Some Results on Entire Functions of Finite Lower Order

Transcription

1 Acta Mathematica Siica, New Series 994, Vol.0, No., pp Some Results o Etire Fuctios of Fiite Lower Order Wu Shegjia Abstract. Let fz) be a etire fuctio of order λ ad of fiite lower order µ. If the zeros of fz) accumulate i the viciity of a fiite umber of rays, the a) λ is fiite; b) for every arbitrary umber k >, there exists k > such that T k r, f) k T r, f)for all r r 0. Applyig the above results, we prove that if fz) is extremal for Yag s iequality p = g, the c) every deficiet value of fz) is also its asymptotic value; d) every asymptotic value of fz) is also its deficiet value; e) λ = µ; f) δa, f) kµ). a. Itroductio This paper is devoted to etire fuctios. All fuctios i this paper, whe o metio is made to the cotrary, are etire fuctios i the plae. There are may papers cocerig the study of etire fuctios of fiite lower order. Especially, uder the coditios that the umber of Julia directios or, more geerally, the zero accumulatio lies is fiite, Zhag Guag-huo [] obtaied a lot of iterestig results which are maily cocered with the relatios amog the lower order, the umber of Julia directios, the deficiet values ad asymptotic values. I this paper we shall maily discuss some problems related to etire fuctios of fiite lower order with zeros accumulated i the viciity of a fiite set of rays. We shall prove some geeral properties of these kids of fuctios. From these results we ca easily see that lots of proofs of the theorems i [] ca be simplified. As a applicatio, we shall discuss a class of fuctios which are extremal for Yag s iequality. We assume the reader is familiar with the fudametal cocepts of Nevalia theory ad i particular with its most usual symbols see [5], [0] ad []). Received December 4, 99.

2 Wu Shegjia Some Results o Etire Fuctios of Fiite Lower Order 69. Some Basic Results Proofs of most of the previous results eed the fact that the fiiteess of the lower order of a fuctio fz) implies the fiitess of the order of fz). I this directio, we have the followig: Theorem. Suppose that fz) is a etire fuctio of order λ ad of lower order µ. Let arg z = θ k 0 θ <θ < <θ N < ; θ N+ = θ +, N < + ) be N rays such that for every ε>0, r log + { N k=ωθ k + ε, θ k+ ε; r),f =0} log r ρ<+,.) where ρ isafixedumber.theλ ad µ are cofiite. Proof. What we eed to prove is that if µ<+ the λ<+. Now we suppose that µ<+. Usig the method i [], we may fid a iteger p>maxρ, µ) such that cos pθ k >, k =,, N..) From.), there exists a umber η 0 > 0 such that cos pθ > for θ Θ= N k=θ k η 0,θ k + η 0 ). Deotig t, Θ, 0) = N k=ωθ k η 0,θ k + η 0 ; t),f =0 ) ad repeatig the argumet i [4], we have t, Θ, 0) t p+ < +..3) Noticig p>ρ, we deduce from.) that t, f ) t, Θ, 0) t p+ < +..4) Combiig.3) ad.4), we kow that the order of r, f ) is at most p. Notigthat µ is fiite, we ca make our assertio by a elemetary argumet. Let fz) be a meromorphic fuctio ad k > ) ad k > ) be two costats. We say that r is k,k )ormal [8] for fz) simplyk,k ) ormal) if T k r, f) k T r, f). I may problems, the existece of the arbitrarily large ormal value of r for some k,k ) is ecessary. Our followig result shows that the growth of the etire fuctios whose zeros accumulate i the viciity of a fiite set of rays is completely ormal. Theorem. Suppose that fz) is a etire fuctio of fiite lower order µ. Let arg z = θ k 0 θ <θ < <θ N < ; θ N+ = θ +, N < + ) be N distict rays such that the

3 70 Acta Mathematica Siica, New Series Vol. 0 No. iequalities log + { N k=ωθ k + ε, θ k+ ε; r),f =0} <µ, if µ>0,.5) r log r N Ωθ k + ε, θ k+ ε; r),f =0) k= < +, if µ =0,.6) log r r hold for every ε>0. The for ay k >, there exists a costat k > which depeds oly o µ, θ k k =,,,N) ad k such that r is k,k ) ormal for all r>r 0. Proof. If all the zeros of fz) are located o these half lies ad the order λ of fz) is fiite, the theorem is exactly the Theorem i [6]. The proof of Theorem eeds some further cosideratios. Without loss of geerality we suppose that fz) is trascedetal. From Theorem, we kow that λ<+ ad that there exists a iteger p>λsuch that cos pθ k >, k =,,,N..7) We take a fixed η 0 > 0 agai such that cos pθ >.8) for θ Θ= N k=θ k η 0,θ k + η 0 ). Let Θ r) = N k=ωθ k + η 0,θ k+ η 0 ; r),f =0 ), ad let a v = a v e iαv v =,, ) be the zeros of fz). If µ>0, we deduce from.5) that there exists a costat ε 0 > 0 such that Θ r) <r µ ε0 for all sufficietly large r. Thuswehave av > r αv Θ ) p r = pr p a v r av > r αv Θ Θ t) pr p+ t = or µ ε0 )=ot r, f)). p r )p µ+ε0 r Θ t) µ ε0+ t.9) If µ = 0,.6) implies that there exists a costat A such that Θ r) <Alog r. Sice fz) is trascedetal, we have ) p r = pr p Θ t) A log t a v p+ prp r t r t +p.0) = Olog r) =ot r, f)). Thus from.9) ad.0), we coclude that r, ) r p p+6 T r, f)+ f, ) p p+7 T r, f)..) f Therefore Theorem ca be proved i a way similar to that i [6]. From Theorem ad a result of Valiro [,p.6],wehave Corollary. Let fz) be a etire fuctio of lower order µ0 <µ< ). If the umber of Borel directios of order µ of fz) especially the umber of Julia directios) is fiite, the the coclusio of Theorem holds.

4 Wu Shegjia Some Results o Etire Fuctios of Fiite Lower Order 7 As Miles [6] observed, we have the followig Theorem 3. Suppose that fz) satisfies the hypotheses of Theorem or Corollary ). The the Nevalia deficiecy is idepedet of the choice of the origi. 3. Applicatios From [], the followig theorem ca be derived immediately. Theorem A. Suppose that fz) is a etire fuctio of fiite lower order µ. Letq deote the umber of Borel directios of order µ ad p deote the umber of fiite deficiet values of fz), thep q. I this sectio, we shall use the basic results i to ivestigate the properties of the etire fuctios extremal for Yag s iequality p = q. The iterestig result i this sectio is Theorem 7, i which we obtai a precise estimate of the total deficiecy of this kid of fuctios. Before statig our results, we formulate two lemmas first. Lemma [8]. Let fz) be trascedetal meromorphic i the plae ad a be a complex umber may be ) such that δa, f) > 0. Suppose that k > ) ad k > ) are two costats ad that E = {r; r is k,k ) ormal }. For every sufficietly large r E,Er) deotes the set of the values of argumet ϕ which satisfies log fre iϕ ) a > 4 δa, f)t r, f), a, log fre iϕ ) > 4 δa, f)t r, f), a =. The we have meser) K>0, where K is a costat depedig oly o k,k ad δa, f) ad ot depedig o r. Lemma [,p.433]. Suppose that fz) is aalytic o Ω θ, θ)0 <θ ) ad satisfies the followig coditios: ) There exist two distict fiite complex umbers a ad b such that the spherical distaces betwee a, b ad are larger tha d 0 <d< ) ad that log + {Ω θ, θ; r),f = X} ν<+, X = a, b. 3.) r + log r ) There exists a poit set E α z; z = R, θ + ε arg z θ ε)0 <ε<θ,r R R ) such that mese α HR H ε ) ad fz) α <e N 3.) for z E α, where α is a fiite complex umber ad N 0 is a costat. The for every η>0, we have { ) 6 [ R ) fz) α exp Aε, θ) R θ ν+η) R R R ν+η log R ] R +log + α Bε, θ) Cθ)log R R + Dε, θ) ) R θ R N } 3.3)

5 7 Acta Mathematica Siica, New Series Vol. 0 No. for z Ω θ + ε, θ ε; R,R ), provided that R is sufficietly large, where Aε, θ) < +,Bε, θ) > 0 ad Dε, θ) < + are costats depedig oly o ε ad θ, adcθ) is a costat depedig oly o θ. Now we prove Theorem 4. Suppose that fz) is extremal for Yag s iequality, i.e., fz) is a etire fuctio of lower order µ<+ ad satisfies p = q where p p<+ ) deotes the umber of fiite deficiet values ad q deotes the umber of Borel directios of order µ of fz). The for every deficiet value a i i =,,,p) there exists a correspodig agular domai Ωθ ki,θ ki+) such that for every ε>0 the iequality log fz) a i >Aθ k i,θ ki+,ε,δa i,f))t z,f) 3.4) holds for z Ωθ ki +ε, θ ki+ ε, r ε, + ), where Aθ ki,θ ki+,ε,δa i,f)) is a positive costat depedig oly o θ ki,θ ki+,ε ad δa i,f). Proof. Suppose that arg z = θ k k =,,,q)areq distict Borel directios of order µ of fz). By Theorem, there exists r 0 > 0 such that r is,k ) ormal for all r>r 0,where k > ) is a costat ot depedig o r. From Lemma, we deduce that mese θ;0 θ<, log fre iθ ) a i > δa ) i,f) T r, f) >Ka i,k ) 3.5) 4 for every ii =,,,p)ad mese θ;0 θ<, log fre iθ ) > 4 ) T r, f) >K,k ), 3.6) provided that r > r > r 0 ), where, from ow o, Kx, y) deotes a positive costat depedig oly o x ad y. Now we write K = mi Ka i,k ),K,k )) > 0. By a result of Valiro [,p.6] ad i p the Heie-Borel theorem, there exist two distict fiite complex umbers b ad b such that for every ε 0 <ε< 8p K ), r log + l= { p i= Ωθ i + ε, θ i+ ε; r),f = b l } log r τ<µ. 3.7) Sice the lower order of fz) equalsµ, settig η = µ 4 τ, we have T r, f) >r µ η for r r >r ). By 3.5), for every a i ad every sufficietly large r, there exists a correspodig agular domai Ωθ ki,θ ki+) such that mese θ; θ ki + ε θ θ ki+ ε, log fre iθ ) a i > δa ) i,f) T r, f) > K 4 4p. Thus by 3.7) ad Lemma, log fz) a i >Aε, θ k i,θ ki+,δa, f))t r, f) 3.8)

6 Wu Shegjia Some Results o Etire Fuctios of Fiite Lower Order 73 holds for z Ω θ ki + ε, θ ki+ ε, r ), r i =,,,p). Our ext goal is to prove that k i is idepedet of the values of r. From the argumet i the proof of the theorem i [], it is ot hard to see that if i j, Ωθ ki,θ ki+) adωθ kj,θ kj+) caot be eighbourig agles. Thus we may assume, without loss of geerality, that for some fixed r >r 3, 3.8) holds with k i =i i =,,,p). The we claim that 3.8) holds with k i =i for all r [r,r + ]. If our assertio is ot true, there must exist r r,r + ] such that, for some deficiet value a i0, 3.8) holds with k i0 i 0. It is obvious that k i0 caot be odd. Thus a similar argumet leads us to kow that for every Ωθ k,θ k+ ), there exists a deficiet value a ik such that log fz) a ik >Aθ k,θ k+,ε,δa ik,f))t r,f) 3.9) for z Ωθ k + ε, θ k+ ε, r, r )k =,,,p). Especially for z z = r ), 3.8) ad 3.9) hold. This cotradicts 3.6) ad the fact that ε< 8p K, ad hece our assertio follows. Replacig r by r + ad repeatig the above argumet successively, we ca prove that 3.8) holds with k i =i for all r>r 3. Theorem 4 is completely proved. As a direct cosequece, we have Corollary [9]. Suppose that fz) satisfies the coditios of Theorem 4. The every deficiet value of fz) is also its asymptotic value. I the sequel, we always assume that the deficiet value a i correspods to Ωθ i,θ i )i =,,,p) i the sese of 3.8). The followig theorem ca be derived from Theorem 4. Theorem 5. Suppose that fz) satisfies the coditios of Theorem 4. The λ = µ, where λ is the order of fz). Theorem 6. Suppose that fz) satisfies the coditios of Theorem 4. The ay asymptotic value of fz) is also its deficiet value. Proof. If Theorem 6 is false, the fz) has a fiite asymptotic value a such that δa, f) =0. Suppose Γ is the asymptotic curve correspodig to a. By Theorem 4, it is clear that for ay ε>0, if r is sufficietly large, we have Γ { z >r} Ωθ k0 ε, θ k0+ + ε) for some fixed k 0. Settig ε 0 =mi 4µ, 4 θ k 0 θ k0 ), ) 4 θ k 0+ θ k0+), we see that r freiθ k 0 ε 0) )=a k0 ad r freiθ k 0 ++ε 0) )=a k0+. Without loss of geerality, we may assume Γ Ωθ k0 ε 0,θ k0++ε 0 ). Thus Γ :argz = θ k0 ε 0, Γ :argz = θ k0++ε 0, ad Γ divide Ωθ k0 ε 0,θ k0+ +ε 0 ) ito two simply coected domais D ad D. Settig Mr, f)= max fre iθ ) ad otig that θ k0+ θ k0 = θ k0 ε 0 θ θ k0 ++ε 0 µ [9], we deduce by a similar argumet i [,p.3] that log log Mr, f) 4 µ. 3.0) r log r 3 This cotradictio the proves the theorem. Before formulatig the mai result i this sectio, we ote that recetly Drasi ad Weitsma proved the followig result. Theorem B []. For a measurable set of [, ] let St, E) =St, E, f) = log fte iθ ) dθ. 3.) E

7 74 Acta Mathematica Siica, New Series Vol. 0 No. If f is a meromorphic i the plae of order λ0 <λ<+ ), the where Nr, 0) + Nr, ) kλ) Sr, E)+Nr, ))/T r, f), 3.) r r T r, f) si λ q kλ) = q + si λ, λ q + ), si λ +q, q + <λ q + ). I [], Drasi ad Weitsma also ivestigated a extremal problem. Usig quasicoformal deformatios, they proved a result which is a applicatio of Theorem B. Now we cosider the case of etire fuctios ad prove Theorem 7. If fz) satisfies the coditios of Theorem 4, the we have δa, f) kµ), a where si µ q kµ) = q + si µ, µ q + ), si µ +q, q + ) <µ q +. Proof. The proof of the theorem cotais several steps. i) By Theorem 4 ad Cauchy iequality, for every ε>0, we have ad for j =,,,p. Settig Ω j,ε = Ωθ j ε, θ j+ + ε) admr, Ω j,ε )= I fact, sice N = r f re iθj ε) )=0 3.3) r f re iθj++ε) )=0 3.4) max f z), weprovethat z=re iθ z Ω j,ε log log Mr, Ω j,ε ) = µ j =,,,p). 3.5) r log r sup θ=θ j ε or θ j+ +ε f re iθ ) < +, there exists z j Ω j,ε such that f z j ) >N. Thus we ca fid N N,eN) such that there is o zero of f z) o the curve defied by f z) = N, i.e., the curve is aalytic. Cosiderig E = Ez; f z) >N) ad deotig the compoet cotaiig z j by Ω j,ε, we see that Ω j,ε Ω j,ε ad that Ω j,ε is ubouded. Deotig z = t) Ω j,ε by θ j,t ad its liear measure by tθ j t), the as i the proof of Theorem 6, we obtai log log Mr, Ω j,ε ) log r +log9 + ε µ

8 Wu Shegjia Some Results o Etire Fuctios of Fiite Lower Order 75 ad Sice for every ε 0 <ε <ε), ad Mr, Ω j,ε ) Mr, Ω j,ε ), therefore log log Mr, Ω j,ε ) r log r log log Mr, Ω j,ε ) r log r µ + ε. µ + ε log log Mr, Ω j,ε ) log log Mr, Ω j,ε ) r log r r log r. µ + ε As ε ca be arbitrarily small, our assertio follows. ii) Usig the same method as i [, p ], we ca prove that there exist p distict curves Γ j j =,,,p) such that log log f z) = µ 3.6) z Γ j log z z ad Γ j Ω j,ε provided that z is sufficietly large. iii) Deotig Er) ={θ, f re iθ ) }, the Er) p j= Ω j,ε for all sufficietly large r by Theorem 4 ad Cauchy iequality. Let E c r) = p j= θ j,θ j+ )\Er). Next we prove If 3.7) is false, the for a fixed ε 0 > 0wehave r mesec r) =0. 3.7) r mesec r) ε ) By a result of Milloux [0,p.95], for etire fuctios of positive fiite order λ every Borel directio of order λ of f z) is also a Borel directio of order λ of fz). Accordig to the coditios of the theorem together with Theorem 5, there is o Borel directio of order µ of f z) i p j= Ωθ j,θ j+ ). Thus there exist two distict fiite complex umbers b ad b such that for ε>0, p log Ωθ j + ε, θ j+ ε; r),f = b l r j= log r τ<µ, l =, ). 3.9) By 3.8), there exists a agular domai Ωθ j0 + ε, θ j0+ ε) ad a sequece of positive umbers r )withr such that mese c r ) θ j0 + ε, θ j0+ ε) ε 0 4p. 3.0)

9 76 Acta Mathematica Siica, New Series Vol. 0 No. Now we take a fixed η 0 > 0) such that 6η 0 θ j0+ θ j0 ε +µη 0 + µ 3η µ η 3.) ad µ η <µ η, 3.) η 0 where η = µ 5 τ. By Lemma, settig N =0,α=0,R = r η0,r = r, we coclude that ) 6η 0 f θ z) exp Aε, θ j0,θ j0 + j0+)r θ j ε +µ 4η)η0+µ 4η 0 expar µ 3η ), for z Ωθ j0 + ε, θ j0+ ε, r η0,r ). This implies that log MΩθ j0 + ε, θ j0+ ε; r η0,r ),f ) Ar µ 3η. 3.3) O the other had, 3.6) implies that if z is sufficietly large, the log f z) > z µ η for z Γ j j =,,,p). Deotig the last itersectio poit of Γ j0 with z = r η0 )by z, i which we omit the subscripts j 0 ad for the simplicity of otatio, we coclude that Thus there exists N log f z ) >r η0)µ η) 4 r η0)µ η), r r0)µ η) >r µ η. 3.4) ) such that there is o zero of f z) o the curve defied by log f z) = N, i.e., the curve is aalytic. Cosiderig E = z;log f z) >N, z r ) ad deotig the compoet cotaiig z by Ω, the from the priciple of maximum modulus, it follows that Ω z = r ) is ot empty. Agai deotig z, z = t, r η0 t r ) Ωbyθ t ad its liear measure by tθt), we deduce from 3.3), 3.5), 3.3) ad 3.4) that =θt) θt) ε θt). Therefore r ε r η 0 t r r η 0 By a estimate for harmoic measure [7,p.4],wehave log f z ) log N +9 e r r η 0 tθt). tθt) log Mr,f). Thus Therefore r η0)µ η) 36 e r r η 0 tθt) log Mr,f). r ε r η 0 t log log Mr,f) log r ) η 0 )µ η)+o)

10 Wu Shegjia Some Results o Etire Fuctios of Fiite Lower Order 77 ad η 0 ε log r log log Mr,f) log r ) η 0 )µ η)+o). 3.5) Dividig every term of 3.5) by log r, lettig ad takig the upper it, we get η 0 ε µ η 0)µ η) =η 0 µ + η 0 )η. 3.6) As ε ca be arbitrarily small, this cotradicts the fiiteess of µ. 3.7) is proved. iv) Deotig E r) =Er) p j= Ωθ j,θ j ), by Theorem 4 we have r mese r) =0. 3.7) v) Let E = p i= θ j,θ j+ )ad Sr, E) = log f re iθ ) dθ. E Now we use the followig lemma. Lemma 3 [3]. Let gz) be meromorphic. With each r> 0) we associate a measurable set Ir) of value of θ) of measure mesir) =µr). The,for <r<r, mr, g; Ir)) = Ir) log + gre iθ ) dθ R R r T R,g)µr) From Theorem ad Chuag s iequality [0,p.06],wehave [ +log + µr) T r, f ) k T r, f ) 3.8) for all sufficietly large r, wherek is a costat. By 3.7), 3.7), 3.8) ad Lemma 3, we deduce that mr, f ) Sr, E) = log + f re iθ ) dθ + log + f re iθ ) ) dθ E r) E c r) [ ] T r, f )mese r) +log + mese r) [ ] +T r, f )mese c r) +log + mese c r) ]. = ot r, f )). By usig Theorem B ad 3.9), we have kµ) Sr, E)/T r, f )) = kµ) mr, f )/T r, f )) r r N r, f ) = kµ) r T r, f = δ0,f ) ). Sice sup δa, f) δ0,f ) [5,p.04], this together with 3.30) gives Theorem 7. a 3.9) 3.30) Ackowledgemet. This paper is part of the author s doctoral dissertatio Chapter 4) writte uder the directio of Professor Yag Lo ad submitted to the Istitute of Mathematics, Academia Siica.

11 78 Acta Mathematica Siica, New Series Vol. 0 No. Refereces [] Drasi, D. ad Weitsma, A., Bouds o the sums of deficiecies of meromorphic fuctios of fiite order, Complex Variables, 3989), 3. [] Edrei, A. ad Fuchs, W.H.J., O the maximum umber of deficiet values of certai class of fuctios, Airforce Techical Report AFOSR TN, April 960). [3] Edrei, A. ad Fuchs, W.H.J., Bouds for the umber of deficiet values of certai classes of meromorphic fuctios, Proc. Lodo Math. Soc., 96), [4] Goldberg, A.A., Meromorphic fuctios with separated zeros ad poles i Russia), Izv Vyss, Uceb. Zaved. Matemat., 7:4960), [5] Hayma, W.K., Meromorphic Fuctios, Oxford, 964. [6] Miles, J., O the growth of meromorphic fuctios with radially distributed zeros ad poles, Pacific J. Math., 986), [7] Tsuji, M., Potetial Theory i Moder Fuctio Theory, Maruze, Tokyo, 959. [8] Wu Shegjia, Agular Distributio ad Borel Theorem of Etire ad Meromorphic Fuctios, Dissertatio, Istitute of Mathematics, Academia Siica, Beijig, 99. [9] Wu Shegjia, Somce problems related to the Borel directios of meromorphic fuctios i Chiese), Acta Math. Siica, 6993), [0] Yag Lo, Theory of Value-Distributio ad Its New Research, i Chiese), Sciece Press, Beijig, 98. [] Yag Lo, Deficiet values ad agular distributio of etire fuctios, Tras. Amer. Math. Soc., ), [] Zhag Guag-huo, The Theory of Etire ad Meromorphic Fuctios-Deficiet Value, Asymptotic Value ad Sigular Directios, i Chiese) Sciece Press, Beijig 986. Wu Shegjia Departmet of Mathematics Pekig Uiversity Beijig, 0087 Chia