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Abstract
In this paper we prove a recent conjecture formulated by Dmitrishin, Smorodin and Stokolos about that certain polynomials are univalent in the unit disk. As a consequence we get an upper estimate for the Koebe radius of univalent polynomials.
Results & Lemmas (4)
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Lemma 1
Lemma 1. The following inequalities are valid (2) (3),, (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (15)
Lemma 1. The following inequalities are valid
$$\sin y \le y \qquad \forall y \ge 0,$$
(2)
$$\cos y \ge \cos x + (x - y)\sin x - \frac{(x - y)^2}{2}\cos x, \qquad 0 \le x \le y \le \pi,$$
(3)
$$\sin y \ge \sin x - (x - y)\cos x - \frac{(x - y)^2}{2}\sin x + \frac{1}{6}(x - y)^3\cos x$$
, $0 \le x \le y \le \pi$ ,
(4)
$$\cos 2y \le 1 - 2y^2 + \frac{2y^4}{3} \quad \forall y \ge 0,$$
(5)
$$\cos y + \cos 2y \ge -\frac{9}{8}, \qquad \forall y \ge 0,$$
(6)
$$\sin y \le \sin x - (x - y)\cos x, \qquad 0 \le y \le x \le \pi,$$
(7)
$$\cos y \le \cos x + (x - y)\sin x - \frac{(x - y)^2}{2}\cos x, \qquad 0 \le y \le x \le \pi,$$
(8)
$$\cos y \le \cos x - (y - x)\sin x, \qquad 0 \le x \le y \le \frac{\pi}{2},$$
(9)
$$\sin y \ge \sin x - (x - y)\cos x - \frac{(x - y)^2}{2}\sin x, \qquad 0 \le y \le x \le \frac{\pi}{2},$$
(10)
$$\cos ky \ge \cos kx + k(x-y)\sin kx - k^2 \frac{(x-y)^2}{2}\cos kx - k^3 \frac{(x-y)^3}{6}\sin kx, \quad 0 \le y \le x \le \frac{\pi}{2k},$$
(11)
$$\tan \frac{\pi y}{x} \le \frac{\pi}{x} (y - x), \qquad 0 \le \frac{x}{2} \le y \le x \le \pi,$$
(12)
$$\tan \frac{\pi y}{x} \ge \frac{\pi}{x} (y - x), \qquad 0 \le x \le y \le \frac{3x}{2} \le \pi,$$
(13)
$$\cot \frac{\pi y}{x} \ge \frac{x}{\pi(y-x)} - \frac{\pi(y-x)}{3x}, \qquad 0 \le \frac{x}{2} \le y \le x \le \pi,$$
$$\cot y \le \frac{1}{y} - \frac{y}{3}, \qquad 0 \le y \le \frac{\pi}{2},$$
(15)
$$2y\cos y - 3\sin y \le 0, \qquad 0 \le y \le \frac{\pi}{2}.$$
Lemma 2
Lemma 2. The set is a simple curve.
Lemma 2. The set $\Gamma_1$ is a simple curve.
Lemma 3
Lemma 3. The set is a simple curve.
Lemma 3. The set $\Gamma_2$ is a simple curve.
Lemma 4
Lemma 4. The set is a simple curve.
Lemma 4. The set $\Gamma_3$ is a simple curve.
Coefficient bounds & claims (2)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
Koebe radius of P_n ≤ (1/4) * sec(pi/(n+2))**2 for class S (polynomials of degree n) [Theorem]
function_family
Class S: Class of analytic and univalent functions in D with Taylor series f(z) = z + ...