Abstract
The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which poly
Results & Lemmas (5)
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Theorem 1.1
Theorem 1.1 (Koebe’s 1 4 theorem). For every f ∈S, the range f(D) contains the disk w: |w| < 1 4. 2010 Mathematics Subject Classification.…
Theorem 1.1 (Koebe’s 1 4 theorem). For every f ∈S, the range f(D) contains the disk {w : |w| < 1 4}. 2010 Mathematics Subject Classification. 30C10, 30C25, 30C55, 30C75. Key words and phrases. Koebe’s one-quarter theorem, Koebe radius, univalent polynomial. The second author was supported in part by grants MTM2014-51834-P and MTM2017-83499- P from El Ministerio de Econom´ıa y Competitividad (Spain), and by grant 2017-SGR-358 from AGAUR (Generalitat de Catalunya). 1
Theorem 1.1
Theorem 1.1 were also touched upon, and this has largely spurred our interest in the problem. While the above discussion seems to provide…
Theorem 1.1 were also touched upon, and this has largely spurred our interest in the problem. While the above discussion seems to provide evidence in favor of a “yes” answer, we now disprove the conjecture (at least in the UN,R setting) by showing that the actual answer is a resounding “no,” already for N = 3. As a
Lemma 4.1.
Lemma 4.1. For (a2, a3) ∈R2, the polynomial (4.1) is univalent in D if and only if (a2, a3) belongs to V. Obviously enough, the univalence…
Lemma 4.1. For (a2, a3) ∈R2, the polynomial (4.1) is univalent in D if and only if (a2, a3) belongs to V . Obviously enough, the univalence region V is also symmetric with respect to the a3 axis (as is Γ). This is due to the fact that the polynomial (4.1) and its reflection p∗(z) := −p(−z) = z −a2z2 + a3z3 are, or are not, univalent simultaneously. The Suffridge polynomial (2.3) corresponds to the vertex 2 √ 2 3 , 1 3 = γ2 ∩γ3
Lemma 4.3.
Lemma 4.3. Let p be a polynomial of the form (4.1) with real coefficients and with a3 ̸= 0. In order that p be of type II, it is necessary…
Lemma 4.3. Let p be a polynomial of the form (4.1) with real coefficients and with a3 ̸= 0. In order that p be of type II, it is necessary and sufficient that a3 > 0 and
Theorem 5.1.
Theorem 5.1. The only extremal polynomials for the class U3,R are p3, as defined by (2.5), and its reflection p∗ 3.
Theorem 5.1. The only extremal polynomials for the class U3,R are p3, as defined by (2.5), and its reflection p∗ 3.
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