T(r,ez)+O(1),d*(P2)T(r,ez)+O(1)<T(r,P2(z,ez))=4rπ+O(1)<d(P2)T(r,ez)+O(1),d*(P3)T(r,ez)+O(1)<T(r,P3(z,ez))=3rπ+O(1)=d(P3)T(r,ez)+O(1),d*(P4)T(r,ez)+O(1)=T(r,P4(z,ez))=3rπ+O(1)=d(P4)T(r,ez)+O(1).
This example shows that (9) is best possible.Example 2.
Considerf(z)=ez again. Then the homogeneous case P4(z,f) in Example 1 also illustrates Theorem 9(a). And the nonhomogeneous differential-difference polynomials Pi(z,f),  i=1,2,3, in Example 1 also illustrate Theorem 9(b), where δ(α,P1(z,f))=0, δ(α,P2(z,f))≤1/4<2/3=1-((2d*(P2)-d(P2))/d*(P2)), and δ(α,P3(z,f))≤1/3<1/2=1-((2d*(P3)-d(P3))/d*(P3)). Next, we consider the nonhomogeneous differential-difference polynomial
(92)P5(z,f)=f′(z)f(z+log⁡2)-f2(z)+f′(z+log⁡3)-3f(z)+1,
where d(P5)=2,d*(P5)=0. Clearly, δ(1,P5(z,f))=δ(1,e2z+1)=1. Note that 2d*(P5)>d(P5) fails; then this example shows that the assumption “2d*(P)>d(P)’’ cannot be omitted in Theorem 9(b).Example 3.
We consider the differential-difference polynomials(93)Q1(z,f)=(f2)2P6(z,f)=f4(z)(f′(z+π2)f(z+π)f′′(z+2π)00000000000+f2(z+π)π2),Q2(z,f)=f2P6(z,f)=f2(z)(f′(z+π2)f(z+π)f′′(z+2π)0000000000+f2(z+π)π2),
and the function f(z)=sinz. On the one hand, N(r,f)=S(r,f), σ2(f)=0<1, and uvQ1>d(P6) and vQ2+2d*(P6)>d(P6) hold, where vQ1=vQ2=u=2 and d(P6)=3>2=d*(P6). On the other hand, δ(α,Q1(z,f))≤1-(11/14)<1-(1/14)=1-(u-1)(uvQ1-d(P))/u(uvQ1+d(P))<1 and δ(α,Q2(z,f))<1 hold. This example shows that Theorems 10 and 11 may hold.Example 4.
We consider the differential-difference polynomials(94)Q4(1)(z,f)=(f2)4+P7(z,f)=f8+P7(z,f)=f8(z)+f′(z+π2)f(z+π)f′′(z+2π),Q4(2)(z,f)=f2+P7(z,f)=f2(z)+f′(z+π2)f(z+π)f′′(z+2π),Q4(3)(z,f)=2f3+P7(z,f)=2f3(z)+f′(z+π2)f(z+π)f′′(z+2π),Q4(4)(z,f)=f4+P7(z,f)=f4(z)+f′(z+π2)f(z+π)f′′(z+2π),
and the function f(z)=sinz again. On the one hand, Q4(1)(z,f) satisfies (u-1)uvQ4(11)/(2u-1)>d(P7) and vQ4(12)-2>(vQ4(12)-1)vQ4(12)/(2vQ4(12)-1)>d(P7), respectively, where u=4, vQ4(11)=2, vQ4(12)=8, d(P7)=d*(P7)=3, and, for i=2,3,4, Q4(i)(z,f) satisfies 2min⁡{d*(P7),vQ4(i)}>max⁡{d(P7),vQ4(i)}, where vQ4(i)=i. On the other hand, δ(α,Q4(i)(z,f))<1,  i=1,2,3,4, hold. This example shows that Theorems 13–15 may hold. Moreover, this example also shows the assumption “N(r,1/f)=S(r,f)’’ is not necessary to Theorems 14(c) and 15, but it is regrettable for us not removing it in our proofs.Example 5.
We consider the differential-difference polynomials(95)R1(z,f)=f2P8(z,f)=f2(z)(f2(z+π)+1sin22zf′2(z+π2)),R2(z,f)=f7+P9(z,f)=f7(z)+sin2zf′(z+π2)f2(z+π2)+f′2(z+π2)f(z+3π2)+zf(z)f(z+π2),
and the function f(z)=esin2z. On the one hand, R1(z,f) satisfies vR1+2d*(P8)>d(P8), and   R2(z,f) satisfies vR2-2>(vR2-1)vR2/(2vR2-1)>d(P9), respectively, where vR1=2  and  d(P8)=d*(P8)=2, and vR2=7 and d(P9)=3. On the other hand, δ(e2,R1(z,f))=δ(ez,R2(z,f))=1 hold, showing that Theorems 11 and 14 fail. Noting that the function f(z)=esin2z satisfies σ2(f)=1, we know that the assumption “σ2(f)<1’’ is essential for Theorems 11 and 14. In fact, it is also essential for our other results in the whole paper, but it is unnecessary to give examples one by one.



---

*Source: 378151-2014-02-27.xml*