someunsolved problems on meromorphic functions...

Internat. J. Math. & Math. Scl.Vol. 8 No. 3 (1985) 477-482

477

SOME UNSOLVED PROBLEMS ON MEROMORPHIC FUNCTIONSOF UNIFORMLY BOUNDED CHARACTERISTIC

SHINJI YAMASHITA

Department of MathematicsTokyo Metropolitan University

Fukasawa, Setagaya,Tokyo 158, Japan

(Received February 14, 1985)

ABSTRACT. The family UBC(R) of meromorphic functions of uniformly bounded characte-

ristic in a Rieman surface R is defined in terms of the Shimizu-Ahlfors characte-

ristic function. There are some natural parallels between UBC(R) and BMOA(R), the

fsmily of holomorr.hc f11nctonz of bounded mesn oscilltion in R. After a survey

some open problems are proposed in contrast with BMOA(R).

KEY WORDS AND PHRASES. Riemann surface, Shimizu-Ahlfors’ characteristic function,

UBC(R), BMOA(R), armonic majorant, F. Riesz’ decomposition, Carleson measure,

classification of Riemann surfaces.1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. Primary 30D35, 30D45, 30D50, 30D55,

30F20. Secondary. o0C8{), 30C85, 30F30.

I. INTRODUCTION

In a series ppers [19] [Pl], [3] [5] I have been studying functions

of uniformly bounded characteristic. After a survey I propose some questions which I

have been unable to answer. The adOective "unsolved" in the present title, therefore,

means more precisely "unsolved by the present author".

Let R be a Riemann surface which has the Green functions gE(z,w) with poles

w in R. As usual, each point of R is identified with its local-parametric image

in the complex plane. By D we always mean a subdomain of R such that the closure

D U 8D is compact and the boundary 8D consists of a finite number of mutually dis-

joint, analytic, smple and closed curves in R. For a point w of D we set

r exp{lim(gD(z,w) + loglz w I) },z-w

where z w within the parametric disk of center w.

Let Dt

{z g D; gD(z,w) log(r/t)}, 0 < t < r. For f meromorphic in R

we consider the second-order differential f#(z)edxdy, where f# If’I/(l + Ill2).The Shimizu-Ahlfors characterztc function of f is defined for a pair w, D with

wg D byr

i- t- [f #T(D,w,f) (z)2dxdy]dt"0 D

t

478 S. Yamashlta

T(r,f) T(D,w,f),

the usual Shimizu-Ahlfors characteristic function of f. Returning to general R we

now set

T(R,w,f) lim T(D,w,f)w6D+E

this means that given E > 0 we may find a compact set K c R (w 6 K) such that

IT(R) T(D) < g for all D D K with the obvious change in case T(R) . It is

.nown that

T(R,w,f) n-ill f#(z)2gR(z,wldxdy w 6 R,R

-if#Green potential of the measure (z)2dxdy in R.

If T(R,w,f) < for a w R, then T(R,w,f) < for all w R. We call f

to be of bounded characteristic, f BC BC(R) in notation, if T(R,w,f) < for

; (hence for each) w R. Thus, f BC if and only if f is Lindelfian and mero-

morphic in R in the sense of Maurice Heins.

By definition, f is of uniformly bounded characteristic, f UBC m UBC(R) in

notation, if

T(R,f) sup T(R,w,f) < .w6R

Each meromorphic f in R can be expressed as a quotient f fl/f2 of holomorphic

functions fl and f2 with no common zero in R. Therefore,

is a finite-valued subharmonic function in R because A(z)dxdy hf#(z)2dxdy. If

f BC, then

^(w) @(w) 2TCR,w,f), w 6 R, (i)CRA

where @R is the least harmonic majorant of in R, the smallest 8nong all the

harmonic majorants of @ in R. Therefore, the function T(R,w,f) of w is the

potential part of the F. Riesz decomposition of @ in R. Apparently, f 6 UBC ifA

and only if @R exists (namely, f BC) and the potential part is bounded in R.

In the special case R we can choose as D the non-Euclidean disks

-Iof center w 6 and the radii tanh r, 0 < r < i, so that

T((w,r),w,f) T(r,fw) (2)

where f (z) f((z + w)/(l + wz)), z 6 4.w

It is not difficult to observe that f 6 UBC(R) fop 6 UBC(A),where p: R is the universal covering projection. Actually, T(R,p(a),f)T(, a,fop). This fact reduces many problems on UBC(R) to those on UBC(), yet,in some cases, there are subtle differences; see the problem (I) below, for example.

Hitherto we have been concerned with meromorphic functions f in R. If f is

UNSOLVED PROBLEMS ON MEMOMORPHIC FUNCTIONS 479

,ot .... ’tee further, ther, we can propose cimilar considet’ati Jns on re:,!ncing f# by

If’l. Thus, beginning wiLh

T*(D,w,f) -lfrt-l[ff If’(z)12dxdy]dt,0 D

twe obtain T*(R,w,f) and others. If T*(R,w,f) < for a w 6 R, then f 6 H2(R),

^ in R, so that, T*(R,w,f) isthat s, II t st mono maont (Ifl>finite for all w 6 R. The converse is also true. To observe these we remember that

if f 6 H2(R), then

)R(w) -]fl (w) 2r*(R,w,f), w e R, (3)

an analogue of (1) holds [21], [23]. The left-hand side of (3) coincides with

(If f(w) 2)^(w), w 6 R.

The obvious analogue of (2), together with (3), yields in A the identity

if

A(Ifw f()le)A(0) e*(A,f).A ho!omorphic function f in R is called BMOA, f BMOA BMOA(R) in notation,

T*(R,f) sup T*(R,w,f) <w6R

the notion of BMOA(R) was first introduced by Thomas A. Metzger, and the present au-

thor extended it to many-valued functions [23]. In case R A, this coincides with

the known class BMOA(A). The inclusion formula BMOA(R) c UBCA(R) is obvious,

where UBCA(R) is the family of all the holomorphic functions in UBC(R). This is a

consequence of the obvious inequality T(R,f) T*(R,f), yet a better estimate

T(R,f) 2-11og{2T*(R,f) + 1}

can be proved. There exists f UBCA(A) BMOA(A).

It is known that if f 6 BMOA(A), then f is Bloch, f B(A) in notation, in

the sense that

sup( Izl)If’(z)l <zA

Similarly, it is known [19] that if f 6 UBC(A), then f is normal in the sense of

Olli Lehto and Kaarlo I. Virtanen, f 6 N(A) in notation, that is,

sup(l Izl2)f#(z) < .zA

Both B(A) and N(A) can be extended to B(R) and N(R) because R has the hyper-bolic metric. The inclusion formulae BMOA(R) c B(R) and UBC(R) c N(R) are pro-per in the case R A. (See the papers [i] [18],[22], [25], on normal meromorphicfuncti6ns and the related topics.) Algebraically, BMOA(R) is closed for summation,while UBC(R) is not; UBC resembles N at this point.

2. PROBLEMSNow, the problems

(I) For 6 * {Izl } we let n(e,f) be the number of the roots of the

equation f in R. Suppose that there exists an integer k 0 such that

cap{a 6 *; n(a,f) k} > 0,

where "cap" denotes the elliptic capacity. Is it true that f 6 UBC(R) ? This is

480 S. YAMASHITA

valid for R in case k O, and is valid for R A and k >: 0. Th# ,.b ..m

unsolved for k > 0 on general R. Earlier and well-known conclusion is that

Be(R).

(II) Let A(R,f) be the spherical area of the image f(R) of R, that is, the

prjection to * of the Riemannian image of R by f. Is there a constant k > 0

swh that

T(R,f) < kA(R,f) (5)

Since A(R,f) <w always holds if the sphere is considered to have the radius

1/2, the answer is "no" in general. We must therefore add the condition that A(R,f)< ; in this case T(R,f) < is obvious; see (I). The celebrated Herbert J.

Alexander-B. A. Taylor-Joseph L. Ullman inequality teaches us that for holomorphic f,

T*(R,f) <_- (2w)-IA(R,f),where, in this case, A*(R,f) is the Euclidean area of f(R). See the problem (VII)

below.

(III) As usual, let 0X

be the family of open Riemann surfaces R of class 0G

or of those which are hyperbolic with X(R) () It is easy to observe that

OUBCA c OBMOATo prove that the inclusion is proper we make a few modifications to the argument in

[23]. Let E be a compact set of linear measure zero, yet of positive capacity, ly-

ing on the real axis. Let a E and let h(z) i/(z a), z 6 We show that

R * h(E) is of OH2 (hence of OBMOA) yet R 0UBCA First, the function

z is of UBCA(R) because it omits the set h(E) of positive capacity, so that R

g OUBCA Next, let f 6 H2(R). Then foh 6 H2( E). Since E is removable for

H2, fh, and hence f must be a constant. Therefore, R 6 OH2 The problem is to

c 0Xcfind a reasonable X for which OUBCA

# #OBMOA

(IV) If f UBC(R), then f UBC(RI) for each subdomain R1 of R. Considerthe converse in the special case R A. Let0 < r < . Let f be meromorphic in A such that for each w A we may choose0 < r < i such that f UBa((w,rl). Is it tre that f UBC(A) The correspond-ing problem for BOA is solved in the positive in [21; the extensions to the border-ed Riemann surfaces un&er the obvious technical conditions are now easy.

(V) Each f UBC(A) has, as a member of BC(A), the expression: f (BI/B2)Fwhere B1

and B2 are Blaschke products with no common zero an F is holomorphican zero-free in A. We observe that F UBC(A). Conversely suppose that F UBC(A)

for f BC(A). Is it true that f UBC(A) ? The answer is "no" 19. Actually,there exists a nonnormal Blaschke quotient BI/B2 we have only to let F i. Whatis a reasonable condition for BI/B2 UBC(AI ? An attempt is proposed in [25 interms of uniformly separated sequences.

(VI) For W we set

if w=O.

UNSOLVED PROBLEMS ON MEMOMORPHIC FUNCTIONS 4I

The length of the arc SAN SZ(w) is (w) 2w(l lwl). A measureis called a Carleson measure if sup (Z(w))/(w) < , where w ranges over A. Itis known that f 6 BMOA(A) if and only if 1 Izl2)If’(z)12dxdy is a Carlesonmeasure (in the differential form). I proved [2hi that if f 6 UBC(A), then

df(z) (i Izl2)f#(z)2dxdyis a Carleson measure, and gave a partial answer for the converse. 17qe

df is CrZeson mesure. f 6 UBC(A) ? I note that a difficulty lies in thefact that f#2 is not subharmonic in general.

(VII) Let UBC0(R be the family of meromorphic functions f in R such thatT(R,w,f) 0 as w 3R, namely, given e > 0 we may find a compact K c R suchthat T(R,w,f) < e for w E R K this can be read in case R A, T(l,fw) 0as lwl i the holomorphic analogue is VMOA(R) obtained on replacing T by T*in the above. It is not difficult to prove that UBC0(R) c UBC(R) and VMOA(R)BMOA(R). Furthermore, we see

fP UBC0(A) f 6 UBC0(R) and fop 6 VMOA(A) f 6 VMOA(R),

because the projection of a closed disk in A is compact in R. Are the

uZ{ ? We remark that the following are known:

ffAf#(z)2dxdy < f UBC0(A); and ffAIf’(z)12dxdy < f VMOA(A);

see [19] and [2]. Are the eztenson8 te for R ? Metzger, in a communication,informed me some partial answers on the VMOA part. I do not know further information.

Finally we note that the hyperbolic analogue of UBC(R) is possible; see [21],[26], and a forthcoming paper [27]. Let f be holomorphic and bounded, Ill < i, in

R, and let log(l If12). Then is subharmonic with Az)dxdy

hlf’(z)12/(l If(z)12)2dxdy. The analogue of (i), and others, are true on replacing

f# by the hyperbolic derivative If’I/(l If12). We note that (f,0) tanh-llflis the non-Euclidean hyperbolic distance of f and O, and (f,0) is subharmonic

with 2(f,0) + log h. It is a future task to find some problems on this

family.

The present article depends on the lecture on February , 1985, at Mie University,

Tsu, Mie, Japan, where slightly abstract treatment beginning with the F. Riesz decompo-

sition of subharmonic function was discussed.

REFERENCES

YAMASHITA, S.:

i. On K. Yosida’s class (A) of meromorphic functions, Proc. Japan Acad. 5_0(1974),37-349.

2. The derivative of a holomorphic function in the disk, Michigan Math. J. 21(197h),129-132

3. On normal meromorphic functions, Math. Zeits. 141(1975), 139-145.4. Almost locally univalent functions. Monatsh. Math. 81(1976), 235-240.5. Lectures on Locally Schlicht Functions Tokyo Metropolitan University Mathematical

Seminar Reports, iii+ll2 pages, 1977; sold at the library of the Department ofMathematics.

6. Cluster sets of set-mappings, Ann. Polon. Math. 35(1977), 77-98.

482 S. YAMASHITA

7. Conformality and semiconformality of a function holomorphic in the disk, Trans.Amer. Math. Soc. 245(1978), 119-138.

8. The normality of the logarithmic derivative, Comm. Math. Univ. St. Pauli 27(1978),25-28.

i0.

ii.

12.

]3.

16.

20.

2

2R.

25.

26.

A non-normal function whose derivative has finite area integral of order O<p,Ann. Acad. Sci. Fenn. Ser. A. I. Math. 4(1978/1979), 293-298.A non-normal function whose derivative is of Hardy class Hp O<p Canad MathBull. 23(1980), 499-500.The normality of the derivative, Revue Roum. ’lath. Pure Appl. 25(1980), 481-484.

Criteria for functions to be Bloch, Bull. Austral. Math. Soc. 21(1980), 223-227.

Nonnormal Dirichlet quotients and nonnormal Blaschke quotients, Proc. Amer. Math.

Soc. 80(1980), 604-606.

Gap series and -Bloch functions, Yokohama Math. J. 28(1980), 31-36.The order of normality and meromorphic univalent functions, Colloquium Math. 44(1981), 159-163.Bi-Fatou points of a Blaschke quotient, Math. Zeits. 176(1981), 375-377.

Non-normal Blaschke quotients, Pacific J. Math. 101(1982), 247-254.Hardy norm, Bergman norm, and univalency, Ann. Polon. Math. 43(1983), 23-33.

Functions of uniformly bounded characteristic, Ann. Acad. Sci. Fenn. Set. A. [.

Math. 7(1982), 349-367.Exceptional sets, subordination, andHath. Scand. 54(1984), 242-252.

of uniformly bounded characteris"

F. Riesz’s decomposition of a subharmonic function, applied to BMOA, Boll.Math. Ital. 3-A(1984), 103-109.

Area criteria for functions to be Bloch, norm, l, and Yosida, Proc. Japan Acad. 59(1983), 462-464.

Functions of uniformly bounded characteristic on Riemann surfaces, Trans. Amer.Math. Soc. 240(1985), to appear.

Image area and functions of uniformly bounded characteristic, Comm. Mat. Univ.

St. Pauli, to appear.

Criteria for a Blaschke quotient to be of uni1’ormly bounded characteristic, !roc.Amer. Math. Soc., to appear.

Tout ensemble H2-enlevable est BMOA-enlevable, Comptes Rend. Acad. Sci. Paris 298(1984), 509-511.

27. Holomorphic f,nctionsof hyperbolically bounded mean oscillation, i preparations.

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