5. The Open Door Lemma and the Integral Existence Theorem

A natural problem is to extend the result by finding a domain D containing the right half-plane so that

A first attempt might be to try to take D as a half-plane whose boundary is "moved" slightly to the left of the imaginary axis. A counter example to this

possibility was given by Pfaltzgraff, Reade and Umezawa [ 324 ]. They

showed that for any a < 0 and for f defined by