Abstract
We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in \({\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}\). We found minimal disc containing all images of \({\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}\) and maximal disc contained in all images of \({\mathbb {D}}\) through polynomials of degree 3 and 4. Moreover we determine the extremal functions for both problems.
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1 Introduction
Let \({\mathcal {S}}\) be the class of all analytic functions \(f:{\mathbb {D}}\rightarrow {\mathbb {C}}\) normalized by \(f(0)=f'(0)-1=0\) which are univalent. Let \({\mathcal {H}}_d\) be the class of all analytic functions \(f:{\mathbb {D}}\rightarrow {\mathbb {C}}\) normalized by \(f(0)=1\) and such that \(f\ne 0\) in \({\mathbb {D}}\). For \(\alpha , \beta \ge 0\) the Kaplan class \(K(\alpha ,\beta )\) is the set of all functions \(f\in {\mathcal {H}}_d\) satisfying the condition
for \(0<r<1\) and \(\theta _1<\theta _2<\theta _1+2\pi \) (see [9, pp. 32–33]). Kaplan clases were first defined by Sheil-Small in [10] and the restated version as in the condition (0.1) originally comes from [11]. The class \(K(\alpha ,\beta )\) is called the Kaplan class because using the Kaplan method [8], one can show that a normalized function f analytic in \({\mathbb {D}}\) is close-to-convex of order \(\alpha \ge 0\) if and only if \(f'\in K(\alpha ,\alpha +2)\). Let \({\mathcal {C}}\) be the class of functions in \({\mathcal {S}}\) that are close-to-convex (see [4]). In particular \(f\in {\mathcal {C}}\) if and only if \(f'\in K(1,3)\). The analytical properties and nice geometric interpretation make Kaplan classes find many applications to this day. In [7] Jahangiri and Ponnusamy use Kaplan classes both as a tool in assumptions as well as part of the result. In this article we will need the following theorem by Sheil-Small (see [12, p. 248], in [5] and [6] one can find additional information about polynomials with all zeros of derivative in \({\mathbb {T}}\) in the context of Kaplan classes).
Theorem A
(Sheil-Small) Every polynomial of degree n with all zeros on \({\mathbb {T}}\) belongs to \(K(1,2\pi /\lambda -n+1)\), where \(\lambda \) is the minimal arclength between each pair of zeros.
Define \({\mathbb {D}}(r):=\{z\in {\mathbb {C}}:|z|<r\}\) for \(r>0\),
and
for any \({\mathcal {F}}\subset {\mathcal {S}}\). Bieberbach in 1916 proved the following theorem (see [1]).
Theorem B
(Koebe Quarter Theorem) For every \(f\in {\mathcal {S}}\), \({\mathbb {D}}(1/4)\subset f({\mathbb {D}})\), i.e. \(\mathrm r({\mathcal {S}})=1/4\). Moreover the function
is extremal for this inclusion.
The problem of determining \(r({\mathcal {F}})\) or \(R({\mathcal {F}})\) for any class \({\mathcal {F}}\subset {\mathcal {S}}\) is well known in the literature. Goodman devoted an entire chapter (see [4, pp. 113–120]) for so called Koebe domains. The concept of the Koebe domain for a set \({\mathcal {F}}\) is similar to the concept of \(r({\mathcal {F}})\) (usually they are equivalent). Directly from Theorem B we conclude that \(r({\mathcal {F}})=1/4\) for \({\mathcal {F}}\) being the class of starlike functions as well as the class of close-to-convex functions since the Koebe function is starlike and in the consequences close-to-convex. Hence, the Koebe domain for starlike functions and close-to-convex functions is a disc \({\mathbb {D}}(1/4)\).
Let \(n\in {\mathbb {N}}\). Denote by \({\mathcal {U}}_n\) the class of all polynomials of degree n belonging to \({\mathcal {S}}\). Additionally \({\mathcal {U}}_{n,{\mathbb {R}}}\) be the subclass of \({\mathcal {U}}_n\) containing only the polynomials with real coefficients. An interesting problem of determining \(\mathrm r({\mathcal {U}}_n)\) and \(\mathrm r({\mathcal {U}}_{n,{\mathbb {R}}})\) was stated in [2]. It is easy to show that \(\mathrm r({\mathcal {U}}_{1,{\mathbb {R}}})=\mathrm r({\mathcal {U}}_1)=1\) and \(\mathrm r({\mathcal {U}}_{2,{\mathbb {R}}})=\mathrm r({\mathcal {U}}_2)=1/2\), but cases for \(n\ge 3\) are far from trivial. Dmitrishin et al. [2] proved that
with the extremal function given by the formula
Moreover they proposed a class of functions which can be extremal at the problem of determining \(\mathrm r({\mathcal {U}}_{n,{\mathbb {R}}})\) for \(n\ge 4\).
Let \(n\in {\mathbb {N}}\). Denote by \({\mathcal {C}}_n(X)\) the class of all polynomials of degree n belonging to \({\mathcal {C}}\) with all zeros of derivative on \(X\subset {\mathbb {C}}\setminus {\mathbb {D}}\). Inspired by [2] and [3] we state the problem of determining \(\mathrm r({\mathcal {C}}_n({\mathbb {T}}))\) and \(\mathrm R({\mathcal {C}}_n({\mathbb {T}}))\). The classes \({\mathcal {C}}_n({\mathbb {T}})\) are fairly large, \({\mathcal {C}}_n({\mathbb {T}})\subset {\mathcal {U}}_n\) and \({\mathcal {C}}_n({\mathbb {T}})\not \subset {\mathcal {U}}_{n,{\mathbb {R}}}\). Let us notice that cases \(n=1\) and \(n=2\) are trivial. It is easy to show that \(\mathrm r({\mathcal {C}}_1({\mathbb {T}}))=\mathrm R({\mathcal {C}}_1({\mathbb {T}}))=1\), \(\mathrm r({\mathcal {C}}_2({\mathbb {T}}))=1/2\) and \(\mathrm R({\mathcal {C}}_2({\mathbb {T}}))=3/2\) with the extremal functions the same as in [2] for the problem of determining \(\mathrm r({\mathcal {U}}_1)\) and \(\mathrm r({\mathcal {U}}_2)\). In this article we determine \(\mathrm r({\mathcal {C}}_n({\mathbb {T}}))\) and \(\mathrm R({\mathcal {C}}_n({\mathbb {T}}))\) for \(n= 3\) and \(n=4\).
2 Main results
In this section we solve the problem of determinig \(\mathrm r({\mathcal {C}}_n({\mathbb {T}}))\) and \(\mathrm R({\mathcal {C}}_n({\mathbb {T}}))\) for \(n=3\) and \(n=4\).
Theorem 1.1
The following equalities hold:
and
Moreover the function
is extremal for (1.1) and (1.2).
Proof
Let \(P\in {\mathcal {C}}_3({\mathbb {T}})\). Then by Theorem A,
for certain \(t_1,t_2\in {\mathbb {R}}\) such that \(\pi /2\le |t_1-t_2|\le \pi \). Without loss of generality we can assume that \(t_1:=0\) and \(\pi /2\le t_2\le \pi \). Therefore
Since P is a polynomial, so there exists the holomorphic extension of P on \({\mathbb {T}}\). Setting \(z:=\mathrm {e}^{\mathrm {i}\theta }\) for \(\theta \in [0,2\pi )\) we get
Therefore
and as a consequence
Function g has the minima at 0 and \(t_2\) and the maxima at \(t_2/2\) and \(t_2/2+\pi \). Since
and
so for every \(z\in {\mathbb {T}}\),
which leads to (1.1) and (1.2).
Now consider the function
Then \(f_{|{\mathbb {D}}}\in {\mathcal {C}}_3({\mathbb {T}})\),
and
\(\square \)
Fix \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\). Define
Now we prove the following lemma.
Lemma 1.2
The function h has exactly three zeros, one in each interval \((0,t_2)\), \((t_2,t_3)\) and \((t_3,2\pi )\).
Proof
Since
for \(\theta \in [0,2\pi )\), so \(h(\theta )\ne 0\) if
Therefore \(h(\theta )\ne 0\) if \(\theta \in [0,2\pi )\setminus (\Omega _1\cup \Omega _2\cup \Omega _3\cup \Omega _4)\). For \(\theta \in [0,2\pi )\) we obtain
Let us notice that
and
Assume that \(\theta \in \Omega _1\). Since \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\), so by using (1.4) and (1.5) we get
Therefore
Hence \(h'(\theta )>0\) for \(\theta \in \Omega _1\).
Assume that \(\theta \in \Omega _2\). Since \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\), so by using (1.4) and (1.5) we get
Therefore
Hence \(h'(\theta )<0\) for \(\theta \in \Omega _2\).
Assume that \(\theta \in \Omega _3\). Since \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\), so by using (1.4) and (1.5) we get
Therefore
Hence \(h'(\theta )>0\) for \(\theta \in \Omega _3\).
Assume that \(\theta \in \Omega _4\). Since \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\), so by using (1.4) and (1.5) we get
Therefore
Hence \(h'(\theta )<0\) for \(\theta \in \Omega _4\).
Let us notice that
Since \(t_2\in [2/5\pi ,2/3\pi ]\) and \(t_3\in [2t_2,\pi +t_2/2]\), so \(\Omega _1\subset [0,t_2)\), \(\Omega _3\cup \Omega _4\subset (t_3,2\pi )\) and \(t_2<\min (\Omega _2)<t_3\). Since \(t_2\notin \Omega _1\cup \Omega _2\cup \Omega _3\cup \Omega _4\), so by (1.3) we obtain \(|h_2(\theta )|<3/4\) for \(\theta \in \Omega _1\cup \Omega _2\cup \Omega _3\cup \Omega _4\). Therefore
and
Moreover
Hence and by monotonicity of h on intervals \(\Omega _1\), \(\Omega _2\), \(\Omega _3\) and \(\Omega _4\) we conclude that h has exactly three zeros, one in each interval \((0,t_2)\), \((t_2,t_3)\) and \((t_3,2\pi )\). \(\square \)
Using Lemma 1.2 we prove the following theorem.
Theorem 1.3
The following equalities hold:
and
Moreover the function
is extremal for (1.6) and (1.7).
Proof
Let \(P\in {\mathcal {C}}_4({\mathbb {T}})\). Then by Theorem A,
for certain \(t_1,t_2,t_3\in {\mathbb {R}}\) such that \(2\pi /5\le |t_1-t_2|\le |t_2-t_3|\le 4\pi /5\). Without loss of generality we can assume that \(t_1:=0\), \(2\pi /5\le t_2\le 2\pi /3\) and \(2t_2\le t_3\le t_2/2+\pi \). Therefore P is given by the formula
Since P is a polynomial, so there exists the holomorphic extension of P on \({\mathbb {T}}\). Setting \(z:=\mathrm {e}^{\mathrm {i}\theta }\) for \(\theta \in [0,2\pi )\) we get
Therefore
The function
has exactly three zeros at points 0, \(t_2\) and \(t_3\). Hence by Lemma 1.2 we conclude that the function \(g'\) has exactly six zeros. Therefore g has three minima at points 0, \(t_2\) and \(t_3\) and three maxima one in each interval \((0,t_2)\), \((t_2,t_3)\) and \((t_3,2\pi )\). Moreover
and
Since \(g(t_2)\) is increasing with respect to \(t_2\) and \(t_3\), so
and the equality holds if and only if \(t_2=2/5\pi \) and \(t_3=4/5\pi \). This ends the proof of (1.6).
Now we prove (1.7). Let us notice that for \(z\in {\mathbb {D}}\cup {\mathbb {T}}\),
and the equality holds if and only if \(z=\mathrm {e}^{\frac{7}{5}\pi \mathrm {i}}\), \(t_2=2/5\pi \) and \(t_3=4/5\pi \). This ends the proof of (1.7).
Moreover from the above considerations we conclude that the extremal function for (1.6) and (1.7) is
\(\square \)
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Ignaciuk, S., Parol, M. On the Koebe Quarter Theorem for certain polynomials. Anal.Math.Phys. 11, 68 (2021). https://doi.org/10.1007/s13324-021-00501-8
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DOI: https://doi.org/10.1007/s13324-021-00501-8