Functions of Uniformly Bounded Characteristic on Riemann Surfaces

A characteristic function T(D, w, f) of Shimizu and Ahlfors type for a function / meromorphic in a Riemann surface R is defined, where D is a regular subdomain of R containing a reference point w G R. Next we suppose that R has the Green functions. Letting T(w,f) = Mthdir T{D,w,f), we define / to be of uniformly bounded characteristic in R, f G UBC(i?) in notation, if sup^g/j T(w, f) < oo. We shall propose, among other results, some criteria for / to be in UBC(ii) in various terms, namely, Green's potentials, harmonic majorants, and counting functions. They reveal that UBC(A) for the unit disk A coincides precisely with that introduced in our former work. Many known facts on UBC(A) are extended to UBC(fl) by various methods. New proofs even for R = A are found. Some new facts, even for A, are added. 0. Introduction. We shall extend the notion of UBC and UBCo from the unit disk A = {|z| < 1} (see [Yi and Y2]) to hyperbolic Riemann surfaces, prove some results analogous to those in A, and add some facts, new, even for A. A hyperbolic Riemann surface S is one possessing Green functions; thus, its universal covering surface S°° must be conformally equivalent to A, so that S°° and A are identified. Our study begins with how to define the Shimizu-Ahlfors characteristic function T(D, w, f) on "good" subdomains D containing a point w of a Riemann surface R, hyperbolic or not, on which / is meromorphic. Each point of R is identified with its local-parametric image in the complex plane C = {|z| < oo}. By D we always mean a relatively compact subdomain of R, whose boundary dD consists of a finite number of mutually disjoint, analytic, simple and closed curves on R. If we refer to a pair D and w E R we always assume that w E D. The radius r = r(D, w) > 0 of D is defined by logr = \im(gD(z, w) + \og\z w\) as z —+ w within the parametric disk of center w, where go(z,w) is the Green function of D with pole at w. We now set dt, T(D,w,f) = n-1 /V1 \f f f*(z)2dxdy Jo U J Dt where Dt = {z E D;gD(z,w) > log(r/i)}, 0 < t < r, and (Q1] f#(z]_i\f'(z)\/(l + \f(z)\2), if /(*) 7^00, (U ] ' {Z) \ (l/f)*(z), if/W = oo, Received by the editors June 6, 1983 and, in revised form, May 10, 1984. 1980 Mathematics Subject Classification. Primary 30D30, 30D50, 30F99; Secondary 30F15. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 395 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 396 SHINJI YAMASHITA is not a function on R, yet the second-order differential f*(z)2 dxdy, z = x + iy E R, is well defined on R. The Green-potential expression (0.2) T(D,w,f) = n-1 j j f#(z)2gD(z,w)dxdy, w E R, will be proved later. The nomenclature of T is justified because for R = {\z\ < p}, D = {\z\ < r} with 0 < r < p < oo, and w = 0, we have the usual one because gD(z,Q) = log|r/z|. Henceforth we always assume that R is hyperbolic and we set T(w, f) = T(R, w, f) = lim T(D, w, f) < oo. Dî R This means that given e > 0 we can find a compact set K, w E K c R, such that |T T(D)\ < e for all D D K, with the obvious change in case T — oo. Lebesgue's convergence theorem applied to (0.2) yields (0.3) T(w,f)=-K~1 j j f*(z)2g(z,w)dxdy, where g = gR is the Green function on R; (0.2) can be regarded as the case R — D. A meromorphic / on R is said to be of uniformly bounded characteristic, / G UBC = UBC(i2) in notation, if the function T(w, f) is bounded on R, while, / G UBCo = UBCo(Ä) if limw^QRT(w, f) = 0, that is, for s > 0 there exists a compact K c R such that T(w, f) < e in R\K. In §1 we extend our study from the family M = M(R) of meromorphic functions on R to Me = Me(R) consisting of multiple-valued meromorphic functions with single-valued moduli on R. We can easily extend the definition of UBC (UBCo, respectively) for / G M to UBCe = \JBCe(R) (UBCe0 = UBCe0(Ä), resp.) for fEMe. In §2, (0.3) for / G Me is proved. Thus, criteria are obtained in terms of the Green potentials (Corollary 2.2). The families BMOAe = BMOAe(Ä) and VMOAe = VMOAe(Ä) are defined for pole-free members of Me; these are extensions of BMOA and VMOA in the disk. For the definition of BMOA(Ä) see [M]; note that BMOA(Ä) = M(R) n BMOAe(Ä). The formulae BMOAe C UBCe and VMOAe C UBCeo are now obvious. An expression of T in terms of the limit (D î R) of the mean of \ log(l + |/|2) on 3D and the limit of N(D,w,f)= Y 9D(w,b) f{b)=oo,beD will be of use to compare T with L. Sario's characteristic function Ts (see [SN]). Sario's class MeB(R) coincides with that of / G Me for which T(w, f) is finite for aw — w(f) G R. §3 is devoted to the study of the least harmonic majorant R is considered in §5, and the identity T(R,ir(6), f) = T(A,6,f o 7t), é g A, for / G M(R) is proved. As applications we obtain: (1) If / G UBC(S) and h: R -> S is an analytic map, then f ohE VBC(R). (2) If h: R —> S is of type B/ in the sense of M. Heins [Hi], and if / o h G UBC(Ä), then / G UBC(S). Finally, a contribution is made to the classification of Riemann surfaces: Otjbc § Obmoa1. Families UBCe and UBCeoThe functions on R, which we shall actually study, are, for the most part, the "generalized" meromorphic functions on R. Let Me = Me(R) be the family of multiple-valued functions / = exp(u + iu*) on R, where u is a single-valued function harmonic on R except for countably many logarithmic singularities an clustering nowhere in R, such that u(z) — kn log \z — an\ is harmonic in the parameter disk of center an with the integral coefficient kn. The multiple-valuedness of / arises from that of the conjugate function u* of u on R. It is natural to regard the constant zero as a member of Me. The modulus |/| of / G Me is single-valued throughout R, and each branch of nonconstant / in the parametric disk of each point w G R is single-valued there, and has the Laurent expansion (1.1) cx(z w)x + cx+i(z w)x+1 + ■ ■ ■ , where A is an integer with c\ ^ 0; the branches differ by multiplicative constants of moduli one. Therefore \c\\ is definite and (1.2) |/(«)|.<|ca| if|/M|¿oo. We call w E R a zero of / of order A if |/(w)| = 0. Similarly, w G R is a pole (or, co-point) of / of order —A if |/(w>)| = oo. The family M = M(R) of single-valued members of Me consists of all the meromorphic functions on R. For a EC we call w E R an o-point of order A of / G M if w; is a zero of order A of / — a. It is now easy to extend T(D,w,f) and T(w,f) = T(R,w,f) to / G Me. Actually, \f'(z)\ for |/(z)| ^ oo, as well as f#(z) defined by (0.1), is definite, so that the differentials \f'(z)\2 dxdy for pole-free / and f#(z)2 dxdy for arbitrary / are defined on R. The definitions of UBCe = UBCe(fi) and UBCe0 = UBCe0(Ä) are thus clear; just extend those of UBC = UBC(ñ) and UBC0 = UBC0(£) in the introduction to f E Me. 2. The Shimizu-Ahlfors characteristic function. We begin with the Green-potential expression (0.3) for / G Me. THEOREM 2.1. The identity (2.1) T(w,f) = ir~1 J J f*(z)2g(z,w)dxdy (< oo) holds for each f E Me(R) and each w E R. PROOF. It suffices to establish (2.2) T(D,w,f) = v-1 j f f*(z)2gD(z,w)dxdy License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 398 SHINJI YAMASHITA for each pair D,w. Set Ct(Z) iizEDt, iîzE D\Dt Then the identity together with /' ./(i ct(z)t l dt = go(z,w), zED, T(D,w,f) J JD [Jo ct(z)rldt f*(z)2dxdy proves (2.2). REMARK. Suppose that w is in the parametric disk Uwi of center w' E R, and define r' > 0 by logr' = \\m(gD(z,w) +log|z w\), where, this time z —* w within Vw>. The same proof as above then shows that T(D,w,f) t: j j f*(z)2dxdy dt, where D[ = {zE D; gD(z, w) > log(r'/£)}, 0 < t < r'. Two corollaries follow from Theorem 2.1. COROLLARY 2.2. For f E Me(R) the following are valid. (I) / G UBCe(R) if and only if (2.3) sup / / f*(z)2g(z,w)dxdy< oo. weDJ JR (II) / G UBCeo(Ä) if and only if (2.4) lim f f f*(z)2g(z,w)dxdy = 0. w^dRJ JR This corollary extends [Yi, Theorem 2.2, p. 352]. A pole-free / G Me(R) is said to be of bounded (vanishing, resp.) mean oscillation on R, f E BMOAe(ñ) (/ G VMOAe(Ä), resp.) in notation, if sup / / \f'(z)\2g(z, w) dxdy < oo weRJ Jr lim w^dR //." (z)\2g(z,w)dxdy = 0, resp. The family BMOA(Ä) = BMOAe(J?) f)M(R) is introduced by T. A. Metzger [M]. An immediate consequence of Corollary 2.2 is the following which extends [Yi, Theorem 7.1, p. 364]. COROLLARY 2.3. BMOAe(Ä) C UBCe(#) and VMOAe(R) C UBCe0(#). Following Sario we shall define the proximity function ms(D, w, /), the counting function Ns(D,w,f), and the characteristic function Ts(D,w,f) for / G Me and w E D. They are extensions of M. Parreau's [P, p. 183 ff.] for / G M. The reader is expected to be familiar with [SN, Chapter III] or with the papers [Si and 82]License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use FUNCTIONS OF UNIFORMLY BOUNDED CHARACTERISTIC 399 Let log+ x = max(logx,0) for 0 < x < oo, and set ms(D,W,f) = -(27rY1 [ log+ \f(z)\dgh(z,w), JdD where 3D is oriented positively with respect to D, and the star means the conjugate. The comparison (2.5) ms(D,w,J))l ^ oo. Therefore N(D,w,f) [ rln(t,f)dt Jo for all w E D because N(D,w,f) = oo if |/(w)| = oo. The characteristic function for / is now defined by Ts(D,w,f) = ms(D,w,f) + Ns(D,w,f). For X = m,N,ms,Ns, and Ts, we set X(w,f) — \imD-\RX(D,w,f). We shall compare Ts(w, f) with T(w, f) in Corollary 2.5 below. THEOREM 2.4. If \f(w)\¿ oo for f EMe(R), then (2.7) T(w, f) = m(w, f) + N(w, /) i log(l + |/M|2). U \f(w)\ = oo, then (2.8) T(w, f) = m(w, f) + Ns(w, f) log |cA|, where c\ is defined in (1.1). REMARK. If / G Me(R) is bounded, |/| < K, then T(w, f) < \ log(l + K2) by (2.7), so that /GUBCe(ñ). COROLLARY 2.5. If f E Me(R) is nonconstant, then (2-9) |TK/)-Ts(W,/)| 0, we let ~jw — {\z — w\ < e} and lb = {\z b\ < e}. Apply the Green formula to the function ip = (1/2) log(l + |/|2) on the domain D£ = D\~fw\(JbibSince A^ = (32/3x2 + 32/3y2)vj = 2/#2 in D£, it follows that (2.11) j j gD{z,w)Arl>(z)dxdy = -/ gD(z,W)d-^l\dz\+f ̂ (zf^^ldzl, JdDe 0V JdDe öv where the normal derivatives 3/3v are considered in the direction of the inner normal. As to the first integral in the right-hand side of (2.11), that on 3D is zero, and those on 3^w and d-^i, tend to zero and 2irk(b)gD(b,w) as e —+ 0, respectively. Furthermore, as to the second, that on 3D equals 2nm(f), and those on 3^w and ¿9-75 tend to — 2ttiP(w) and 0 as s —> 0, respectively. The resulting identity divided by 2tt, together with go(b,w) = go(w,b), yields (2.12) 7T-1 / / gD(z,w)f#(z)2 dxdy = m(f) ^(w) +Yk(b)gD(w,b). J Jd b In view of (2.2) we immediately observe that (2.7') is true. Suppose now that 3D contains at least one pole of /. For t, 0 < t < r, sufficiently near r, we obtain (2.7') for Dt\3Dt instead of D. Observing that T,m, and hence TV, all are continuous in t, one obtains (2.7') for D by letting t]r. For the proof of (2.8') we quote Jensen's formula (2.13) log|cA| = Ts(/)-Ts(l//), valid for / without any assumption on |/(w)| (see [SN, (7), p. 77],). Now, for / with |/H| = oo, we first note that (2.7') for 1// yields (2.14) Ns(l/f) = N(l/f) = T(l/f) m(l/f) = T(f) m(l//). On the other hand, (2.13) for the present /, together with log |/| = log+ |/| log+11//|, yields. log|cA| = -(27T)-1 / \og\f(z)\dgD(z,w) + Ns(f) Ns(l/f), JdD License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use FUNCTIONS OF UNIFORMLY BOUNDED CHARACTERISTIC 401