ABSTRACT
The function g satisfies all of the conditions of Lemma 2.2a and so by part (i)
By combining parts (i) and (ii) of Lemma 2.2a we obtain the inequality
and since
we obtain
(7.2-3)
7.2 EXTENSIONS 351
By using the summation form of A, conditions (i) and (ii) follow from (7.2-2)
and (7.2-3). Finally, from the fact that
IA(Df(z0 )z0 )1 ~ IIDf(zo)zoll • we obtain II Df(z0 )z0 II :?: mil f(z0 ) II which leads to condition (iii). D
G. Kohr and P. Liczberski in 1998 extended this last lemma from the
supremum norm case f : ~ n -7 en to the arbitrary norm case f : B -7 en . We next state their result, but omit the proof since it is similar to the proof of
If r0 E (0, 1) and Zo E Br0 are chosen such that
llf(z0 )11 = Max { llf(z)ll : z E Br0 },
such that s :<:: m :<:: 1 and the following relations hold:
(ii) 11Df(z0 )z0 II = sllf(zo)ll. and
Next we consider en as an n-dimensional Hilbert space with the Euclidean inner product and norm as given in (7.2-1). The following extension of the
Jack-Miller-Mocanu lemma on the open ball Br of en was obtained in 1995
by P. Curt and C. Varga.
