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Additional resources for Romanian-Finnish Seminar on Complex Analysis: Proceedings, Bucharest, Romania, June 27–July 2, 1976

Example text

Springe~--Verlag, ~ittaker, E. , Cambridge, 1958. and G. N. Watson, Ziemer, W. P. Ex~remal Trans. mer. Math. Berlin, Soc. Michigan State University Department of Mathematics East Lansing, Michigan 48824 USA 1971. Modern Analysis. length and conformal capacity. 126 (1967), 460-473. Some problems in quasiconfqrmalit~f Cabiria Andreian Cazacu §i. The fi2st part oZ this paper contains some problems related to the theory of qc(quasiconformal)mappings. In 1969 st the flzst Romanian-Finnish p~esented a paper [ 5 ] concerning Seminar in Bra@ov I the influence of the orientation of the characteristic ellipses ~n the properties in the plane, of the qc mappings a subject which preoccupied me for a long time.

We introduce was obtained Following the ring in i l-n [ 2 ] Gehring R E = RE(b) whose For [iO] complementary are the segment C O = ~(x I ..... ~ i, xj = O, and Cl = the complement {(Xl ..... Xn) of an open the work of G e h r i n g b cited. be the rings bounded : 2 xI ~ coshZb "prolate" remark that the parameter R" ~n ~0 (23) components and fGD[ (l+cos2u lower bound [6 ], [II]) ~4 for n ~ 3 , O'n_ 1 • n = 3 and ~ 2 0n______~2 ~/2 cos n-2 u log ~ (23) upper bounds section we show that when Theorem (22) a set of improved 1 + sinh2b ellipsoid 2 ~ j ~_ n > n j2 ~ x ~i~ j=2 in above corresponds Next, for to We sinh-la sinh b > 4 , let by the segment CO n 2 S' = C(x !

Now, and employing /0 for integrand ~n of gives an For (22) and gives n-2 [ (coth v) n-i _ i] dv of variable (n-l) t = (coth v)i/(n-l) /i~D t 2n-4 _ t n-2 dt t 2n-2 - 1 is the upper bound previously known , ([6 ], [ii]) we get 27 5. Numerical For purposes of numerical we make the change in the inner integral. (22) following (33) log - - When If n n is even, and m a k i n g integral, equivalent the inner Proof. follows n = 3 The from in of the upper bound above. Theorem to change the change of variable we are able to rewrite in (33) is elementary.

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