About a problem of Hermite and Biehler

Todor Stoyanov
2002 Journal of the Australian Mathematical Society  
A point of departure for this paper is the famous theorem of Hermite and Biehler: If / (z) is a polynomial with complex coefficients a* and its zeros Zk satisfy Im Zt > 0, then the polynomials with coefficients Re a t and Im a k have only real zeros. We generalize this theorem for some entire functions. The entire functions in Theorem 2 and Theorem 3 are of first and second genus respectively. 2000 Mathematics subject classification: primary 30D20. PROOF. We have/ (z) = «(z) + iv(z) = cz n UZi(
more » ... + iv(z) = cz n UZi( l ~ z/z k )exp(z/z*). Let zo be such that v(zo) = 0 or u(zo) = 0. Then M(ZO) + iv(zo) = K(ZO) -iv(zo) or u(zo) + iv(zo) =
doi:10.1017/s1446788700003608 fatcat:dmmwvw5xwrdgbg2puw7ztw6j2a