Research Article Filling Disks of Hayman Type of Meromorphic Functions

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1 Joural of Fuctio Spaces Volume 206, Article ID , 5 pages Research Article Fillig Disks of Hayma Type of Meromorphic Fuctios Na Wu ad Zuxig Xua 2 Departmet of Mathematics, School of Sciece, Chia Uiversity of Miig ad Techology, Beijig 00083, Chia 2 Beijig Key Laboratory of Iformatio Service Egieerig, Departmet of Geeral Educatio, Beijig Uio Uiversity, No. 97 Bei Si Hua Dog Road, Chaoyag District, Beijig 000, Chia Correspodece should be addressed to Zuxig Xua; zuxigxua@63.com Received 4 Jauary 206; Accepted 0 April 206 Academic Editor: Jaeyoug Chug Copyright 206 N. Wu ad Z. Xua. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We obtai the existece of the fillig disks with respect to Hayma directios. We prove that, uder the coditio (T(r, f)/(log r) 3 )=, there exists a sequece of fillig disks of Hayma type, ad these fillig disks ca determie a Hayma directio. Every meromorphic fuctio of positive ad fiite order ρ has a sequece of fillig disks of Hayma type, which ca also determie a Hayma directio of order ρ.. Itroductio ad Results For a meromorphic fuctio f, disks of the form Δ(z,ε z ) = {z : z z <ε z }, () where z ad ε 0,arecalledfilligdisksoff if f takes every exteded complex value with at most two exceptios ifiitely ofte i ay ifiite subcollectio of them, which are also called the fillig disks of Julia type. We ca prove that if a meromorphic fuctio f satisfies T(r,f) =, 2 (2) (log r) the f must possess a sequece of fillig disks of Julia type, ad these fillig disks ca determie a Julia directio. A directio θ [0,2π] is said to be a Julia directio for a meromorphic fuctio f,if, giveε > 0, f takes all complex valuesifiitelyofteitheregiod={z: arg z θ <ε} except possibly two exceptios. For a meromorphic fuctio f of order 0<ρ<,disks of the form (), where z ad ε 0,arecalledfillig disks of Borel type of f if f takes every exteded complex value at least z ρ δ times, except some complex values which ca be cotaied i a disk whose spherical radius is e z ρ δ,whereδ. A meromorphic fuctio with positive ad fiite order must possess a sequece of Borel type fillig disks, which determie a ρ-order Borel directio, ad a ρ-order Borel directio also determies a sequece of Borel type fillig disks. A directio θ [0,2π]is said to be a Borel directio for a meromorphic fuctio f of order ρ, if,give ε>0, log (r,z ε (θ),f=a) =ρ (3) log r for all complex values a,atmostwithtwopossibleexceptios. Here ad throughout the paper, (r, Z ε (θ), f = a) deotes the umbers of the roots of f=ai the regio Z ε (θ) = {z : arg z θ <ε}. For the case of Hayma directio, we pose a questio whether there exist fillig disks of Hayma type. I this paper, we maily obtai the followig two theorems. Theorem. Let f(z) be a meromorphic fuctio i the plae satisfyig T(r,f) =. 3 (4) (log r) The, there exists a sequece of fillig disks with the form (), where z ad ε 0;ieachdisk,f takes all complex values at least k times or else there exists a iteger umber k

2 2 Joural of Fuctio Spaces such that f (k) takes all complex values, except possibly zero at least k times, where (k / log z ) =.Moreover, these fillig disks ca determie a Hayma directio arg z= θ, such that for arbitrary small ε>0,positiveitegerk, ad complex umbers a ad b =0we have ( (r, Z ε (θ),f=a)+(r,z ε (θ),f (k) =b)) =. Theorem 2. Let f(z) be a meromorphic fuctio i the plae of order 0<ρ<. The, there exists a sequece of fillig disks with the form (), where z ad ε 0;ieachdisk,f takesallcomplexvaluesatleast z ρ δ times or else there exists a iteger umber k such that f (k) takes all complex values, except possibly zero at least z ρ δ times, where δ 0. Moreover, these fillig disks ca determie a Hayma directio arg z=θ,suchthat,forarbitrarysmallε>0,positiveiteger k, ad complex umbers a ad b =0,wehave log ( (r, Z ε (θ),f=a)+(r,z ε (θ),f (k) =b)) log r =ρ. Remark. The fillig disks i Theorem are called the fillig disks of Hayma-Julia type. Moreover, if we add the growth coditio 0<ρ< to f, we ca obtai the fillig disks of Hayma-Borel type i Theorem 2. Hayma-Borel type fillig disks are more precise tha Hayma-Julia type fillig disks. 2. Proof of Theorem First of all, let us recall the defiitio of Ahlfors-Shimizu characteristic i a agle (see []). Let f(z) be a meromorphic fuctio defied i a agle Ω={z:α arg z β}.set Ω(r)=Ω {z:< z <r}.defie S (r,ω,f)= π ( f (z) Ω(r) + f (z) 2 ) dσ, r T (r,ω,f)= S (t, Ω, f) dt. t To prove our theorems, we eed the followig lemma which was first established by Che ad Guo [2] ad essetially comes from the Hayma iequality ad the estimatio of primary values appeared i the iequality ad was used to cofirm the existece of Hayma T directios of meromorphic fuctios by Zheg ad the first author [3]. Lemma 3. Let f be meromorphic i z < R ad let 2 (5) (6) (7) N=(R, f = a) +(R, f (k) =b) (8) for two complex umbers a ad b with b =0.The,wehave S ( R 256,f)<C k {N + log R+ 3 }, (9) 2 where C k is a positive costat depedig oly o k. Lemma3cabeobtaiedbyusigLemma8ofChead Guo [2] to the fuctio (br k ) (f(z) a) i the uit disk {z : z < } ad oticig the followig: S (r, Ω, f) = π z (r,ω,f=z)dω(z), π dω (z) =, z π log z x, z dω (z) = 2, (0) where dω(z) is the area elemet o the Riema sphere ad x, z is the chordal distace betwee x ad z. The followig lemma comes from [4], which is very useful for our study. Lemma 4. Let f be meromorphic i the plae, ad let η, R, ad M be arbitrary positive umbers with η < ad R > 2.If f satisfies T(r,f) =, 3 () (log r) the there exists a positive umber r>rsuch that S (Γ (( η) r, (+η) r),f) >Mlog r, (2) where Γ(( η)r, ( + η)r) = {z : ( η)r < z < ( + η)r}. I view of Lemmas 3 ad 4, we ca prove Theorem. ProofofTheorem. For each N,letM >0, 0<η <, ad R >2. The, there exists r >R,suchthat S (Γ (( η )r,(+η )r ),f)>m log r. (3) Divide Γ(( η )r,( + η )r ) ito domais D j (j =,2,..., ),where D j ={z:( η )r < z <(+η )r, 2jπ 2(j+)π }. arg z (4) The, there exists at least oe j 0 =j 0 () ( j 0 ),such that S (D j0,f) M log r, D j0 B ={z: z z < π z } B ={z: z z < 256π z }, where z =r exp(i((2j 0 + )/ )π).iviewof S (B,f)<C k { (B,f=a)+(B,f (k) =b) + log 256πr }, (5) (6)

3 Joural of Fuctio Spaces 3 for large eough,wehave M (B,f=a)+(B,f (k) =b) <C k. (7) log r Here we choose the proper,makigm /.The proof is complete. Here we poit out that, uder the curret techology, we caot replace the growth coditio (4) with (2). Rossi [4] also obtaied that if the growth coditio (4) was replaced by (2), the the iequality i Lemma 4 should be replaced by S(Γ(( η)r, ( + η)r), f) > M.Ufortuately,theiequality of Lemma 3 has the term log R; owig to this term, we caot replace the growth coditio, because we otice that if the growth coditio is (2) the the term log R should be bouded. If we choose =r, the the term log R ca be bouded but we caot assure that M / as. 3. Proof of Theorem 2 The followig lemma ca be proved by the same method of Lemma 4, ad we prove it for the completeess. Lemma 5. Let f be meromorphic i the plae with order 0< ρ <, ad let η, R, ad M be arbitrary positive umbers with η<ad R>2.The,forayε>0, there exists a positive umber r>rsuch that Proof of Theorem 2. For each N,letM >0, 0<η <, ε >0,adR >2. The, there exists r >R,suchthat S (Γ (( η )r,(+η )r ),f) >M. (23) r ρ ε Divide Γ(( η )r,( + η )r ) ito domais D j (j =,2,..., ),where D j ={z:( η )r < z <(+η )r, 2jπ 2(j+)π }. arg z (24) The, there exists at least oe j 0 =j 0 () ( j 0 ),such that S(D j0,f) M r ρ ε, D j0 B ={z: z z < π z } B ={z: z z < 256π z }, where z =r exp(i((2j 0 + )/ )π).iviewof (25) S (Γ (( η) r, ( + η) r), f) r ρ ε >M. (8) Proof. Suppose that the lemma is ot true. The, for every t>r, S (B,f)<C k { (B,f=a)+(B,f (k) =b) + log 256πr }, (26) S (Γ (( η) t, ( + η) t), f) t ρ ε M. (9) Choose r larger tha 2R ad set β = ( η)/( + η). The, there exists α>0such that R<(+η)β α r<2r. (20) Let r 0 =rad r =βr. Combiig (9) with (20), we have S (Γ (2R, r),f) [α]+ j=0 S (Γ (( η) r j,(+η)r j ),f) (2) whe is large eough, we have C k M r ρ ε <(B,f=a) +(B,f (k) =b). (27) Choosig = M, we ca obtai the result. Hece, we complete the proof of Theorem 2. It is ot difficult to see that these fillig disks ca determie a Borel directio of order ρ. Zhag ad Yag [5] also obtaied a sequece of fillig disks, whose result is as follows: let f be a meromorphic fuctio of positive ad fiite order ρ. The, there exists a sequece M([α] +2) r ρ ε. It follows from (20) that α log r+log ( + η). (22) log β Substitutig (22) ito (2) leads to the order of f beig ρ ε by the fact that R ad η are fixed. Γ j : z z j <ε j z j, lim ε j =0, j z j+ >2 z j, arg z j =θ, j =, 2, 3,... (28)

4 4 Joural of Fuctio Spaces such that at least oe of the followig holds: () f(z) i Γ j takesallcomplexvaluesatleast z j ρ ε j times, with some exceptioal values which ca be cotaiediasphericaldiskwithceter adradius e z j ρ ε j. (2) For ay fixed positive iteger k, f (k) (z) i Γ j takes all complex values at least z j ρ ε j times, with some exceptioal values which ca be cotaied i two spherical disks with ceter ad 0 ad radius e z j ρ ε j,wherelim j ε j =0. We ca see that our result is differet from theirs. Divide Γ(( η )r,( + η )r,z δ (θ)) ito domais D j (j=0,2,..., ),where D j ={z:( η )r < z <(+η )r,θ δ + 2jδ arg z θ δ + 2(j+)δ }. (32) The, there exists at least oe j 0 =j 0 () ( j 0 ),such that S(D j0,f) M r ρ ε, 4. Fillig Disks o a Agular Regio D j0 B ={z: z z < δ z } B (33) I Yag s book [6], he said that a Borel directio of 0 < ρ< ca determie a sequece of Borel type fillig disks. I this paper, we ca see that a ρ order Borel directio ca determie ot oly a sequece of Borel type fillig disks but also a sequece of Hayma type fillig disks. Actually, if a meromorphic fuctio f o ay agular regio Ω(α, β) satisfies some growth coditios, it possesses a sequece of fillig disks. Zhag [7] obtaied the followig lemma, which established the growth coditio o ay agular domai Ω(α, β) cotaiig the Borel directio arg z=θ. Lemma 6. Let f be a meromorphic fuctio i the plae of order 0<ρ<.The,ahalflieL:arg z=θis a ρ-order Borel directio if ad oly if it satisfies log T (r, Z ε (θ),f) log r log S (r, Z = ε (θ),f) =ρ. log r IviewofLemma6,wehavethefollowigresult. (29) Theorem 7. Let f(z) be a meromorphic fuctio i the plae ad let arg z=θbe a Borel directio of order ρ. The,there exists a sequece of disks of the form C ={z: z z < 2δ K z }, (30) where z =r exp[i(θ δ +((2j 0 +)/ )δ )],. I C, f takes all but possibly two exteded complex values with expoet (M / ) z ρ ε ad f takes a ad f (k) takes b with expoet (M / ) z ρ ε,wherem ad δ 0. Proof. For each N,letM >0, 0<η <, ε >0,ad R >2. It follows from (29) ad Lemma 5 that there exists r >R,suchthat S (Γ (( η )r,(+η )r ),Z δ (θ),f) >M. (3) r ρ ε ={z: z z < 256δ z }, where z =r exp[i(θ δ + ((2j 0 + )/ )δ )].Iviewof S (B,f)<C k { (B,f=a)+(B,f (k) =b) + log 256πr }, whe is large eough, we have (34) M r ρ ε <(B C k,f=a)+(b,f (k) =b). (35) Choosig =M, we ca obtai the result. Atlast,weposeaquestio: Does a Hayma directio of order 0<ρ< have a sequece of fillig disks of Hayma type? Competig Iterests The authors declare that they have o competig iterests. Ackowledgmets Na Wu is supported i part by the grats of NSF of Chia (os , , ad 37363). Zuxig Xua is supported i part by NNSFC (o ), Beijig Natural Sciece Foudatio (o. 3203), ad the Project of Costructio of Iovative Teams ad Teacher Career Developmet for Uiversities ad Colleges Uder Beijig Muicipality (CIT ad TCD , IDHT ). Refereces [] M. Tsuji,Potetial Theory i Moder Fuctio Theory, Maruze, Tokyo, Japa, 959. [2] H. H. Che ad H. Guo, O sigular directios cocerig precise order, Joural of Najig Normal Uiversity,vol.5, o. 3, pp. 3 6, 992 (Chiese).

5 Joural of Fuctio Spaces 5 [3] J.-H. Zheg ad N. Wu, Hayma T directios of meromorphic fuctios, Taiwaese Joural of Mathematics,vol.4,o.6,pp , 200. [4] J. Rossi, A sharp result cocerig cercles de remplissage, Aales Academiae Scietiarum Feicae, Series A: I. Mathematica,vol.20,o.,pp.79 85,995. [5] Q. D. Zhag ad L. Yag, New sigular directios of meromorphic fuctios, Sciece i Chia, Series A, vol. 4, o. 26, pp , 984. [6] L. Yag, Value Distributio ad New Research, Spriger, Berli, Germay, 993. [7] X. L. Zhag, A fudametal iequality for meromorphic fuctios i a agular domai ad its applicatio, Acta Mathematica Siica, New Series, vol. 0, o. 3, pp , 994.

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