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A new example of meromorphic functions sharing four values and auniqueness theoremMartin Reinders aa Institut für Mathematik , Universität Hannover , Welfengarten 1, Hannover 1, D-3000, Federai Republic of GemanyPublished online: 29 May 2007.

To cite this article: Martin Reinders (1992) A new example of meromorphic functions sharing four values and a uniqueness theorem, Complex Variables,Theory and Application: An International Journal: An International Journal, 18:3-4, 213-221

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Complex Variables, 1992, Vol. 18, pp. 213-221 Reprints available directly from the publisher Photocopying permitted by license only @ 1992 Gordon and Breach Science Publishers S.A. Printed in the United States of America

A New Example of Meromorphic Functions Sharing Four Values and a Uniqueness Theorem

MARTIN REINDERS lnstitut fiir Mathematik, Universita"t Hannover, Welfengarten 1, 0-3000 Hannover 1, Federai Republic of Gemany

Lct f and g be nonconslant meromorphic functions sharing four values a,, ..., a4. If every simple a,- point o f f is an a,-point of g with multiplicity at least three and every simplc a,-point of g is an a,-point o f f with multiplicity at least three (v = 1, ..., 4), then f = L o F o h and g = L o C o h , whcre L is a Mobius transformation. 11 is an entire Function and F and G are a given pair of elliptic functions sharing the values 0, 1, oo and - 1 DM.

AMS No. 30D35 Communicated: K. Habetha (Received August 7, 1991)

1. INTRODUCTION AN6 RESGLTS

In this paper the term "meromorphic" will always mean meromorphic in the com- plex plane C. Two meromorphic functions f and g share the value a E e iff -'({a)) = g-'({a)). We say that f and g share the value a CM (counting multiplicities) if f and g share the value a and every p-fold a-point zo o f f is also a p-fold a-point of g. f and g share the value a DM (different multiplicities) if f and g share a and every p-fold a-point zo o f f is a q-fold a-point of g with p # q. Here, p and q depend on 20.

We will use the standard notations and results of the Nevanlinna theory (see [3] for example).

The following two well-known uniqueness theorems are due to R. Nevanlinna.

THEOREM A (Five-Point Theorem [5, 61) Iff and g are nonconstant merornorphic functions sharing jive distinct values then f = g . THEOREM B (Four-Point Theorem [5, 61) Iff and g are distinct nonconstant mero- morphic functions sharing four distinct values al , . . .,a4 CM, then f = T o g with a Mobius transformation T . Moreover, two of the four values, a3 and ad, say, are Picard exceptional values off and g , and the cross-ratio (al,a2,a3,a4) equals -1.

The conclusion of Theorem B can be stated in the following equivalent form: there exist a Mobius transformation L and an entire function h such that f = L o exp(h) and g = L o exp(-h).

Gundersen [2] and Mues [4] proved that in Theorem B the condition that all four values are shared CM can be relaxed considerably.

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214 M. REINDERS

However, Gundersen showed that the condition CM cannot be dropped com- pletely in Theorem B. In [I] he gave the following example: let

eZ + 1 ( r Z + 1)2 j ' (z) := and g ( z ) :=

(ei - 1): 8(ez - 1) '

Then j' and g share the values 0, 1. m and -118 DM. In particular, j' is not a Mobius transformation of g , hence the conclusion of Theorem B does not hold.

This was the first example of two meromorphic functions sharing four values DM. All zeros and I-points are simple for f and double for g , whereas all -118-points and poles are double for f and simple for g . f and g are (up to trivial transforma- tions) the only functions sharing four values DM with the additional property that all shared values are taken only with the multiplicities one and two. Slightly more general, we have

THEOREM C ([7]) Let j and g be noncorutarzt meromorpltlc f~i~zctzorzs shanng lour distinct values al , , a4 I f every simple a,-point of f is a dolible a,-point of g and every smple a,-point of g is a double a,-poirtl o f f (v = 1 , . . ,4), then

f = ~ . o f o i z and g = L . n t o h

with a Mobius transformation L and an entire function h.

In this paper, we give a new example of meromorphic functions F and G sharing four values DM and prove a uniqueness theorem for F and G.

where U is a nonconstant solution of the differential equation

Then F and G share the values 0, 1, ca and - 1 . Every 0-, I - , m- and - 1-point of F and G is either simple for F and triple for G or triple for F and simple for G.

In contrast to the functions f and g in Gundersen7s example, F and G have both simple and multiple a-points for every shared value a (see Lemma 3(b) in the proof of Theorem 2). Up to trivial transformations, F and G are the only functions sharing four values DM and taking the shared values only with multiplicities one and three. A little more general, we have the fo!lowing "analogue" to Theorem C:

THEOREM 2 Let f and g be nonconstunr meromorphic functions sharing four dis- tinct values al , . . ., ad. I f every simple a,-point o f f is art a,-point of g with multiplic- ity at least three and every simple a,-point of g is an a,-point o f f with multiplicity at least three (v = 1,. . ., 4), then

f = L o F o h and g = L o G o l t ,

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MEROMORPHIC FUNCTIONS SHARING VALUES 215

where L is a M6biu.s transformation, h is an entire function and F and G are dejined in Theorem 1.

Remarks It follows from a well-known theorem of Rellich [8] tha? every noncon- stant solution of (4) is an elliptic function, so F and G are elliptic functions.

Steinmetz [9] gave a complete characterization of all triples of distinct mero- morphic functions sharing four values. In the simplest case, these are also elliptic functions. Every two of the three functions share four values, but neither CM nor DM. Together with the functions j and g of Gundersen and the functions F and G defined in Theorem 1 this are the only known pairs of meromorphic functions sharing four values such that the conclusion of Theorem B does not hold.

In [I] and [2] Gundersen stated some questions concerning distinct nonconstant rneromorphic function f and g sharing four values a,, . = .; ad.

side a set of finite linear measure? (d) If the cross-ratio of the snared values is equal to - 3 , then do tht: functions

necessarily share all four values CM?

A theorem of Collingwood (see [lo, p. 1011) states that m ( r , w ) = O(1) for every elliptic function w . Hence Theorem 1 as well as [9, Theorem 21 show that (a), (b) and (c) are to be answered in the negative. Theorem 3 gives a counterexample to (d).

The results of this paper are part of the author's dissertation [7].

2. THE PROOF OF THEOREM 1

From (2), (3) and (4) we obtain

The last equation implies that F and G share the values 0, 1, m and -1. Now let F(zo) = 0 with multiplicity p and G(zo) = 0 with multiplicity q. Because

of the differential equation (4), U takes the values 0 and -4 only with multiplicity two and the values 2 and -2 only with multiplicity one. Comparing this with ( 5 ) we see that F - G has only simple zeros, hence min{p,q} = 1. From (6) we get p + q = 4, thus either p = 1 and q = 3 or p = 3 and q = 1.

A similar reasoning applies to the I-, cc- and -1-points of F and G.

3. PRELIMINARIES FOR THE PROOF OF THEOREM 2

Let f and g be distinct nonconstant meromorphic functions sharing four distinct values al , az, a3 and a4 = m. For r > 0 let T(r) := max{T(r, f) ,T(r,g)). We write

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216 M. REINDERS

$(r) = S(r) for every function $ : ( 0 , ~ ) + R satisfying $(r) /T(r) + O for r --t x possibly outside a set of finite Lebesgue measure.

Following [4], we define

?,/I = J 'g l ( f -g)' (f a l ) ( f - a2)(f - a3)(g - u ~ ) ( R - QZ)@ - ~ 3 ) '

In Lemma 1, some well-known facts are collected (cf. [2, 41).

Proof We prove only the first part. Let

The lemma of the logarithmic derivative gives m(r ,p) = S(r), and it is easy to verify that cp has no multiple poles. Together with ~ e m r n a l(c) this gives the conclusion.

rn

4. THE PROOF OF THEOREM 2

Without loss of generality we assume that the shared values are 0, 1, m and c with some c E C, c # O,1. Let N(l,3)(r,a,) (N(3,1)(r,a,)) denote the counting function of the a,-points that are simple for f and triple for g (triple for f and simple for g, resp.), each point counted once.

LEMMA 3 For v = 1,. . . , 4 we have:

Proof The assumptions of Theorem 2 imply that

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218

If R $0, then

M. REINDERS

for v = 1,. . ., 4, which is a contradiction to Lemma 3(b).

Now let zp be a pole which is triple for f and simple for g . Lemma 3(b) guaran- tees that such a point exists. Let the Laurent expansions o f f and g at z p be

Substituting the expansions (15) and (16) into ( I l j gives, together with 62 - 0,

Substituting the expansions (15) and (16) into (12) gives, together with fi -= 0,

and from (13) we get

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MEROMORPHIC FUNCTIONS SHARING VALUES 219

Elimination of the unknown coefficients A-3, A-2: A-1, A. and B-l , Bo, B1, B2 from the equations (17)-(22) yields

and 2 ~ 3 - 3 ~ 2 - 3~ + 2 = 0.

The last equation has the solutions c = -1, c = 2 and c = l j2 . 'W.1.o.g. we assume that c = -1 since the other cases can be reduced to this case with the aid of a Mobius transformation. (23) then gives a = 16 and (13) reads

To solve the algebraic equation (24) we define a meromorphic funci i~n u by

We want to eliminate g from the equations (24) and (25). To this aim, we write both equations as polynomial equations with respect t o g :

The resultant of the two polynomials must be identically zero:

1 12f - 16f3 6 f 1 2 f 3 - 16f f 0

0 1 12f - 16f3 6 f 12 f3 - 16f f ( u + l ) f ( 2 - u ) f 2 + u f 3 - ( u + 4 ) f 0 0 0

0 ( u + l ) f ( 2 - u ) f 2 + u f 3 - ( u + 4 ) f 0 0

0 0 ( u + l ) f ( 2 - u ) f 2 + u f 3 - ( U t 4 ) f 0

0 0 0 ( u + l ) f ( 2 - u ) f 2 + u f 3 - ( u + 4 ) f

= 16f ' ( f - l ) * ( f - 2f - 112(f + 2 f - 1 ) ~ ( 1 6 ( ~ + 1) f - u3(u + 4) ) 0.

f is not constant, so we conclude

It follows that u is not constant and takes the values 0, -1, -4 and m only with even multiplicities. Hence the function u t 2 / ( u ( u + l ) ( u + 4)) has only zeros of even multiplicity and no poles. Thus there exists an nonconstant entire function h satis- fying

d2 = 1 2 h ' ~ u ( u + 1)(u + 4). P 9 )

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220 M. REINDERS

Now let U be a nonconstant solution of (4) and define F and G by (2) and (3) , respectively. It follows from a well-known theorem of Rellich [8] that u ( z ) =

U ( h ( z ) + d ) where d is a constant. By redefining h we may assume that

From (28) and (29) we get

If necessary, we replace h by -h to obtain

Together with (30), this impiies f = F o h .

To prove tha: g = G o 12, we write g = f w in (27) and use (28) to eiimiriate S from the resulting equation. This gives a quadratic equation in w

which has the solutions u + 4 w = ---

u(u + 1)

and

Now assume that (33) holds and let zo be a zero of u3 - 16(u + 1). Then f (20) f 0, ca and w(zo) = 0 which implies g ( z o ) = 0. This is not possible since f and g share the value 0 .

So (32) must be true. Combining this with (30) and (31), we get

This completes the proof.

References

G. Gundersen, Merornorphic functions that share three or four values, J. London Math. Soc. 20 (1979), 45746 . G. Gundersen, Meromorphic functions that share four values, Trans. Arne,: Math. Soc. 277 (1983), 545-567. W. K. Hayrnan, Meromorphic Functions, Clarendon Press, Oxford 1964. E. Mues, Merornorphic functions sharing four values, Complex Varinbles Theory Appl. 12 (1989), 169-179.

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MEROMORPHIC FUNCTIONS SHARING VALUES 221 (

15) R. Nevanlima, Einige Eindeutigkeitssatze in der Theorie der meromorphen Funktionen, Acta Math. ( 48 (1926). 367-391. I

I [6] R. Nevanlinna, Le ;hkor2me de Picard-Burel et la rhtorie des functions mkrumorphes, Gauthier- 1

Villars, Paris, 1E9. I I [7] M. Reinders, Eindeutigkirssarzefir meromnrphe Funktionen, die vier Werte teilen, Dissertation, Uni- (

versitiit Hannover, 1990. I

[8] E Rellich, EUiptische Funktionen und die ganzen Liisungen von y" = f ( y ) , Marh. Z. 47 (1942),: 153-160. I

[Y] N. Steinrnetz, A uniqueness theorem for three meromorphic functions, Ann. Acad. Sci. Fenn. Ses A( I Math. 13 (1988), 93-110. I

I

[lo] 0. Teichmiiller, Eine Umkehrung des zweilen Hauptsatzes der Wertverteilungslehre, Deutsche Math.( 2 (1937), S107. I I

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