DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS

Transcription

1 Journal o Applied Analysis Vol. 14, No , pp DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS B. BELAÏDI and A. EL FARISSI Received December 5, 2007 and, in revised orm, July 14, 2008 Abstract. In tis paper, we study ixed points o solutions o te dierential equation + A 1 z + A 0 z = 0, were A j z 0 j = 0, 1 are transcendental meromorpic unctions wit inite order. Instead o looking at te zeros o z z, we proceed to a sligt generalization by considering zeros o g z ϕ z, were g is a dierential polynomial in wit polynomial coeicients, ϕ is a small meromorpic unction relative to, wile te solution is o ininite order. 1. Introduction and main results Trougout tis paper, we assume tat te reader is amiliar wit te undamental results and te standard notations o te Nevanlinna s value distribution teory and wit te basic Wiman Valiron teory as well see 2000 Matematics Subject Classiication. Primary: 34M10, 30D35. Key words and prases. Linear dierential equations, meromorpic solutions, yper order, exponent o convergence o te sequence o distinct zeros, yper exponent o convergence o te sequence o distinct zeros. ISSN c Heldermann Verlag.

2 260 B. BELAÏDI AND A. EL FARISSI [7], [8], [10], [13], [14]. In addition, we will use λ and λ 1/ to denote respectively te exponents o convergence o te zero-sequence and te polesequence o a meromorpic unction, ρ to denote te order o growt o, λ and λ 1/ to denote respectively te exponents o convergence o te sequence o distinct zeros and distinct poles o. A meromorpic unction ϕ z is called a small unction o a meromorpic unction z i T r, ϕ = o T r, as r +, were T r, is te Nevanlinna caracteristic unction o. In order to express te rate o growt o meromorpic solutions o ininite order, we recall te ollowing deinition. Deinition 1.1 [2], [12], [16]. Let be a meromorpic unction. te yper order ρ 2 o z is deined by ρ 2 = lim r + Ten log log T r,. 1.1 log r Deinition 1.2 [2], [9], [12]. Let be a meromorpic unction. Ten te yper exponent o convergence o te sequence o distinct zeros o z is deined by log log N r, 1 λ 2 = lim, 1.2 r + log r were N r, 1/ is te counting unction o distinct zeros o z in { z < r}. Consider te second order linear dierential equation + A 1 z + A 0 z = 0, 1.3 were A j z 0 j = 0, 1 are transcendental meromorpic unctions wit inite order. Many important results ave been obtained on te ixed points o general transcendental meromorpic unctions or almost our decades see [17]. However, tere are a ew studies on te ixed points o solutions o dierential equations. It was in year 2000 tat Z. X. Cen irst pointed out te relation between te exponent o convergence o distinct ixed points and te rate o growt o solutions o second order linear dierential equations wit entire coeicients see [2]. In [15], Wang and Yi investigated ixed points and yper order o dierential polynomials generated by solutions o second order linear dierential equations wit meromorpic coeicients. In [11], Laine and Rieppo gave improvement o te results o [15] by considering ixed points and iterated order. In [3], Cen Zongxuan and Son Kwang Ho ave studied te dierential equation + A 1 z e az + A 0 z e bz = 0 1.4

3 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 261 and ave obtained te ollowing results: Teorem A [3]. Let A j z 0 j = 0, 1 be meromorpic unctions wit ρ A j < 1 j = 0, 1, a, b be complex numbers suc tat ab 0 and arg a arg b or a = cb 0 < c < 1. Ten every meromorpic solution z 0 o te equation 1.4 as ininite order. In te same paper, Z. X. Cen and K. H. Son ave investigated te ixed points o solutions, teir 1 st, 2 nd derivatives and dierential polynomial and ave obtained: Teorem B [3]. Let A j z j = 0, 1, a, b, c satisy te additional ypoteses o Teorem A. Let d 0, d 1, d 2 be complex constants tat are not all equal to zero. I z 0 is any meromorpic solution o equation 1.4, ten: i,, all ave ininitely many ixed points and satisy λ z = λ z = λ z = +, ii te dierential polynomial g z = d 2 + d 1 + d 0 as ininitely many ixed points and satisies λ g z = +. In tis paper, we study te relation between te small unctions and solutions o equation 1.3 in te case wen all meromorpic solutions are o ininite order and we obtain te ollowing results: Teorem 1.1. Let A j z 0 j = 0, 1 be transcendental meromorpic unctions wit inite order suc tat all meromorpic solutions o equation 1.3 are o ininite order, let d j j = 0, 1, 2 be polynomials tat are not all equal to zero, ϕ z 0 be a meromorpic unction o inite order satisying d 1 d 2 A 1 ϕ d 2 A 2 1 d 2 A 1 d 2 A 0 d 1 A 1 + d 0 + d 1 ϕ I 0 is a meromorpic solution o equation 1.3 wit λ 1/ < +, ten te dierential polynomial g z = d 2 + d 1 + d 0 satisies λ g ϕ = +. Teorem 1.2. Suppose tat A j z 0 j = 0, 1, ϕ z 0 satisy te ypoteses o Teorem 1.1. I z 0 is a meromorpic solution o 1.3 wit ρ = + and ρ 2 = ρ, ten satisies λ ϕ = λ ϕ = λ ϕ = ρ = +, 1.6 λ 2 ϕ = λ 2 ϕ = λ 2 ϕ = ρ 2 = ρ. 1.7

4 262 B. BELAÏDI AND A. EL FARISSI Corollary. Let A 0 z be a transcendental entire unction wit ρ A 0 < 1, let d j j = 0, 1, 2 be polynomials tat are not all equal to zero, ϕ z 0 be an entire unction o inite order. I is a nontrivial solution o te equation + e z + A 0 z = 0, 1.8 ten te dierential polynomial g z = d 2 + d 1 + d 0 satisies λ g ϕ = Auxiliary lemmas We need te ollowing lemmas in te proos o our teorems. Lemma 2.1 [6]. Let be a transcendental meromorpic unction o inite order ρ, let Γ = {k 1, j 1, k 2, j 2,..., k m, j m } denote a inite set o distinct pairs o integers tat satisy k i > j i 0 or i = 1,..., m and let ε > 0 be a given constant. Ten te ollowing two statements old: i Tere exists a set E 1 [0, 2π tat as linear measure zero, suc tat i ψ [0, 2π\E 1, ten tere is a constant R 1 = R 1 ψ > 1 suc tat or all z satisying arg z = ψ and z R 1 and or all k, j Γ, we ave k z j z z k jρ 1+ε. 2.1 ii Tere exists a set E 2 1, + tat as inite logaritmic measure lm E 2 = + 1 χ E2 t dt, t were χ E2 is te caracteristic unction o E 2, suc tat or all z satisying z / E 2 [0, 1] and or all k, j Γ, we ave k z j z z k jρ 1+ε. 2.2 Lemma 2.2. Let A j j = 0, 1 and A 2 0 be meromorpic unctions wit ρa j < 1 j = 0, 1, 2. We denote ψ 2 z = A 0 + A 1 e z. Let ψ 21, ψ 20 ave te orm o ψ 2, and ϕ 0 be a meromorpic unction o inite order. Ten i ψ 2 z + A 2 e 2z 0; ϕ ii ψ 21 ϕ + ψ 20 + A 2 e 2z 0.

5 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 263 Proo. Suppose tat te claim ails. As or i, i A 2 e 2z +A 1 e z +A 0 0, ten 2T r, e z T r, A 2 e 2z 1 + T r, A 2 = T r, A 1 e z 1 + A 0 + T r, T r, e z + A 2 2 T r, A j + O 1, 2.3 ence ρ e z < 1, a contradiction. As or ii, te let and side can be written as ollows ψ 21 ϕ ϕ + ψ 20 + A 2 e 2z j=0 ϕ = A 00 + A 01 A ϕ + ϕ 10 + A 11 ϕ e z + A 2 e 2z, 2.4 were A 00, A 01, A 10, A 11 are meromorpic unctions o order < 1. Since ϕ is o inite order, ten by te lemma o logaritmic derivative [7] m r, ϕ = O ln r. 2.5 ϕ Tereore, by a reasoning as to above, but using te proximity unctions instead o te caracteristic, a contradiction ρ e z < 1 again ollows. Lemma 2.3 [4]. I A 0 z is a transcendental entire unction wit ρ A 0 < 1, ten every solution z 0 o equation 1.8 as ininite order. Lemma 2.4 Wiman-Valiron, [8], [14]. Let z be a transcendental entire unction, and let z be a point wit z = r at wic z = M r,. Ten te estimate k z z ν r, k = 1 + o 1 k 1 is an integer, 2.6 z olds or all z outside a set E 3 o r o inite logaritmic measure, were ν r, denotes te central index o. To avoid some problems caused by te exceptional set we recall te ollowing lemma.

6 264 B. BELAÏDI AND A. EL FARISSI Lemma 2.5 [5]. Let g : [0, + R and : [0, + Rbe monotone non-decreasing unctions suc tat g r r or all r / E 4 [0, 1], were E 4 1, + is a set o inite logaritmic measure. Let α > 1 be a given constant. Ten tere exists an r 0 = r 0 α > 0 suc tat g r αr or all r r 0. Lemma 2.6. Let z be a meromorpic unction wit ρ = + and te exponent o convergence o te poles o z, λ 1/ < +. Let d j z j = 0, 1, 2 be polynomials tat are not all equal to zero. Ten satisies ρ g = +. g z = d 2 z + d 1 z + d 0 z 2.7 Proo. We suppose tat ρ g < + and ten we obtain a contradiction. First, we suppose tat d 2 z 0. Set z = w z / z, were z is canonical product or polynomial ormed wit te non-zero poles o z, λ = ρ = λ 1/ < +, w z is an entire unction wit ρ w = ρ = +. We ave z = w 2 w, z = w 2 2 w w 2.8 and Hence z = w 3 2 w z =w z w w 3 w w w 2 w w , 2.10 z z =w w 2 w w , 2.11 z z =w w. 2.12

7 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 265 Dierentiating bot sides o 2.7, we obtain g = d 2 + d 2 + d 1 + d 1 + d 0 + d Writing g = g /gg and substituting g = d 2 + d 1 + d 0 into 2.13, we get d 2 + d 2 g g d 2 + d 1 + d 1 g g d 1 + d 0 + d 0 g g d 0 = Tis leads to d 2 d + 2 g g d 2 + d 1 d + 1 g g d 1 + d 0 d + 0 g g d 0 = Substituting into 2.15, we obtain w d 2 w w 3 w w w + + d 2 g g d w 2 + d 1 w w 2 w w w + d 0 g g d 0 + d 1 g g d 1 + d = By Lemma 2.1 ii, tere exists a set E 1 1, + tat as inite logaritmic measure, suc tat or all z satisying z = r / E 1 [0, 1], we ave g z gz = O rα, j z z = O rα j = 1, 2, 3, 2.17 were 0 < α < + is some constant. By Lemma 2.4, tere exists a set E 2 1, + wit logaritmic measure lm E 2 < + and we can coose z satisying z = r / E 2 [0, 1] and w z = M r, w, suc tat w j z w z ν r, w j = 1 + o 1 j = 1, 2, z Now we take point z satisying z = r / E 1 E 2 [0, 1] and w z = M r, w, by substituting 2.17 and 2.18 into 2.16, we get ν r, w = O r β, 2.19 were 0 < β < + is some constant. By Lemma 2.5, we conclude tat 2.19 olds or a suiciently large r. Tis is a contradiction by ρ w = +. Hence ρ g = +.

8 266 B. BELAÏDI AND A. EL FARISSI Now suppose d 2 0, d 1 0. Using a similar reasoning as above, we get a contradiction. Hence ρ g = +. Finally, i d 2 0, d 1 0, d 0 0, ten we ave g z = d 0 z z and by d 0 is a polynomial, ten we get ρ g = +. Lemma 2.7 [1]. Let A 0, A 1,..., A k 1, F 0 be inite order meromorpic unctions. I is a meromorpic solution wit ρ = + o te equation k + A k 1 k A 1 + A 0 = F, 2.20 ten λ = λ = ρ = +. Lemma 2.8. Let A 0, A 1,..., A k 1, F 0 be inite order meromorpic unctions. I is a meromorpic solution o te equation 2.20 wit ρ = + and ρ 2 = ρ, ten satisies λ 2 = λ 2 = ρ 2 = ρ. Proo. By equation 2.20, we can write k 1 = 1 F + A k 1 k A 1 + A I as a zero at z 0 o order α > k and i A 0, A 1,..., A k 1 are all analytic at z 0, ten F as a zero at z 0 o order at least α k. Hence, n r, 1 k n r, 1 + n r, 1 k 1 + n r, A j 2.22 F and N r, 1 k N r, 1 + N r, 1 k 1 + N r, A j F By 2.21, we ave m r, 1 k m r, j j=1 m r, A j + m k 1 + j=0 j=0 j=0 r, 1 + O F Applying te lemma o te logaritmic derivative see [7], we ave m r, j = O log T r, + log r j = 1,..., k, 2.25 olds or all r outside a set E 0, + wit a inite linear measure m E < +. By 2.23, 2.24 and 2.25, we get T r, =T r, 1 + O 1

9 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 267 k N r, 1 k 1 + T r, A j + T r, F + O log rt r, j=0 z = r / E Since ρ = +, ten tere exists {r n} r n + suc tat log T r lim n, r n + log r n = Set te linear measure o E, m E = δ < +, ten tere exists a point r n [r n, r n + δ + 1] E. From it ollows tat log T r n, log T r n, log r n log r n + δ + 1 log T r = n, log r n + log 1 + δ + 1, 2.28 r n log T r n, lim = r n + log r n Set σ = max {ρ A j j = 0,..., k 1, ρ F }. Ten or a given arbitrary large β > σ, olds or suiciently large r n. 0 < 2ε < β σ, we ave T r n, r β n 2.30 On te oter and, or any given ε wit T r n, A j rn σ+ε j = 0,..., k 1, T r n, F rn σ+ε or suiciently large r n. Hence, we ave { T rn, F max T r n,, T r } n, A j j = 0,..., k 1 rσ+ε n 0, T r n, Tereore, T r n, F 1 k + 3 T r n,, r β n 2.31 r n T r n, A j 1 k + 3 T r n, j = 0,..., k olds or suiciently large r n. From O log r n + log T r n, = o T r n,, 2.34

10 268 B. BELAÏDI AND A. EL FARISSI we obtain tat O log r n + log T r n, 1 k + 3 T r n, 2.35 also olds or suiciently large r n. Tus, by 2.26, 2.33, 2.35, we ave T r n, k k + 3 N r n, It yields λ 2 = λ 2 = ρ 2 = ρ. 3. Proo o Teorem 1.1 First, we suppose d 2 0. Suppose tat 0 is a meromorpic solution o equation 1.3 wit ρ = + and λ 1/ < +. Set w = g ϕ = d 2 + d 1 + d 0 ϕ, ten by Lemma 2.6 we ave ρ w = ρ g = ρ = +. In order to te prove λ g ϕ = +, we need to prove only λ w = +. Substituting = A 1 A 0 into w, we get w = d 1 d 2 A 1 + d 0 d 2 A 0 ϕ. 3.1 Dierentiating bot a sides o equation 3.1, we obtain w = d 2 A 2 1 d 2 A 1 d 2 A 0 d 1 A 1 + d 0 + d 1 Set Ten we ave Set + d 2 A 1 A 0 d 1 A 0 d 2 A 0 + d 0 ϕ. 3.2 α 1 = d 1 d 2 A 1, α 0 = d 0 d 2 A 0, β 1 = d 2 A 2 1 d 2 A 1 d 2 A 0 d 1 A 1 + d 0 + d 1, β 0 = d 2 A 1 A 0 d 1 A 0 d 2 A 0 + d 0. We divide it into two cases to prove. Case 1. I 0, ten by , we get α 1 + α 0 = w + ϕ, 3.3 β 1 + β 0 = w + ϕ. 3.4 = α 1 β 0 β 1 α α 1 w β 1 w = α 1 ϕ β 1 ϕ = F. 3.6

11 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 269 By te ypoteses o Teorem 1.1, we ave α 1 ϕ β 1 ϕ 0 and ten F 0. By α 1 0, F 0, and Lemma 2.7, we obtain λ w = λ w = ρ w = +, i.e., λ g ϕ = +. Case 2. I 0, ten by , we get = α 1w + ϕ β 1 w + ϕ. 3.7 Substituting 3.7 into equation 1.3 we obtain α 1 w + φ 2 w + φ 1 w + φ 0 w α1 ϕ β 1 ϕ α1 ϕ β 1 ϕ α1 ϕ β 1 ϕ = + A 1 + A 0 = F, 3.8 were φ j j = 0, 1, 2 are meromorpic unctions wit ρ φ j < + j = 0, 1, 2. By te ypoteses o Teorem 1.1, we ave α 1 ϕ β 1 ϕ 0 and by α1 ϕ β 1 ϕ ρ < +, it ollows tat F 0. By Lemma 2.7, we obtain λ w = λ w = ρ w = +, i.e., λ g ϕ = +. Now suppose d 2 0, d 1 0 or d 2 0, d 1 0 and d 0 0. Using a similar reasoning to tat above we get λ w = λ w = ρ w = +, i.e., λ g ϕ = Proo o Teorem 1.2 Suppose tat 0 is a meromorpic solution o equation 1.3 wit ρ = + and ρ 2 = ρ. Set w j = j ϕ j = 0, 1, 2. Since ρ ϕ < +, ten we ave ρ w j = ρ = +, ρ 2 w j = ρ 2 = ρ j = 0, 1, 2. By using a similar reasoning to tat in te proo o Teorem 1.1, we obtain tat λ ϕ = λ ϕ = λ ϕ = ρ = and by Lemma 2.8, we get λ 2 ϕ = λ 2 ϕ = λ 2 ϕ = ρ 2 = ρ. 4.2

12 270 B. BELAÏDI AND A. EL FARISSI 5. Proo o Corollary Suppose tat 0 is a solution o equation 1.8. Ten by Lemma 2.3, is o ininite order. Let ϕ 0 be a inite order entire unction. Ten by using Lemma 2.2, we ave d1 d 2 e z ϕ Hence, ϕ d 2 e 2z d 1 + d 2 d 2 e z d 2 A 0 + d 0 + d d1 d 2 e z ϕ d 2 e 2z d 1 + d 2 d 2 e z d 2 A 0 + d 0 + d 1 ϕ By Teorem 1.1, te dierential polynomial g z = d 2 +d 1 +d 0 satisies λ g ϕ = +. Acknowledgement. Te autors would like to tank te reerees or teir elpul comments and suggestions to improve our paper. Reerences [1] Cen, Z. X., Zeros o meromorpic solutions o iger order linear dierential equations, Analysis Oxord , [2] Cen, Z. X., Te ixed points and yper order o solutions o second order complex dierential equations in Cinese, Acta Mat. Sci. Ser. A Cin. Ed , [3] Cen, Z. X., Son, K. H., On te growt and ixed points o solutions o second order dierential equations wit meromorpic coeicients, Acta Mat. Sin. Engl. Ser , [4] Gundersen, G. G., On te question o weter + e z + Q z = 0 can admit a solution 0 o inite order, Proc. Roy. Soc. Edinburg Sect. A , [5] Gundersen, G. G., Finite order solutions o second order linear dierential equations, Trans. Amer. Mat. Soc , [6] Gundersen, G. G., Estimates or te logaritmic derivative o a meromorpic unction, plus similar estimates, J. London Mat. Soc , [7] Hayman, W. K., Meromorpic Functions, Clarendon Press, Oxord, [8] Hayman, W. K., Te local growt o power series: a survey o te Wiman-Valiron metod, Canad. Mat. Bull , [9] Kinnunen, L., Linear dierential equations wit solutions o inite iterated order, Souteast Asian Bull. Mat , [10] Laine, I., Nevanlinna Teory and Complex Dierential Equations, Walter de Gruyter, Berlin-New York, [11] Laine, I., Rieppo, J., Dierential polynomials generated by linear dierential equations, Complex Var. Teory Appl ,

13 DIFFERENTIAL POLYNOMIALS GENERATED BY EQUATIONS 271 [12] Liu, M. S., Zang, X. M., Fixed points o meromorpic solutions o iger order Linear dierential equations, Ann. Acad. Sci. Fenn. Mat , [13] Nevanlinna, R., Eindeutige Analytisce Funktionen in German, Zweite Aulage. Reprint. Grundleren Mat. Wiss. 46, Springer-Verlag, Berlin-New York, [14] Valiron, G., Lectures on te General Teory o Integral Functions, Celsea, New York, [15] Wang, J. and Yi, H. X., Fixed points and yper order o dierential polynomials generated by solutions o dierential equation, Complex Var. Teory Appl , [16] Yang, C. C., Yi, H. X., Uniqueness teory o meromorpic unctions, Mat. Appl. 557, Kluwer Acad. Publ. Group, Dordrect, [17] Zang, Q. T., Yang, C. C., Te Fixed Points and Resolution Teory o Meromorpic Functions in Cinese, Beijing Univ. Press, Beijing, Benarrat Belaïdi Abdalla El Farissi Department o Matematics Department o Matematics Laboratory o Pure Laboratory o Pure and Applied Matematics and Applied Matematics University o Mostaganem University o Mostaganem B.P. 227 Mostaganem, Algeria B.P. 227 Mostaganem, Algeria el.arissi.abdalla@caramail.com belaidi@univ-mosta.dz belaidibenarrat@yaoo.r