On the Extension of a Vector Function so as to Preserve a Lipschitz Condition

ing the origin, the theorem from Polya-Szegö may be applied with the F(z) of the theorem taken as A (z). Theorem 111(b) then follows immediately. As an application of Theorem III, let us consider the polynomial F(z)=jy~oakG(k+p)z where p>0 and G(z)=T(z)~ = ^ n * = i ( l + n~'z)e~} the reciprocal of the gamma function. Since J> = 0 and all the zeros of G(z-\-p) are negative, any sector wi ̂ arg z ̂ C02 S 7T—coi containing all the zeros of A (z) will also contain all the zeros of F(z). For example, if A{z) = (2 — 2)(z+l — i), then F(z)=0.5z-(l+i)z-2 + 2i, which has the zeros (3.058 + 0.514*) and ( —1.058 +1.486i), both thus being in the sector O^arg 3^135° containing the zeros of A(z).