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E . S . Katsoprinakis and V. Nestoridis

circles Cj, j 6 J. We have card С = с and С с [jj^ACj) with card J < с. If С is different from each CjJ e J, then card(C n Cj) ^ 2; it follows that с < c, a diction. Therefore, С coincides with one of the circles Cj,jeJ. D

Proposition 5. Let s ^ 1, Л cz Z* a finite set, к у e<£^ for each 7 = (71,. . . , у^) 6 Л, and T and T^ be as above. Consider thefiinction f(zi,. . . , z^) = YjyeA K^V- • -^Г and suppose that there is a family of circles {Cj}jçj, with card J < c, such that /(T*) c: (Jjgj(Cj). Then we have the following:

i . The function f is of the form ''constant plus one term''; i.e. there is ay^ e A, 7^ Ф (0,. . . , 0), such that ky = 0, for all y in A - {(0,. . . , 0), 7^} and f{Zu . . . , Z,) = /C(o, ,0) + fcyoz^J. . . zf^.

ii . There exists a circle C, such that /(T*) с С.

iii . // feyo Ф 0, t/ien /(T^) = С and the circles С is unique. The center of С is /C(o, ,0) ^nd its radius is \kyo\ > 0. Furthermore, С coincides with one of the circles Cj,JE J, the coefficient /C(o, ,o) i^ one of the centers of Cj, and \ kyo\ is one of the radii of the above circles.

Proof For s = 1 the function / is holomorphic in С — {0} and 0 is not an essential singularity of/; Lemma 4 gives the result.

By induction on s we suppose that the statement holds for s and will prove it for 5+1. We have

( M fn= -M

According to the induction hypothesis, for each fixed z^+i g T, this function is of the form "constant plus one term".

Suppose /cyj, ,b,m = 0, for all (71,. . . ,7^ Ф (0,. . . ,0) and m = — M,. . . , M; then

M

f { Zi , . . . , Zs , Zs + i ) = g{Zs+i)= 2^ /Co, ,0,m^T+l

m= —M

and the result follows from Lemma 4.

Therefore , we can assume that there exists (7?,. . . , 7^) Ф (0,. . . , 0) and mo G {— M,. . . , M}, such that feyO ^o^^ ф 0. Then the function ^(Zs+1 ) = X m = - M '^ y?, , y«, m ^?+1 îs not identically zero and its zeros form a finite set ß с С Let z^^ i be fixed in T — ß. Then, according to the induction hypothesis, the constant term Хт=-м^о ,о,т^7+1 is one of the centers of the circles CjJeJ.

Therefore , for every z^+i g T — ß, the function

M

6 ( ^s + l ) = Z^ fco, ,0,т^?+1 m= —M

takes values in a set with cardinahty strictly less than c. It easily follows that 6(Zs+i) is constant and

• z:

( I )

K , ,o,m = 0, for all m Ф 0.