# Contributions to Non-Standard Analysis by W.A.J. and Robinson, A. (Ed.) Luxemburg By W.A.J. and Robinson, A. (Ed.) Luxemburg By W.A.J. and Robinson, A. (Ed.) Luxemburg

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P d-isometries B w(a) = b. wE z E 52, 52 d p W-invariant {f o w : f E F wE = F. a). 0 = K. D. STROYAN 56 8, conformally invariant. by 0 The following are equivalent for a W-invariant family I;: F is a normal family. M(s/d)f ( z ) is ajinite for every z E *B and every f E *F. M(s/d)f ( z ) < K (a standard constant) for every z E B and f E F. Every f E *F is S-continuous on all of *B and hence uniformly Scontinuous in the metric d. Proof. => f E *F zE w(a) = z, w E * W. 2. 1. W s / d l f ( z ) = [ ~ ( ~ / d ) *m[Wd/d)w(a)I l = MWlf =- W(4Y O f E *F Mf(z) z E so (2) *B bound K.

1 Ip < 2. 1 s p < 2, E : C(T) < coo} < co 5. 1. 3. THEOREM. in L ~ ( T ) . E 2, x E T} L1(T) C(T). E *(C(T)) on *T is, *(C(T)), f E - C(T) {f> 22 E *T) = 0, {E(n) : n E * Z } - *f, 2 IF(n) - n@Z A IF(t) = 2n - *f (t)I2 = 0. 13 cnnZ - A = 0. W. A. J. 4. (Hausdorfs-Young Inequalities). 10. 1 1 (s,,,)~. m. IIsmllq by p. 6. f 0, + l), x E f(x) = by f 6.

2, I Kolb f -4 (3) = 0, (1). 1 *C. 2. > E Ix - yI S- d 52 E *C:lzl < l}. on B b E B. 3. DEFINITION. We say b is the center of an S-disk of 52 with respect to d provided there exists an internal conformal (1-1) mapping cp: *U + 52 satisfying: (1) 440) = b, (2) cp(0) = OAb), (3) M(d/p)p(z) exists, isfinite and non-infinitesimalfor z in the monad of zero. 4. 1(3) are taken with respect to d. Proof. f(cp(z)) f(w) d b M(s/p)f 0 q(z) = M(d/p)cp(z)* M(s/d)f(w). ) 4. 3. 5. 8 Riemann surface d D a f D c R.