ON THE VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE

Transcription

1 Математичнi Студiї. Т.34, 2 Mateatychni Studii. V.34, No.2 УДК A. Fenández ON THE VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE A. Fenández. On the value distibution o eoophic in the punctued plane, Mat. Stud. 34 2, Soe well-known acts o the value distibution theoy o eoophic unctions in the plane will be consideed in the punctued plane and also soe esults o entie unctions will be extended to eoophic unctions in the plane with initely any poles. A. Фернандес. О распределении значений мероморфных функций в проколотой плоскости // Мат. Студiї. 2. Т.34, 2. C Некоторые хорошо известные факты теории распределения значений для мероморфных функций в плоскости рассматриваются в проколотой плоскости. Обобщаются некоторые результаты о целых функциях на мероморфные в плоскости функции с конечным числом нулей.. Intoduction. The punctued plane C\{}, can be consideed as a degeneated annulus A = {z : R < z < R 2 }, that is, A = A R R 2,with R =, R 2 =, whee A R R 2 = {z : R < z < R 2 }. Value distibution theoy on ultiply connected doains was aleady consideed by soe authos in the ist decades o the 2th centuy, Biebebach [2], Hällsto [4] and in oe ecent ties S. Bank and I. Laine [], in connection with dieential equations, paid also attention to this question. In the last yeas, two aticles o R. Kohonnen [6] and A.Ya. Khistiyanin and A.A. Kondatyuk appeaed [7], [8], whee the two ain theoes o Nevanlinna o eoophic unctions on annuli ae poved and whee they indicate the way to extend the value distibution o Nevanlinna theoy to eoophic unctions in annuli. Hee we shall ake use o the tools developed in these papes to analyze o eoophic unctions in the punctued plane soe well-known and elevant esults about eoophic unctions in the entie plane. In paticula, we shall pay attention to the Boel and deicient values o eoophic unctions o inite ode. 2. Nevanlinna theoy on annuli. We shall ake use o the deinitions and notation o Khystiyanin and Kondatyuk, see [7], [8]. Given a eoophic unction z, deined on { } A = z : < z < R, R donde < R, the poxiity, counting and chaacteistic unctions, a, N, a and T, ae deined in such a way that the coesponding analogues to both ain theoes in the plane can be poved o annuli. 2 Matheatics Subject Classiication: 3D5. c A. Fenández, 2

2 VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE37 Siilaly to the case o the entie coplex plane, it is intoduced the notion o deiciency o the case o the punctued plane, which coesponds to the case o an annulus with R =. Deinition. Let be a non-constant eoophic unction on C\{} = {z : < z < }, and a C. Then the value R, δ a, = li in a R T R,, is called the deiciency o deect o the unction o the value a. Fo a =, we set R, δ, = li in R T R,. Theoe Deiciency elationship o eoohic unctions in the punctued plane. Khystiyanin and Kondatyuk. Let z be a non-constant eoophic unction in the punctued plane C\{} = {z < z < }, then thee ae at ost countably any deicient values o z, that is, values a ν Ĉ, o which δ a ν, >, and δ a ν, 2. ν= 3. The Picad-Boel theoe o eoophic unctions in the punctued plane. Siilaly to the case o the entie plane C, we deine the concept o Boel exceptional value in the case o a eoophic unction in the punctue plane in tes o n, a by equiing od n, a < od, in act o λ >, the ollowing equality holds od n, a = od N, a, so that we can also oulate in this case the deinition with the aveage counting unction N, a. Theoe 2 Picad-Boel theoe in the punctued plane. Fo a eoophic unction : C\{} Ĉ, deined in the punctued coplex plane o inite ode, thee ae at ost two Boel exceptional values. In the case o non-integal ode thee can only exist one Boel exceptional value. Poo. Fo a geneal λ >, the esult ollows o the undaental inequality in Nevanlinna second ain theoe exactly as in the plane case. Let us assue that λ = od T, is non-intege and suppose that o two values a, a 2, the coesponding N, a, N, a 2 ae o ode less than λ. By a actoization due oiginally to G. Valion and which has been ade pecise in its hypotheses by S. Bank and I. Laine [] and inally by R. Kohonen [6], naely z = z Φzuz, whee Z, Φz is eoophic in Ĉ\{} = {z : < z }, and uz is eoophic in C = {z : z }. Also we shall ake use o the ollowing estiate, see Hayan [5], p. 2, o eoophic unctions h in the plane. We set n = n, h + n, n, h n,, h h

3 38 A. FERNÁNDEZ and then it holds T, h c q {q q N = nt dt, t Nt Nt } dt + q + q+ tq+ t dt + O q, 2 q+2 q = [λ], λ = od T, h, and c q is a constant depending only on q. We shall apply 2 to the eoophic unctions Φ z = Φ, uz. z I we denote by N Φ, N u unctions N coesponding to Φ and u espectively, we obseve o the deinitions that N Φ = N, + N,, N u = N 2, + N 2,, so that setting we get N = N, + N,, N Φ + N u = N, + N, = N. 3 We apply now 2 to Φ and u, and obtain T, Φ c q {q q N Φ t dt + q + q+ t q+ T, u c q {q q Making use o 3, 4 and 5, we conclude T, c q {q q N u t dt + q + q+ tq+ N t t q+ dt + q + q+ N Φ t } dt + O q, 4 t q+2 N u t t q+2 dt } + O q. 5 N t t q+2 dt } +O q. And aguing as in the case o the entie plane i od N < od T,, we shall get a contadiction. Fo eoophic unctions o inite ode λ in the plane, whee λ is non intege, a elated esult involving deiciencies is due to R. Nevanlinna, see Hayan [5]. It can also be extended to the punctued plane C\{}, aking use o he ideas o Theoe 6. Theoe 3. Suppose that z is a eoophic unction with a inite nube o poles in the punctue plane C\{}, o inite ode λ, whee λ is not a positive intege, then we have δ a, kλ, a whee kλ is a quantity independent o z depending on λ only.

4 VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE39 4. Functions with axial deiciency su. Finally, we shall conside a possible extension to the punctued plane o the ollowing theoe o Pluge []. Theoe 4. Given an integal unction in the plane z o inite ode o which a ν δa ν, =, 6 then z ust be o integal ode q and the deiciencies ust be o the o δa ν, = k ν q, k ν N. 7 In geneal, o eoophic unctions D. Dasin [3], poved that axial deiciency su, that is, δa ν, = 2, 8 ν= iplies that the ode o z ust be ultiple o q, whee q is a positive intege o + 2 and that 7 also holds. Hee we shall pove the coesponding esult to Theoe 4 in C\{} unde the ollowing uthe assuption. Let z = z Φzuz, be the Valion actoization o z, so that uz and Φz ae anlytic, and set Φ z = Φ, then we ipose od Φ z < loweod u, what is equivalent, in the case o integal ode q, to by a theoe o Edei and Fuchs. od Φ < od u = q, 9 Theoe 5. Given an analytic unction z in C\{}, o inite ode, with axial su o deiciencies, that is, satisying 6, then z is o integal ode q, and assuing that z also eets condition 9, then 7 also holds, that is, the deiciencies ae o the o δa ν, = k ν q, k ν N, and as a consequence, thee can be a inite nube o deicient values only. The sae aguent as that o the poo o Theoe 5 yields the ollowing stateent. Theoe 6. Given a eoophic unction z with initely any poles, satisying 8, then we can conclude that z is o integal ode q and its deiciencies satisy 7. In act, witing the deivative as z = z Φ d zu d z, whee Φ d z is a ational unction and u d z is an entie unction, then one has that δa ν, = δa ν, U, whee Uz is the entie unction such that U z = u d z. This esult is a paticula case o Dasin s esult, but on the othe hand, the poo is quite siple and we get the additional inoation that in the case o initely any zeos the ode ust be an intege. Poo o Theoe 5. The act that 6 iplies od to be an intege ollows iediately o Theoe 3, the coesponding analogue in the punctued plane to Nevanlinna s theoe.

5 4 A. FERNÁNDEZ The poo o the pat o Theoe 7 coesponding to the value o the deiciencies is inspied by the behaviou o the unction z = z e ζq dζ. The asyptotic behaviou o this unction is, in soe way, geneal o entie unctions o integal ode. Given a poxiate ode ρ o an integal unction Gz, we shall call the unction V = ρ, a copaison unction o Gz when log M li sup V =. It can be shown that thee exists such a copaison unction. Theoe 7. Pluge. Let Gz be an entie unction o integal ode q > and assue that the sequence o zeos o Gz is easuable with espect to the copaison unction V and let Nφ be the easue unction, then we have log Ge iφ V = hφ + [ C πi q + q S ] e iqφ + ε, φ, V whee 2π 2π hφ = θ sin qθdθ, C = e iqθ dnθ, S does not depend on φ and ε, φ, unioly in the whole inteval φ 2π. As a consequence o Theoe 7 applied to Gz = z, whee z is entie satisying 6, we obtain the ollowing lea. Lea. Let z be an entie unction o integal ode q, then whee log e iφ < H, φ + ov, H, φ = A cosqφ α, A = q S, α = ag q S V V. Poo Theoe 7. Now we can decopose the cicle C,, z = e iφ, in 2q subacs α, α2,.., αq, β, β2,.., βq o length π, the decoposition depending on, such that q H, φ, on the acs α ν, ν =,..., q, H, φ, on the acs β ν, ν =,..., q. Fo, by an aguent involving integation, it is deived as in [] log + e iφ a dφ = ov, α ν and log + dφ = ov, 2 β ν e iφ a

6 VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE4 Fo and 2 and the act that V is a copaison unction o, we can ind a sequence n o which T n > 2π V n, witing αν n = αν n, βν n = βν n and then setting D n ν = T n, we should have o 2 and 3 α n ν log + dφ, ν =,..., q, 3 n e iφ a li in n q Dν n δa. ν= Fo these consideations and ou hypotheses, one can conclude that thee can be only a inite nube o deicient values a i, i =,..., p and δa i = k i q, k i ultiplicity o a i, so that k + k k p =, whence we obtain Theoe 7. Now we conside again analytic unctions in the punctued plane : C\{} Ĉ with axial deiciency su o which we have aleady poved that they have integal ode q. By application o Valion theoe, we can decopose z in the o z = z Φzuz, whee uz and Φz ae analytic, and ecall that we ae assuing condition 9. Then we set Gz = z and conside the Valion decoposition o Gz, that is, o the deivative o, Gz = z d Φd zu d z then again Gz is a unction o ode q, o which od Φ d, < od u d = q, 4 whee Φ d, z = Φ d z. Fo this decoposition we obtain o a copaison unction V ud z o u d z log Ge iφ = d log + log u de iφ + log Φ de iφ. 5 Since Φ d z is analytic at, we have log Φ d e iφ = Olog, so that log Φ d e iφ, unioly in φ when, and we conclude log Ge iφ = log u de iφ + o, 6 as.

7 42 A. FERNÁNDEZ Siilaly, since u d is analytic at z =, we also obtain log G eiφ = log Φ d,e iφ + o, 7 when. Now we eak that is siultaneously a copaison unction o u d,, u, by ou hypothesis 9. By Theoe 7 applied to u d and taking into account 6, we conclude o Lea applied to u d log e iφ = log u d e iφ + o < H ud, φ + o. 8 Recalling, =, +,, i we had the elation we ight deive, a δ a, = li in T, Then o 8, 9 and 2, we obtain li, a =, 9 T, = li in, a T,. 2 δ a, = δa, U = k q, 2 with U an entie unction o ode q, naely U is an entie unction o which U = u d, aking use o the aguent in [] and descibed above, whee now the aguent is also valid o z. Making use o the hypothesis 4, 9 ollows o 2 in [7], page 25, in the poo o the ist ain theoe in annuli. In act, accoding to this, we get o, < <,, + N, =, + N, + Olog, and aking use o the elation log + a log + log + a + log 2, we also get that is, + N, =, + N a, + Olog, a, = a, a = O, + N,. 22

8 VALUE DISTRIBUTION OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE43 Fist o all, we eak N, = N, Φ T, Φ + O = ot, u = ot,, 23 on the othe hand, whee, =, uφ, u +, Φ + log 2, 24, u = Olog, 25 since u is eoophic at z =. By ou hypothesis, Φ =, Φ T, Φ = ot, u = ot,. 26 Then 9 ollows o Theeoe, we conclude 9 and as a consequence 2 holds. Poo o Theoe 6. The elationship 9 holds in the paticula case that Φz is ational, which coesponds to the case o z eoophic with a inite nube o poles, since in this case = z uφ is also analytic o has a pole at z =, so that z a = z gz, g, Z, whence = O a, as, o what clealy iplies 9. log + a = Olog, This conclusion is contained in Dasin s theoe but we obtain, in addition, the inoation 2. Futheoe, by Dasin s theoe the case od = kq has to be allowed. It is clea that 2 2 cannot hold o eoophic unctions with a inite nube o poles and ode kq with 2 k odd. We shall conclude by ecalling that the validity o Pluge s esult in the punctued plane eains open in geneal, that is, i we eove condition 9. REFERENCES. Bank S., Laine I. Repesentation o solutions o peiodic second ode linea dieential equations// Jounal ü die Reine und Angewandte Matheatik V.344. P Biebebach L. Theoie de Dieentialgleichungen. Spinge Velag, Belin, Dasin D. Poo o a conjectue o F.Nevanlinna concening unctions which have deiciency su two// Acta Matheatica V.58. P. 94.

9 44 A. FERNÁNDEZ 4. Hällstö G. Übe eoophe Funktionen it ehach zusaenhängenden Existenzegebieten// Acta Acadeiae Aboensis. 94. V.2, Hayan W.K. Meoophic unctions. Oxod Claendon Pess Kohonen R. Nevanlinna Theoy in an annulus// Value distibution theoy and elated topics, Adv. Coplex Anal. and Applic. Kluwe Acadeic Publishes. 24. P Khystiyanin A.Ya., Kondayuk A.A. On the Nevanlinna theoy o eoophic unctions on annuli. I // Mat. Stud. V.23,. P Khystiyanin A.Ya., Kondayuk A.A. On the Nevanlinna theoy o eoophic unctions on annuli. II. // Mat. Stud. V.24,. P Nevanlinna R. Le théoee de Picad-Boel et la théoie de onctions eoophes. Chelsea Publishing Copany, Nevanlinna R. Analytic Functions. Spinge Velag, 97.. Pluge A. Zu Deektelation ganze Funktiones endliche Odnung// Coentaii Matheatici Helvetici V.9. P Dpto de Mateáticas Fundaentales UNED Avda de Senda del Rey 9, 284 Madid, Spain aenan@at.uned.es Received 6.7.2