SECTOR ANALOGUE OF THE GAUSS-LUCAS THEOREM

BLAGOVEST SENDOV, HRISTO SENDOV
2019 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The classical Gauss-Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows. We show that if the coefficients of a monic polynomial p(z) are in the sector {te iψ : ψ ∈ [0, φ], t ≥ 0}, for some φ ∈
more » ... 0}, for some φ ∈ [0, π), and the zeros are not in its interior, then the critical points of p(z) are also not in the interior of that sector. In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result. 2010 Mathematics Subject Classification. Primary 30C10. Key words and phrases. Gauss-Lucas theorem and polynomial and zeros and critical points of polynomials and polynomial with coefficients in a sector and interlacing polynomials and nonconvex.
doi:10.4153/s0008414x19000609 fatcat:d3cgr2gjuzcrddqkq42hsiw4wm