Extremal polynomials on arcs of the circle with zeros on these arcs

Trigonometric polynomials of least deviation from zero in measure and related problems

“…His solution is expressed in terms of harmonic measure. Lukashov and Tyshkevich also discuss problem (3.9) for several arcs of the circle in their recent paper [7]. As a special case, [7] contains a solution of problem (3.10) (for one arc).…”

“…As a special case, [7] contains a solution of problem (3.10) (for one arc). Note that methods of [16,7] are different from that of [8][9][10]; in fact, [16,7] continue investigations by Lukashov [6]. Problem (3.9) on an arc of the circle without any restrictions on the arrangement of zeros was studied earlier in [15].…”

On Polynomials and Rational Functions Normalized on Circular Arcs

“…where T n (z) is the Chebyshev polynomial of the first kind of degree n, and ε is an arbitrary number such that |ε| = 1 (see [7,9]). 580 …”

“…Lately, considerable attention has been paid to polynomials with constraints on arcs of the unit circle [3][4][5][6][7][8][9][10]. In particular, in [9] it was demonstrated how new covering and distortion theorems, as well as estimates on the absolute value of the product of the leading coefficient and constant term of an algebraic polynomial with constraints on circular arcs, can be derived from the majorization principles for meromorphic functions [11][12][13].…”

“…Let w, |w| = r > r λ , ϕ(w) = 0, be a point of the set f (|ζ| > 1). The left-hand-side inequality in (7) and the fact that the function y = (x + 1) 2 /x is strictly increasing on the ray x > 1 imply that (|F (w)| + 1) 2 |F (w)| ≥ λ (r + 1) 2 r > λ (r λ + 1) 2 r λ = 4.…”