R E S E A R C H
Open Access
On the iterated exponent of convergence of
zeros of
f
(
j
)
(
z
) –
ϕ
(
z
)
Jin Tu
1, Zu-Xing Xuan
2*and Hong-Yan Xu
3*Correspondence:
xuanzuxing@ss.buaa.edu.cn
2Beijing Key Laboratory of
Information Service Engineering, Department of General Education, Beijing Union University, No. 97 Bei Si Huan Dong Road Chaoyang District, Beijing, 100101, China Full list of author information is available at the end of the article
Abstract
In this paper, the authors investigate the iterated exponent of convergence of zeros of
f(j)(z) –
ϕ
(z) (j= 0, 1, 2,. . .), wherefis a solution of some second-order linear differential equation,ϕ
(z)≡0 is an entire function satisfyingσ
p+1(ϕ
) <σ
p+1(f) ori(ϕ
) <i(f)(p∈N). We obtain some results which improve and generalize some previous results in (Chen in Acta Math. Sci. Ser. A 20(3):425-432, 2000; Chen and Shon in Chin. Ann. Math. Ser. A 27(4):431-442, 2006; Tuet al.in Electron. J. Qual. Theory Differ. Equ. 23:1-17, 2011) and provide us with a method to investigate the iterated exponent of convergence of zeros off(j)(z) –
ϕ
(z) (j= 0, 1, 2,. . .).MSC: 34A20; 30D35
Keywords: linear differential equations; iterated order; iterated type; iterated exponent of convergence of zero-sequence
1 Introduction
In this paper, we assume that readers are familiar with the fundamental results and stan-dard notation of Nevanlinna’s theory of meromorphic functions (see [, ]). First, we in-troduce some notations. Let us define inductively, forr∈[,∞),expr=erandexp
j+r= exp(expjr),j∈N. For all sufficiently larger, we definelogr=lograndlogj+r=log(logjr),
j∈N; we also denoteexpr=r=lograndexp–r=logr. Moreover, we denote the linear measure and the logarithmic measure of a setE⊂[, +∞) bymE=EdtandmlE=
Edt/t
respectively. Letf(z), a(z) be meromorphic functions in the complex plane satisfying
T(r,a) =o{T(r,f)}except possibly for a set ofrhaving finite logarithmic measure, then we call thata(z) is a small function off(z). We usepto denote a positive integer through-out this paper, not necessarily the same at each occurrence. In order to describe the infi-nite order of fast growing entire functions precisely, we recall some definitions of entire functions of finite iterated order (e.g., see [–]).
Definition . Thep-iterated order of a meromorphic functionf(z) is defined by
σp(f) = lim r→∞
logpT(r,f)
logr . (.)
Remark . Iff(z) is an entire function, thep-iterated order off(z) is defined by
σp(f) = lim r→∞
logpT(r,f)
logr =rlim→∞
logp+M(r,f)
logr . (.)
It is easy to see thatσp(f) =∞ifσp+(f) > . Ifp= , the hyper-order off(z) is defined by
Definition . The finiteness degree of the iterated order of an entire functionf(z) is defined by
Definition . The p-iterated exponent of convergence of distinct zero-sequence of
Definition . Ifϕ(z) is an entire function satisfyingσp(ϕ) <σp(f) ori(ϕ) <i(f), then the
finiteness degree of the iterated exponent of convergence of zero-sequence off(z) –ϕ(z) is defined by
iλ(f–ϕ) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
forN(r,
f–ϕ) =O(logr), min{j∈N:λj(f–ϕ) <∞} for somej∈N
withλj(f–ϕ) <∞exists,
∞ ifλj(f–ϕ) =∞for allj∈N.
(.)
Remark . From Definitions . and ., we can similarly give the definitions ofλp(f),
iλ(f–z),iλ(f–ϕ) andiλ(f).
2 Main result
In [], Chen firstly investigated the fixed points of the solutions of equations (.) and (.) with a polynomial coefficient and a transcendental entire coefficient of finite order and obtained the following Theorems A and B. Two years later in [], Chen investigated the zeros off(j)(z) –ϕ(z) (j= , , ) and obtained the following Theorems C and D, where
f(z) is a solution of equation (.) or (.),ϕ(z) is an entire function satisfyingσ(ϕ) <∞. In [], Tu, Xu and Zhang investigated the hyper-exponent of convergence of zeros of
f(j)(z) –ϕ(z) (j∈N) and obtained the following Theorem E, wheref(z) is a solution of (.),ϕ(z) is an entire function satisfyingσ(ϕ) <σ(B). One year later, Xu, Tu and Zheng improved Theorem E to Theorem F in [] from (.) to (.). In the following, we list Theorems A-F which have been mentioned above.
Theorem A [] Let P(z)be a polynomial with degree n(≥).Then every non-trivial solution of
f +P(z)f = (.)
has infinitely many fixed points and satisfiesλ(f–z) =λ(f –z) =σ(f) =n+ .
Theorem B[] Let A(z)be a transcendental entire function with σ(A) =σ <∞.Then every non-trivial solution of
f +A(z)f = (.)
has infinitely many fixed points and satisfiesλ(f–z) =λ(f –z) =σ(f) =σ.
Theorem C[] Let Aj(z) (≡) (j= , )be entire functions withσ(Aj) < .a,b are complex
numbers and satisfy ab= andarga=argb or a=cb( <c< ).Ifϕ(z)≡is an entire function of finite order,then every non-trivial solution f of
f +A(z)eazf +A(z)ebzf = (.)
Theorem D[] Let A(z)≡,ϕ(z)≡,Q(z)be entire functions withσ(A) < and <
σ(Q) <∞.Then every non-trivial solution f of
f +A(z)eazf +Q(z)f= (.)
satisfiesλ(f–ϕ) =λ(f –ϕ) =λ(f –ϕ) =∞,where a= is a complex number.
Theorem E[] Let A(z)and B(z)be entire functions with finite order.Ifσ(A) <σ(B) <∞
or <σ(A) =σ(B) <∞and≤τ(A) <τ(B) <∞,then for every solution f ≡of
f +A(z)f +B(z)f= (.)
and for any entire functionϕ(z)≡satisfyingσ(ϕ) <σ(B),we have
(i) λ(f –ϕ) =λ(f –ϕ) =λ(f –ϕ) =λ(f –ϕ) =σ(f) =σ(B);
(ii) λ(f(j)–ϕ) =σ(f) =σ(B)(j> ,j∈N).
Theorem F[] Let Aj(z) (j= , , . . . ,k– )be entire functions of finite order and satisfy
one of the following conditions:
(i) max{σ(Aj),j= , , . . . ,k– }<σ(A) <∞;
(ii) <σ(Ak–) =· · ·=σ(A) =σ(A) <∞andmax{τ(Aj),j= , , . . . ,k– }<τ(A) <∞.
Then for every solution f ≡of
f(k)+Ak–f(k–)+· · ·+Af= (.)
and for any entire functionϕ(z)≡satisfyingσ(ϕ) <σ(A),we have
λ
f(j)–ϕ=σ(f) =σ(A), j∈N.
The main purpose of this paper is to improve Theorem E from entire coefficients of finite order in (.) to entire coefficients of finite iterated order. And we obtain the following results.
Theorem . Let A(z)and B(z)be entire functions of finite iterated order satisfyingσp(A) < σp(B) <∞or <σp(A) =σp(B) <∞and≤τp(A) <τp(B)≤ ∞.Then for every solution
f ≡of(.)and for any entire functionϕ(z)≡satisfyingσp+(ϕ) <σp(B),we have (i) λp+(f–ϕ) =λp+(f –ϕ) =λp+(f –ϕ) =λp+(f –ϕ) =σp+(f) =σp(B); (ii) λp+(f(j)–ϕ) =σp+(f) =σp(B),j> ,j∈N.
Theorem . Let A(z),B(z)be entire functions satisfying i(A) <i(B) =p.Then for every solution f ≡of(.)and for any entire functionsϕ(z)≡with i(ϕ)≤p,we have
(i) iλ(f(j)–ϕ) =iλ(f(j)–ϕ) =i(f(j)–ϕ) =p+ (j= , , , . . .);
(ii) λp+(f(j)–ϕ) =λp+(f(j)–ϕ) =σp+(f(j)–ϕ) =σp(B)(j= , , , . . .).
Theorem . Under the hypotheses of Theorem.,let L(f) =akf(k)+ak–f(k–)+· · ·+
Corollary . Under the hypotheses of Theorem.,ifϕ(z) =z,we have
(i) λp+(f–z) =λp+(f –z) =λp+(f –z) =λp+(f –z) =σp+(f) =σp(B); (ii) λp+(f(j)–z) =σp+(f) =σp(B),j> ,j∈N.
Corollary . Under the hypotheses of Theorem.,ifϕ(z) =z,we have
(i) iλ(f(j)–z) =iλ(f(j)–z) =i(f(j)–z) =p+ (j= , , , . . .);
(ii) λp+(f(j)–z) =λp
+(f(j)–z) =σp+(f(j)–z) =σp(B)(j= , , , . . .).
Remark . Theorem . is an extension and improvement of Theorem E. As for
Theo-rem C, ifa=cb( <c< ), it is easy to see thatσ(Aeaz) =σ(Aebz) = andτ(Aeaz) =a<
τ(Aebz) =b. By Theorem E, for every solutionf ≡ of (.) and for any entire function
ϕ(z)≡ withσ(ϕ) < , we haveλ(f–ϕ) =λ(f –ϕ) =λ(f –ϕ) = , therefore Theorem E is also a partial extension of Theorem C. Theorem B is a special case of Corollary . for
p= .
Remark . Nevanlinna’s second fundamental theorem is an important tool to investi-gate the distribution of zeros of meromorphic functions. From Nevanlinna’s second fun-damental theorem [, p., Theorem .], we have that
+o()T(r,f) <N r,
f
+N(r,f) +N r,
f–ϕ
+S(r,f),
whereϕ(z) is a small function off(z). For example, setf(z) =ea(z),a(z) is a transcenden-tal entire function withi(a) =p, then we haveλp+(f –ϕ) =σp+(f) =σp(a) ifi(ϕ) <p+ .
Our Theorem . and Theorem . also provide us with a method to investigate the iter-ated exponent of zero sequence off(j)(z) –ϕ(z) (j= , , , . . .), wheref(z) andϕ(z) are entire functions satisfying σp+(ϕ) <σp+(f) ori(ϕ) <i(f). If we can find equation (.) with entire coefficientsA(z),B(z) satisfyingσp(A) <σp(B) or <σp(A) =σp(B) <∞and
≤τp(A) <τp(B)≤ ∞such thatf(z) is a solution of (.), then we haveλp+(f(j)–ϕ) =σp(B)
(j= , , , . . .). By the above example, setf(z) =ea(z),a(z) is transcendental withi(a) =p, thenf(z) is a solution off – (a +a)f= . Sinceσ
p(a +a) =σp(a) and by Theorem .,
we have λp+(f(j)–ϕ) =σp(a) (j= , , , . . .) for any entire function ϕ(z)≡ satisfying σp+(ϕ) <σp(a) ori(ϕ) <p+ .
3 Lemmas
Lemma .[, ] Let f(z)be an entire function withσp(f) =σ,andνf(r)denote the central
index of f(z).Then
lim
r→∞
logpνf(r)
logr =σ. (.)
Lemma . Let f(z)be an entire function withσp(f) =σ, then there exists a set E⊂ [, +∞)with infinite logarithmic measure such that for all r∈E,we have
lim
r→∞
logpT(r,f)
Proof By Definition .limr→∞loglogpT(rr,f)=σ, there exists a sequence{rn}∞n=tending to∞
By Lemma . and the same proof in Lemma ., we have the following lemma.
Lemma . Let f(z)be an entire function withσp(f) =σandνf(r)denote the central index
then we have the following two statements:
Moreover,for anyε> ,we have
logβ(z)≥–expp–rσp(β)+ε, r∈/E, (.)
where E⊂[, +∞)is a set of r of finite linear measure.
Lemma . Let f(z)be an entire function of finite iterated order withσp(f) =σ < +∞.
Then for any givenε> ,there is a set E⊂[,∞)that has finite linear measure such that
for all z satisfying|z|=r∈/[, ]∪E,we have
exp–expp–rσ+ε≤f(z)≤expprσ+ε. (.)
Proof Letf(z) be an entire function of finite iterated order with σp(f) =σ. By
Defini-tion ., it is easy to obtain that|f(z)|<expp{rσ+ε}holds for all sufficiently larger|z|=r. By
Lemma ., there exist entire functionsβ(z) andD(z) such that
f(z) =β(z)eD(z), σp(f) =max
σp(β),σp
eD(z).
For anyε> , we have
β(z)≥exp–expp–rσp(β)+ε≥exp–exp
p–
rσp(f)+ε, r∈/E
, (.)
hold outside a setE⊂[, +∞) of finite linear measure. Sinceσp–(D(z)) =σp(eD(z))≤σp(f),
by Definition ., we have that|D(z)| ≤expp–{rσp(f)+ε}holds for all sufficiently larger.
From|eD(z)| ≥e–|D(z)|≥exp{–exp
p–{rσp(f)+
ε
}}and (.), we have
f(z)≥β(z)e–|D(z)|≥exp–expp–rσp(f)+ε
≥exp–expp–rσp(f)+ε, r∈/E, (.)
whereEis a set ofrof finite linear measure. By Definition . and (.), we obtain the
conclusion of Lemma ..
Remark . Lemma . gives the modulus estimation of an entire function with finite iterated order and extends the conclusion of [, p., Lemma ].
Lemma . Let f(z)be an entire function of finite iterated order withσp(f) =σ> (p≥),
and let L(f) =af +af +af,where a,a,aare entire functions of finite iterated order
which are not all equal to zero and satisfy b=max{σp–(aj),j= , , }<α,thenσp(L(f)) = σp(f) =σ.
Proof L(f) can be written as
L(f) =f a
f f +a
f f +a
. (.)
By the Wiman-Valiron lemma (see [, ]), for allzsatisfying|z|=rand|f(z)|=M(r,f), we have
f(k)(z)
f(z) =
νf(r)
z
k
whereEis a set of finite logarithmic measure. From (..) in [, p.], for any given
holds outside a setEwith finite logarithmic measure, whereμf(r) is the maximum term
off. By the Cauchy inequality, we haveμf(r)≤M(r,f). Substituting it into (.), we have νf(r) <
logM(r,f)+ε, r∈/E. (.)
By Lemma ., there exists a setEhaving infinite logarithmic measure such that for all
|z|=r∈E, we have
Lemma . Let f(z)be an entire function withσp(f) =σ > (p≥)and L(f) =akf(k)+
ak–f(k–)+· · ·+af,where a,a, . . . ,akare entire functions which are not all equal to zero
satisfying b=max{σp–(aj),j= , , . . . ,k}<σ.Thenσp(L(f)) =σp(f) =σ.
Remark . By a similar proof to that of Lemma . in [] or Lemma in [], we can easily get the following lemma which is a better result than that in [] or [] by allowing
τp(f) =∞.
Lemma . Let f(z)be an entire function satisfying <σp(f) =σ<∞, <τp(f) =τ≤ ∞,
then for any givenβ<τ,there exists a set E⊂[, +∞)that has infinite logarithmic measure
such that for all r∈E,we have
logpM(r,f) >βrσ. (.)
Lemma .[] Let f(z)be a transcendental meromorphic function andα> be a given constant,for any givenε> ,there exists a set E⊂[,∞)that has finite logarithmic
mea-sure and a constant B> that depends only onαand(m,n) (m,n∈ {, . . . ,k})with m<n such that for all z satisfying|z|=r∈/[, ]∪E,we have
ff((mn))((zz))
≤B T(αr,f) r
logαrlogT(αr,f)
n–m
. (.)
Lemma .[] Let Aj(z) (j= , , . . . ,k– )be entire functions with finite iterated order
satisfyingmax{σp(Aj),j= }<σp(A),then every solution f ≡of(.)satisfiesσp+(f) = σp(A).
Lemma .[, ] Let Aj(z) (j= , , . . . ,k– )be entire functions with finite iterated order
satisfyingmax{σp(Aj),j= } ≤σp(A) ( <σp(A) <∞)andmax{τp(Aj),σp(Aj) =σp(A),j=
}<τp(A) <∞.Then every solution f ≡of(.)satisfiesσp+(f) =σp(A).
Remark . The conclusion of Lemma . also holds ifτp(A) =∞.
Lemma . Let A(z),B(z)be entire functions of finite iterated order satisfyingσp(A) <
σp(B).Let G(z),H(z)be meromorphic functions withσp(H)≤σp(A),σp(G)≤σp(B),if f(z)
is an entire solution of equation
f + A+G
G
f + B+H +HG
G
f= , (.)
thenσp+(f)≥σp(B).
Proof From (.), we have
m r,B+H +HG
G
≤m r,f
f
+m r,f
f
+m r,A+G
G
. (.)
By the lemma of logarithmic derivative and (.), we have
whereEis a set having finite linear measure. By Lemma ., there exists a setEhaving infinite logarithmic measure such that for all|z|=r∈E–E, we have
expp–rσp(B)–ε≤
OlogrT(r,f)+ expp–rσp(A)+ε
, (.)
where < ε<σ(B) –σ(A). By (.), we haveσp+(f)≥σp(B).
Lemma . Let A(z),B(z)be entire functions satisfying <σp(A) =σp(B) =σ<∞and
τp(A) <τp(B)≤ ∞ and let G(z),H(z) be meromorphic functions satisfyingσp(H)≤σ,
σp(G)≤σ and|H(j)(z)| ≤expp{(τ(A) +ε)rσ}(j= , )outside of a set E of finite
loga-rithmic measure,where < ε<τp(B) –τp(A).If f(z)is an entire solution of(.),then σp+(f)≥σ.
Proof Without loss of generality, we suppose thatτp(A) <τp(B) <∞. From (.), we have B(z)≤f
f
+
|A|+G
G
ff +H+|H|G G
. (.)
By Lemma ., for any givenβ(τp(A) + ε<β<τp(B)), there exists a setEhaving infinite logarithmic measure such that for all|z|=r∈E, we have
M(r,B) >exppβrσ. (.)
By Lemma ., there exists a setE having finite logarithmic measure such that for all
|z|=r∈/E, we have
ff ≤BT(r,f), f
f
≤BT(r,f),
GG≤BT(r,G)<expp–rσ+ε,
(.)
whereM> is a constant. By the hypotheses, for all|z|=r∈/E, we have
H<exppτ(A) +εrσ, |H|<exp
p
τ(A) +εrσ. (.)
By (.)-(.), for allzsatisfying|B(z)|=M(r,B) and|z|=r∈E– (E∪E), we have
exppβrσ≤exp
p
τp(A) + ε
rσT(r,f). (.)
By (.), we haveσp+(f)≥σ.
By the above proof, we can easily obtain that Lemma . also holds ifτp(B) =∞.
4 Proof of Theorem 2.1
Now we divide the proof of Theorem . into two cases: case (i)σp(A) <σp(B) and case (ii) τp(A) <τp(B) andσp(A) =σp(B) > .
Case (i): () We prove thatλp+(f–ϕ) =σp+(f). Assume thatf ≡ is a solution of (.),
σp+(f) =σp(B),λp+(g) =λp+(f –ϕ). Substitutingf =g+ϕ,f =g +ϕ,f =g +ϕ into
Substituting (.), (.) into (.), we obtain
by (.), we have
can also get the following equation which has a similar form to (.):
By Lemma . and Lemma ., for all sufficiently large |z|=r∈E–E and for anyε
By following the proof of ()-() in case (i) and the proof of () in case (ii), we can obtain
λp+(f(j)–ϕ) =λp+(f(j)–ϕ) =σp+(f) =σp(B) (j≥). 5 Proof of Theorems 2.2-2.3
Using a similar proof to that in case (i) of Theorem . and by Lemma ., we can easily obtain Theorem .. Theorem . is a direct result of Theorem . and Lemma ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
TJ completed the main part of this article, TJ, XZX and XHY corrected the main theorems. All authors read and approved the final manuscript.
Author details
1College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China.2Beijing Key
Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, No. 97 Bei Si Huan Dong Road Chaoyang District, Beijing, 100101, China.3Department of Informatics and Engineering, Jingdezhen
Ceramic Institute, Jingdezhen, Jiangxi 333403, China.
Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article. This project is supported by the National Natural Science Foundation of China (11171119, 61202313, 11261024, 11271045), and by the Natural Science Foundation of Jiangxi Province in China (20122BAB211005, 20114BAB211003, 20122BAB201016) and the Foundation of Education Bureau of Jiang-Xi Province in China (GJJ12206). Zuxing Xuan is supported in part by NNSFC (No. 11226089, 60972145), Beijing Natural Science Foundation (No. 1132013) and the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (IDHT20130513).
Received: 27 November 2012 Accepted: 27 February 2013 Published: 22 March 2013
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