Download Conformal Invariants, Inequalities, and Quasiconformal Maps by Glen D. Anderson PDF

oo where an = :I: -k1 - log n. n k= I The constant y has the following equivalent definitions. 10. Lemma. 9) may be written as (I) (2) y = I f0 oo (3) y = J dt (1 - e- 1 - e- 1 / 1) t ' ( 1 1 -t ---1 - - ) e dt. 1 - et Proof. Part (1 ) of this lemma is given in [WW, Example 2, p.

For x = (x 1 , x2 , . . , Xn ) , a = (a 1 , a 2 , . . , a n ), x; :::: 0, a ; :::: 0, i = I , . . 4 1 ) g (t) = S, (x) = (I>: n ) 1/r i= I is decreasing on (0, oo) [BhB ] . For O < r < t < oo these two monotone properties yield the very useful inequalities M, (a, x) ::: M, (a, x) and S, (x) :::: S, (x). The expression M, (a, x) is called the weighted mean of the numbers x ; , whereas S, (x) is called the t -norm or t -sum of the numbers x; . Using l ' Hopital's Rule, we can show that lim M, (a, x ) l->-0 = n i=I xt .

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