ABSTRACT
Since this contradicts (2.5-8), we see that (2.5-11) holds and from Theorem
A version of this lemma for the case n = 1 was given in [ 216 ], while the version for arbitrary n was proved in [ 199 ].
The condition Re P(z) > 0 for z E U implies condition (2.5-8) and so here we also have the conclusion Re p(z) > 0. This "right-half' condition on P played a crucial role in several previous investigations on differential
inequalities. The new condition (2.5-8) involving the slit mapping Rc,n
opens the left half-plane through the opening ( "open door" ) between the
two slits of Rc,n. This essentially doubles the region of variability of P and
is a significant extension of earlier results.
