By Geertrui K. Immink (auth.)

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**Example text**

2. 1. 61) are again valid. 4). ,N} such that Eh = O. The corresponding point sh on the curve ~* is the only point in S(R) at a distance R from the origin. 62) " Let Xg(S h) = th and let £(t) be the directed segment from t h to to Since g < p , Xp(S(R)) is convex, which implies that £(t) c Xg(S(R)) and, consequently, C(s) ~ S(R). 63) we find + - ~ + c h ~ arg(t- th ) - a r g th < ~ + e h , tEXg(S(R)). 11) this implies that . 64) larg(t- th)- arg thl ~ ~ - mln{£h,- Ch} < ~ . 12 (case (iv), with e=arg(t h - t)) we obtain the estimate S(R) lllIIg,a,r+1_g ~ K IIfIl, R~I, provided that g>k and a>0, or g=k and a>b', where + -1 b' = [sin(min{~h,-Ch} )] max{0,-min Re{e i arg(th-t)~}}.

The proof is completed by a simple induction argument. ~7. Definition o~ Ac. Introductory remarks. For the definition of a right inverse of a (left) difference operator A several formulas are available in the literature (cf. ~4], ~5])Let A be an n x n matrix function which is holomorphic in a region G with the property that s C G implies s- I EG, and let A be the corresponding left difference operator. Suppose that Yo(S) is a fundamental matrix of the homogeneous equation Ay = O. 3) (s) ~ Yo(~ - I)-I{I -e-2~i(s-~)} -I f(~)d~.

49) in the form II(s)l _< K I S Ieq(tl/p)-q(T1/p)(t)~G f (TI/P)TI/P-ld~ 1, ~(t) s6 S(R). 65) Let sh be defined in the same manner as in a) and let th = Xp (Sh)We choose for Z(t) the directed segment from t h to t. Since Xp(S(R)) is convex, i(t) c Xp(S(R)) and hence C(s) c S(R). 66) y - r - I + p) log u(x), • 2 (x) =~(Re p and ~ ( x ) = % (x) + % ( x ) . 41) and the definitions given above, we obtain an inequality of the form ll(s)]{Fg,a,r+l_0(Isl)} -I < Kllfll ~ o o~(Xo)_~0(X)~rlr ~ ~ U(Xo)~I, fLy, tog u(--~--~JaXl.