STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER

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1 UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: / AM , STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER by Benhaat Belaïdi Abstact The main pupose of this pape is to study the gowth of solutions of the linea diffeential-diffeence equation n m Lz, f = A ijz f j z + c i = 0, i=0 j=0 whee A ijz i = 0,, n; j = 0,, m ae entie o meomophic functions of finite logaithmic ode and c i 0,, n ae distinct complex numbes We extend some pecedent esults due to Wu and Zheng and othes 1 Intoduction and main esults Thoughout this pape, we assume that eades ae familia with the standad notations and the fundamental esults of the Nevanlinna value distibution theoy of meomophic functions [12, 19] Let f be a meomophic function; we define and N, f = m, f = 1 2π 0 2π 0 log + f e iϕ dϕ, nt, f n0, f dt + n0, f log, t T, f = m, f + N, f > Mathematics Subject Classification 30D35, 34K06, 34K12 Key wods and phases Linea diffeential-diffeence equation, meomophic function, logaithmic ode, logaithmic type, logaithmic lowe ode, logaithmic lowe type

2 16 is the Nevanlinna chaacteistic function of f, whee log + x = max0, log x fo x 0, and nt,, f = nt, f is the numbe of poles of fz lying in z t, counted accoding to thei multiplicity Also, fo a, we define 1 m, = m, a, f = 1 2π log + 1 f a 2π 0 fe iϕ a dϕ, 1 nt, a, f n0, a, f N, = N, a, f = dt + n0, a, f log, f a 0 t whee nt, a, f is the numbe of zeos of the equation fz = a lying in z t, counted accoding to thei multiplicity Also, we use the notations µf, ρf to denote the lowe ode and the ode of a meomophic function f To expess the ate of gowth of meomophic solutions of infinite ode, we ecall the following definition Definition 11 [16, 19] Let f be a meomophic function Then the hype-ode ρ 2 f of fz is defined by ρ 2 f = lim sup + log log T, f log If f is an entie function, then the hype-ode of fz is defined as log log T, f ρ 2 f = lim sup + log = lim sup + log log log M, f, log whee M, f is the maximum modulus of f in the cicle z = Definition 12 [12] Let f be an entie function of ode ρ 0 < ρ < + The type of f is defined as log M, f τf = lim sup + ρ Similaly, the lowe type of an entie function f of lowe ode µ 0 < µ < is defined by log M, f τf = lim inf + µ Definition 13 [12, 19] Fo a C = C, the deficiency of a with espect to a meomophic function f is defined as 1 1 m, f a N, f a δa, f = lim inf = 1 lim sup, a, + T, f + T, f δ, f = lim inf + m, f T, f = 1 lim sup + N, f T, f

3 17 Recently, the diffeence countepats of Nevanlinna theoy have been established The key esult is the diffeence analogue of the lemma on the logaithmic deivative obtained by Halbud Kohonen [10] and Chiang Feng [6], independently Subsequently Halbud and Kohonen [11] showed how all key esults of the Nevanlinna theoy have coesponding diffeence vaiants as well Afte that, it was with a gowing inteest that solutions to diffeence equations in the complex domain have been investigated by making use of this vaiant of the value distibution theoy, see [4, 15, 17, 18, 20] In [15], Laine and Yang consideed complex linea diffeence equations and obtained the following theoem Theoem A [15] Let A 0 z, A 1 z,, A n z be entie functions of finite ode such that among those having the maximal ode ρ = max ρa j, 0 j n thee is exactly one whose type is stictly geate than the othes Then fo any meomophic solution of A n z fz + n + A n 1 z fz + n A 1 z fz A 0 z fz = 0, we have ρf ρ + 1 In [16], Tu and Yi investigated the gowth of solutions of a class of highe ode linea diffeential equations with entie coefficients when most of them ae of the same ode, and obtained the following esult Theoem B [16] Let A 0 z,, A k 1 z be entie functions such that ρa 0 = ρ 0 < ρ < + and τa 0 = τ 0 < τ < +, and let ρa j ρa 0 = ρ j = 1, 2,, k 1, τa j < τa 0 = τ j = 1, 2,, k 1 if ρa j = ρa 0 j = 1, 2,, k 1 Then evey solution f 0 of f k + A k 1 z f k A 1 z f + A 0 z f = 0, satisfies ρf = + and ρ 2 f = ρa 0 = ρ Fom Theoems A and B, we deduce that when thee is exactly one dominant coefficient among those coefficients having the same maximal ode, we may obtain the gowth elation between the solutions and the coefficients of the above complex linea diffeence equation o complex linea diffeential equation In ecent pape [18], Wu and Zheng investigated the gowth of meomophic solutions of the linea diffeential-diffeence equation n m 11 Lz, f = A ij z f j z + c i = 0, i=0 j=0 whee A ij z i = 0,, n; j = 0,, m ae entie o meomophic functions of finite ode and c i 0,, n ae distinct complex numbes, whee thee is only one dominant coefficient Hence, fom Theoems A and B a natual

4 18 question emeges: How to expess the gowth of solutions of 11 when all coefficients A 0 z, A 1 z,, A n z ae entie o meomophic functions and of ode zeo in C? The main pupose of this pape is to make use the concept of finite logaithmic ode due to Chen [5] to extend pevious esults of Wu and Zheng [18] fo meomophic solutions to equation 11 of zeo ode in C We ecall the following definitions Definition 14 [5] The logaithmic ode of a meomophic function f is defined as log T, f ρ log f = lim sup + log log If f is an entie function, then ρ log f = lim sup + log log M, f log log Remak 11 Obviously, the logaithmic ode of any non-constant ational function f is one, and thus, any tanscendental meomophic function in the plane has logaithmic ode no less than one Howeve, a function of logaithmic ode one is not necessaily a ational function Constant functions have zeo logaithmic ode, while thee ae no meomophic functions of logaithmic ode between zeo and one Moeove, any meomophic function with finite logaithmic ode in the plane is of ode zeo Definition 15 The logaithmic lowe ode of a meomophic function f is defined as log T, f µ log f = lim inf + log log If f is an entie function, then µ log f = lim inf + log log M, f log log Definition 16 [3] The logaithmic type of an entie function f with 1 ρ log f < + is defined by τ log f = lim sup + log M, f log ρ logf Similaly the logaithmic lowe type of an entie function f with 1 µ log f < + is defined by log M, f τ log f = lim inf + log µ logf

5 19 Remak 12 It is evident that the logaithmic type of any non-constant polynomial P equals its degee degp ; that any non-constant ational function is of finite logaithmic type, and that any tanscendental meomophic function whose logaithmic ode equals one in the plane must be of infinite logaithmic type Recently, the concept of logaithmic ode has been used to investigate the gowth and the oscillation of solutions of linea diffeential equations in the complex plane [3] and complex linea diffeence and q-diffeence equations in the complex plane and in the unit disc [1, 2, 13, 14, 17] In what follows, we conside the gowth estimates of meomophic solutions of the homogeneous equation 11 with some coefficients having the same maximal ode o maximal lowe ode, and we obtain the following esults Theoem 11 Let A ij z i = 0,, n; j = 0,, m be entie functions such that thee exists an intege s 0 s n satisfying 12 maxρ log A ij : i, j s, 0 ρ log A s0 <, and 13 maxτ log A ij : ρ log A ij = ρ log A s0, i, j s, 0 < τ log A s0 Then evey meomophic solution f 0 of equation 11 satisfies ρ log f ρ log A s0 + 1 Theoem 12 Let A ij z i = 0,, n; j = 0,, m be entie functions such that thee exists an intege s 0 s n satisfying 14 maxρ log A ij : i, j s, 0 µ log A s0 <, and 15 maxτ log A ij : ρ log A ij = µ log A s0, i, j s, 0 < τ log A s0 Then evey meomophic solution f 0 of equation 11 satisfies µ log f µ log A s0 + 1 Theoem 13 Let H be a set of complex numbes satisfying log dens z : z H > 0, and let A ij z i = 0,, n; j = 0,, m be entie functions satisfying maxρ log A ij : i = 0,, n; j = 0,, m ρ with 1 ρ < + If thee exists an intege s 0 s n such that fo some constants 0 β < α and sufficiently small ε 0 < ε < ρ, we have 16 A s0 z exp α [log ] ρ ε and 17 A ij z exp β [log ] ρ ε, i, j s, 0 as z = + fo z H, then evey meomophic solution f 0 of equation 11 satisfies ρ log f ρ log A s0 + 1

6 20 Remak 13 By the assumptions of Theoem 13, we obtain ρ log A s0 = ρ Indeed, we have ρ log A s0 ρ Suppose that ρ log A s0 = µ < ρ Then, by Definition 14 and 16, we have fo any given ε 0 < ε < ρ µ 2 exp α [log ] ρ ε A s0 z exp [log ] µ+ε as z = + fo z H By ε 0 < ε < ρ µ 2 this is a contadiction as + Hence ρ log A s0 = ρ Theoem 14 Let A ij z i = 0,, n; j = 0,, m be entie functions of finite logaithmic ode such that thee exists an intege s 0 s n satisfying m, A ij 18 lim sup + i,j s,0 m, A s0 < 1 Then evey meomophic solution f 0 of equation 11 satisfies ρ log f ρ log A s0 + 1 The following theoems give some popeties of the logaithmic ode of meomophic solutions of 11 in the case when the coefficients ae meomophic functions Theoem 15 Let A ij z i = 0,, n; j = 0,, m be meomophic functions such that thee exists an intege s 0 s n satisfying ρ log A s0 > maxρ log A ij : i, j s, 0 and δ, A s0 > 0 Then evey meomophic solution f 0 of equation 11 satisfies ρ log f ρ log A s0 + 1 Theoem 16 Let A ij z i = 0,, n; j = 0,, m be meomophic functions of finite logaithmic ode such that thee exists an intege s m,a ij i,j s,0 0 s n satisfying lim sup m,a + s0 < 1 and δ, A s0 > 0 Then evey meomophic solution f 0 of equation 11 satisfies ρ log f ρ log A s Some lemmas We ecall the following definitions The linea measue of a set E 0, + is defined as me = + 0 χ E t dt and the logaithmic measue of a set F 1, + is defined by lmf = + χ F t 1 t dt, whee χ H t is the chaacteistic function of a set H The uppe density of a set E 0, + is defined by dense = lim sup + me [0, ] The uppe logaithmic density of a set F 1, + is defined by log densf = lim sup + lmf [1, ] log

7 21 It is easy to obtain the following emak Remak 21 Fo all H [1, + the following statements hold: i If lmh =, then mh = ; ii If densh > 0, then mh = ; iii If log densh > 0, then lmh = Lemma 21 [8] Let fz be a tanscendental meomophic function in the plane, and let α > 1 be a given constant Then thee exist a set E 1 1, + of finite logaithmic measue, and a constant B > 0 depending only on α and m, n m, n 0, 1,, k m < n such that fo all z with z = [0, 1] E 1, we have f n z T α, f n m f m z B log α log T α, f Fom the above lemma, we obtain the following esult Lemma 22 Let fz be a tanscendental meomophic function in the plane with 1 ρ log f = ρ < +, and let ε > 0, α > 1 be given constants Then thee exist a set E 2 1, + of finite logaithmic measue, and m, n m, n 0, 1,, k m < n such that fo all z with z = [0, 1] E 2, we have f n z log ρ+α+ε n m f m z Poof Since fz has finite logaithmic ode ρ log f = ρ < +, so given ε 0 < ε < 2 and sufficiently lage > R, we have 21 T, f < log ρ+ ε 2 Combining 21 with Lemma 21, fo α > 1, thee exist a set E 2 = [0, R] E 1 of finite logaithmic measue and a constant B > 0, such that if z = [0, 1] E 2, we obtain f n z f m z B log α ρ+ ε n m 2 log α loglog α ρ+ ε 2 22 log ρ+α+ε n m Remak 22 It is shown in [7, p 66], that fo an abitay complex numbe c 0, the following inequalities 1 + o1 T c, fz T, fz + c 1 + o1 T + c, fz

8 22 hold as + fo an abitay meomophic function fz Theefoe, it is easy to obtain ρ log f + c = ρ log f, µ log f + c = µ log f Lemma 23 [1] Let η 1, η 2 be two abitay complex numbes such that η 1 η 2 and let fz be a finite logaithmic ode meomophic function Let ρ be the logaithmic ode of fz Then fo each ε > 0, we have m, fz + η 1 fz + η 2 = O log ρ 1+ε Lemma 24 [6] Let f be a meomophic function, η a non-zeo complex numbe, and let γ > 1, and ε > 0 be given eal constants Then thee exist a subset E 3 1, + of finite logaithmic measue, and a constant A depending only on γ and η, such that fo all z = / E 3 [0, 1], we have log fz + η T γ, f fz A whee nt = nt,, f + nt,, 1/f + nγ log γ log + nγ, Lemma 25 [8] Let f be a tanscendental meomophic function, let j be a non-negative intege, let a be a value in the extended complex plane, and let α > 1 be a eal constant Then thee exists a constant R > 0 such that fo all > R, we have 23 n, a, f j 2j + 6 T α, f log α Lemma 26 Let f be a meomophic function with 1 µ log f < + Then thee exists a set E 4 1, + of infinite logaithmic measue such that fo E 4 1, +, we have T, f < log µ logf+ε Poof By definition of the logaithmic lowe ode, thee exists a sequence n n=1 tending to, satisfying n n < n+1 and log T n, f lim = µ log f n log log n Then fo any given ε > 0, thee exists an intege n 1 such that fo all n n 1, T n, f < log n µ logf+ ε 2 Set E 4 = [ ] n n+1 n, n Then fo E 4 1, +, we obtain n=n 1 T, f T n, f < log n µ logf+ ε 2 log n + 1 n µ log f+ ε 2 < log µ log f+ε,

9 23 and lme 4 = n n=n 1 n n+1 n dt t = log n = Thus, Lemma 26 is poved n=n 1 Lemma 27 Let f be a meomophic function, η a non-zeo complex numbe, and ε > 0, β > 1 be given eal constants Then thee exists a subset E 5 1, + of finite logaithmic measue, such that if f has finite logaithmic ode ρ, then fo all z = / E 5 [0, 1], we have log ρ+β+ε 24 exp fz + η fz exp log ρ+β+ε Poof By Lemma 24, thee exist a subset E 5 1, + of finite logaithmic measue, and a constant A depending only on γ and η, such that fo all z = / E 5 [0, 1], we have 25 log fz + η T γ, f fz A + nγ log γ log + nγ, whee nt = nt,, f + nt,, 1/f By using 23 and 25, we obtain log fz + η T γ, f fz A + 12 T αγ, f 12 log γ log + T αγ, f log α log α B T β, f logβ 26 log T β, f, whee B > 0 is some constant and β = αγ > 1 Since fz has finite logaithmic ode ρ log f = ρ < +, so given ε, 0 < ε < 2, fo sufficiently lage, we have 27 T, f < log ρ+ ε 2 Then by using 26 and 27, we obtain log fz + η fz B T β, f logβ log T β, f 28 Blog β ρ+ ε logβ 2 loglog β ρ+ ε 2 Fom 28, we easily obtain 24 log ρ+β+ε Lemma 28 Let η 1, η 2 be two abitay complex numbes such that η 1 η 2 and let fz be a meomophic function of finite logaithmic ode ρ Let

10 24 ε > 0 and β > 1 be given Then thee exists a subset E 6 1, + of finite logaithmic measue such that fo all z = / E 6, we have log ρ+β+ε 29 exp fz + η 1 fz + η 2 exp log ρ+β+ε Poof We can wite fz + η 1 fz + η 2 = fz + η 2 + η 1 η 2 fz + η 2 η 1 η 2 Then by using Lemma 27, fo any given ε > 0, β > 1 and all z + η 2 = R / E 5 [0, 1], such that lme 5 <, we obtain log ρ+β+ε exp exp log z + η 2 ρ+β+ ε 2 z + η 2 = exp log ε Rρ+β+ 2 fz + η 1 R fz + η 2 = fz + η 2 + η 1 η 2 fz + η 2 exp log R ρ+β+ ε 2 R log z + η 2 ρ+β+ ε 2 log ρ+β+ε exp exp, z + η 2 whee z = / E 6 and E 6 is a set of finite logaithmic measue By using Lemmas 24 26, we can genealize Lemma 22 and Lemma 28 into finite logaithmic lowe ode case as following Lemma 29 Let fz be a tanscendental meomophic function in the plane with 1 µ log f = µ < +, and let ε > 0, α > 1 be given constants Then thee exist a set E 7 1, + of infinite logaithmic measue, and m, n m, n 0, 1,, k m < n such that fo all z with z = E 7, we have f n z log µ+α+ε n m f m z Lemma 210 Let η 1, η 2 be two abitay complex numbes such that η 1 η 2 and let fz be a meomophic function of finite logaithmic lowe ode µ Let ε > 0 and β > 1 be given Then thee exists a subset E 8 1, + of infinite logaithmic measue such that fo all z = E 8, we have exp log µ+β+ε fz + η 1 fz + η 2 exp log µ+β+ε

11 25 Lemma 211 [1] Let f be a meomophic function with ρ log f 1 Then thee exists a set E 9 1, + of infinite logaithmic measue such that log T, f lim = ρ + log log E 9 Lemma 212 [1] Let f 1, f 2 be meomophic functions satisfying ρ log f 1 > ρ log f 2 Then thee exists a set E 10 1, + of infinite logaithmic measue such that fo all E 10, we have T, f 2 lim + T, f 1 = 0 Lemma 213 Let f be an entie function with 1 µ log f < + Then thee exists a set E 11 1, + of infinite logaithmic measue such that τ log f = log M, f lim log µ logf E 11 + Poof By the definition of the logaithmic lowe type, thee exists a sequence n n=1 tending to, satisfying n n < n+1, and τ log f = lim n log M n, f log n µ logf Then [ fo any given ε > 0, thee exists an n 1 such that fo n n 1 and any n n+1 n, n ], we have log M n n+1 n, f log n µ logf log M, f log µ logf log M n, f log n n+1 n µ logf It follows that log n n+1 µlog n f log M n n+1 n, f log M, f log n log n n+1 n µ logf log µ logf log M µlog f n, f log n log n µ logf log Set Then we have E 11 = log M, f lim log µ logf = E 11 + n=n 1 n n+1 n [ ] n n + 1 n, n lim n + log M n, f log n µ logf = τ logf

12 26 and lme 11 = E 11 d = + n n=n 1 n n+1 n dt t = + log n = + n=n 1 Lemma 214 [9] Let ϕ : [0, + R and ψ : [0, + R be monotone non-deceasing functions such that ϕ ψ fo all / E 12 [0, 1], whee E 12 1, + is a set of finite logaithmic measue Let γ > 1 be a given constant Then thee exists an 0 = 0 γ > 0 such that ϕ ψγ fo all > 0 3 Poofs of the Theoems Poof of Theoem 11 Let f 0 be a meomophic solution of 11 We suppose ρ log f < ρ log A s0 + 1 < + We divide though equation 11 by fz + c s to get 31 A s0 z n m A ij z i=0 i s + j=0 m A sj z j=1 In elation to 12 and 13, we set and f j z + c i fz + c i f j z + c s fz + c s fz + c i fz + c s ρ = maxρ log A ij : i, j s, 0, τ = maxτ log A ij : ρ log A ij = ρ log A s0 : i, j s, 0 Then fo a sufficiently lage, we have 32 A ij z exp log ρ+ε, i, j s, 0 if ρ log A ij < ρ log A s0, and 33 A ij z exp τ + ε log ρ loga s0, i, j s, 0 if ρ log A ij = ρ log A s0 By Lemma 22 and Remak 22, fo any given ε > 0 and α > 1, thee exists a set E 2 1, + of finite logaithmic measue such that fo all z = / [0, 1] E 2, we have 34 f j z + c i fz + c i log ρ logfz+c i +α+ε j log ρ j logf+α+ε = i = 0, 1,, n, j = 1,, m

13 27 By Lemma 28, thee exists a set E 6 1, + of finite logaithmic measue such that fo all z = / E 6, we have fo any given ε > 0 and β > 1 35 fz + c i fz + c s exp log ρ logf+β+ε i = 0, 1,, n, i s Then we can choose an ε > 0 sufficiently small to satisfy 36 τ + 2ε < τ log A s0, max ρ, ρ log f 1 + 2ε < ρ log A s0 Substituting 32, 33, 34 and 35 into 31, fo z = / [0, 1] E 2 E 6, we get log ρ logf+β+ε M, A s0 exp O exp τ + ε log ρ loga s exp log ρ+ε log ρ m logf+α+ε, whee A s0 z = M, A s0 By 36 and 37 and Lemma 214, we get log M, A s0 τ log A s0 = lim sup + log ρ loga s0 τ + ε < τ loga s0 ε which is a contadiction Hence ρ log f ρ log A s0 + 1 Poof of the Theoem 12 Hee, we use a method simila to the one in the poof of Theoem 11 Let f 0 be a meomophic solution of 11 We suppose µ log f < µ log A s0 + 1 < + In elation to 14 and 15, we set ρ 1 = maxρ log A s0 : i, j s, 0, and τ 1 = maxτ log A ij : ρ log A s0 = µ log A s0 : i, j s, 0 Then fo a sufficiently lage, we have 38 A ij z exp log ρ 1+ε, i, j s, 0 if ρ log A ij < µ log A s0, and 39 A ij z exp τ 1 + ε log µ loga s0, i, j s, 0 if ρ log A ij = µ log A s0 By Remak 22, Lemma 29 and Lemma 210, fo any given ε > 0, α > 1, β > 1, thee exists a set E 8 1, + of infinite logaithmic measue such that fo all z = E 8, we have 310 f j z + c i fz + c i log µ j logf+α+ε i = 0, 1,, n, j = 1,, m

14 28 and 311 fz + c i fz + c s exp log µ logf+β+ε i = 0, 1,, n, i s Then we can choose an ε > 0 sufficiently small to satisfy 312 τ 1 + 2ε < τ log A s0, max ρ 1, µ log f 1 + 2ε < µ log A s0 Substituting into 31, fo z = E 8, we get log µ logf+β+ε M, A s0 exp O exp τ 1 + ε log µ loga s exp log ρ 1+ε log µ m logf+α+ε, whee A s0 z = M, A s0 By 312, 313 and Lemma 213, we get log M, A s0 τ log A s0 = lim inf + log µ loga s0 τ 1 + ε < τ log A s0 ε E 8 which is a contadiction Hence µ log f µ log A s0 + 1 Poof of the Theoem 13 By Remak 13, we know that ρ log A s0 = ρ Let f 0 be a meomophic solution of 11 We suppose ρ log f < ρ log A s0 +1 = ρ+1 < + By the assumptions of Theoem 13, thee is a set H of complex numbes satisfying log dens z : z H > 0 such that fo z H, we have 16 and 17 as z = + Set H 1 = = z : z H, since log dens z : z H > 0 Then by Remak 21, fo H 1 thee is d H 1 = Clealy, 34 and 35 hold fo all z satisfying z = / [0, 1] E 2 E 6, whee E 2 and E 6 ae defined similaly as in the poof of Theoem 11 Substituting 16, 17, 34 and 35 into 31, fo z = H 1 \ [0, 1] E 2 E 6, and any given ε 0 < ε < ρ ρ logf+1 2, we get exp α [log ] ρ ε n exp β [log ] ρ ε log ρ logf+β+ε log ρ m logf+α+ε exp It follows that 314 exp α β [log ] ρ ε log ρ logf+β+ε log ρ m logf+α+ε n exp

15 29 By 0 < ε < ρ ρ logf+1 2 and 314, we obtain a contadiction Hence we get ρ log f ρ + 1 = ρ log A s0 + 1 Poof of the Theoem 14 Let f 0 be a meomophic solution of 11 If ρ log f =, then the esult is tivial Now we suppose ρ log f < + We divide though equation 11 by fz + c s to get 315 n m A s0 z = A ij z f j z + c i fz + c i m fz + c i fz + c s + A sj z f j z + c s fz + c s It follows 316 i=0 i s m, A s0 Suppose that j=0 n m m, A ij + i=0 i s j=0 n m m i=0 j=1 i,j s,0 m m, A sj j=1, f j z + c i fz + c i m, A ij m, A s0 + n m i=0 i s j=1 < 1 = µ < λ < 1 Then fo a sufficiently lage, we have 318 m, A ij < λm, A s0 i,j s,0, fz + c i fz + c s + O1 By Lemma 23, fo a sufficiently lage and any given ε > 0, we have 319 m, fz + c i = O log ρ logf 1+ε, i = 0,, n, i s fz + c s The logaithmic deivative lemma and Remak 22 lead to 320 m, f j z + c i = O loglog ρ logf 1+ε, j = 1,, m fz + c i Thus, by substituting 318, 319 and 320 into 316, fo a sufficiently lage and any given ε > 0, we obtain 321 m, A s0 λm, A s0 + O log ρ logf 1+ε + O log log ρ logf 1+ε

16 30 Fom 321, it follows that λ m, A s0 O log ρ logf 1+ε + O log log ρ logf 1+ε By 322, we obtain ρ log f ρ log A s0 +1 Thus, Theoem 14 is poved Poof of the Theoem 15 Let f 0 be a meomophic solution of 11 If ρ log f =, then the esult is tivial Now we suppose ρ log f < + Set 323 δ, A s0 = lim inf + m, A s0 T, A s0 = δ > 0 Thus fom 323, fo a sufficiently lage, we have 324 m, A s0 > 1 2 δt, A s0 Thus, by substituting 319, 320 and 324 into 316, fo a sufficiently lage and any given ε > 0, we obtain δ n m m 2 T, A s0 < m, A s0 m, A ij + m, A sj n m m i=0 j=1 i=0 i s j=0 n m T, A ij + i=0 i s j=0, f j z + c i fz + c i + j=1 n i=0 i s m, fz + c i + O1 fz + c s m T, A sj + O log ρ logf 1+ε j=1 + O loglog ρ logf 1+ε Since max ρ log A ij : i, j s, 0 < ρ log A s0, then by Lemma 212, thee exists a set E 10 1, + of infinite logaithmic measue such that T, Aij 326 max : i, j s, 0 0, +, E 10 T, A s0 Thus, by 325 and 326, fo all E 10, +, we have 327 δ 2 o1 T, A s0 O log ρ logf 1+ε + O log log ρ logf 1+ε It now follows fom 327 and Lemma 211 that ρ log f ρ log A s0 + 1 Thus, Theoem 15 is poved

17 31 Poof of the Theoem 16 Let f 0 be a meomophic solution of 11 If ρ log f =, then the esult is tivial Now we suppose ρ log f < + As in the poof of Theoem 14, by substituting 318, 319 and 320 into 316, fo a sufficiently lage and any given ε > 0, we have λ m, A s0 O log ρ logf 1+ε + O loglog ρ logf 1+ε By Lemma 211, we have log T, A s0 329 lim = ρ log A s0, + log log E 9 whee E 9 is a set of of infinite logaithmic linea measue Since δ, A s0 = m,a lim inf s0 + T,A s0 > 0, we obtain log m, A s0 330 lim = ρ log A s0 + log log E 9 Thus, by 328 and 330, we obtain ρ log f ρ log A s0 + 1 Thus, Theoem 16 is poved Acknowledgements The autho is gateful to the efeee fo his/he valuable comments which helped to impove this pape Refeences 1 Belaïdi B, Gowth of meomophic solutions of finite logaithmic ode of linea diffeence equations, Fasc Math, , Belaïdi B, Some popeties of meomophic solutions of logaithmic ode to highe ode linea diffeence equations, submitted 3 Cao T B, Liu K, Wang J, On the gowth of solutions of complex diffeential equations with entie coefficients of finite logaithmic ode, Math Repots, 1565, , Chen Z X, Shon K H, On gowth of meomophic solutions fo linea diffeence equations, Abst Appl Anal, 2013, At ID , Chen P T Y, On meomophic functions with finite logaithmic ode, Tans Ame Math Soc, , no 2, Chiang Y M, Feng S J, On the Nevanlinna chaacteistic of fz + η and diffeence equations in the complex plane, Ramanujan J, , no 1, Goldbeg A, Ostovskii I, Value Distibution of Meomophic functions, Tansl Math Monog, 236, Ame Math Soc, Povidence RI, Gundesen G G, Estimates fo the logaithmic deivative of a meomophic function, plus simila estimates, J London Math Soc 2, , no 1, Gundesen G G, Finite ode solutions of second ode linea diffeential equations, Tans Ame Math Soc, , no 1,

18 32 10 Halbud R G, Kohonen R J, Diffeence analogue of the lemma on the logaithmic deivative with applications to diffeence equations, J Math Anal Appl, , no 2, Halbud R G, Kohonen R J, Nevanlinna theoy fo the diffeence opeato, Ann Acad Sci Fenn Math, , no 2, Hayman W K, Meomophic functions, Oxfod Mathematical Monogaphs Claendon Pess, Oxfod Heittokangas J, Wen Z T, Functions of finite logaithmic ode in the unit disc, Pat I J Math Anal Appl, , no 1, Heittokangas J, Wen Z T, Functions of finite logaithmic ode in the unit disc, Pat II Comput Methods Funct Theoy, , no 1, Laine I, Yang C C, Clunie theoems fo diffeence and q-diffeence polynomials, J Lond Math Soc 2, , no 3, Tu J, Yi C F, On the gowth of solutions of a class of highe ode linea diffeential equations with coefficients having the same ode, J Math Anal Appl, , no 1, Wen Z T, Finite logaithmic ode solutions of linea q-diffeence equations, Bull Koean Math Soc, , no 1, Wu S Z, Zheng X M, Gowth of meomophic solutions of complex linea diffeentialdiffeence equations with coefficients having the same ode, J Math Res Appl, , no 6, Yang C C, Yi H X, Uniqueness theoy of meomophic functions, Mathematics and its Applications, 557, Kluwe Academic Publishes Goup, Dodecht, Zheng X M, Tu J, Gowth of meomophic solutions of linea diffeence equations, J Math Anal Appl, , no 2, Received Mach 29, 2016 Depatment of Mathematics Laboatoy of Pue and Applied Mathematics Univesity of Mostaganem UMAB B P 227 Mostaganem Algeia benhaatbelaidi@univ-mostadz