Some properties of extremal polynomials for the Ilieff conjecture

Maximal Polynomials and the Ilieff-Sendov Conjecture

“…In addition, Conjecture 1.2 is known to be true when the convex hull H of the roots of P is a line segment or a triangle [13], when H has all its vertices on the unit circle [12], when at least 2/3 of the coefficients of Pare 0 [13], and for a number of other special cases (see [8,13]). As for Conjecture 1.5, Phelps and Rodriguez have shown [10] that at least two roots of an extremal polynomial must lie on the unit circle.…”

“…In still another attempt to shed light on Conjecture 1.2, Phelps and Rodriguez [10] have made the Definitions 1.4. (1) Sn is the set of all polynomials of degree n which have all their roots in D.…”

On the Sendov Conjecture for Sixth Degree Polynomials

“…Surprisingly this conjecture has been proved only for polynomials of degree n < 5 and in a few special cases [2][3][4][9][10][11][12][13][14][15]. (See Marden [8] for an excellent expository article on this conjecture.)…”

“…Let «^ denote the family of all monic polynomials of the form Define I(zk) = min,^^,., \zk -C;|, I(p) = maxx<k<nI(zk), and I(0>n) = sup e3¡¡ I(p). Phelps and Rodriguez [10] proved that there exists an extremal polynomial pt(z) = Yll=x(z -z*k) £ 3Pn such that I(pJ = I(■&'") (see Lemma A). It thus suffices to prove the conjecture for pt i.e., show I(z*k) < 1 for 1 < k < n .…”

“…Lemma A (Phelps and Rodriguez [10]). There exists an extremal polynomial pt £ ¿Pn such that I(pt) = I(¿P¿) ■ Moreover, pt(z) has a zero on each subarc of \z\ -1 of length n .…”