Covering Problems for Functions \(n\) -Fold Symmetric and Convex in the Direction of the Real Axis II

Bulletin of the Malaysian Mathematical Sciences Society, Oct 2015

Let \({\mathcal {F}}\) denote the class of all functions univalent in the unit disk \(\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}\) and convex in the direction of the real axis. The paper deals with the subclass \({\mathcal {F}}^{(n)}\) of these functions \(f\) which satisfy the property \(f(\varepsilon z)=\varepsilon f(z)\) for all \(z\in \Delta \), where \(\varepsilon =e^{2\pi i/n}\). The functions of this subclass are called \(n\)-fold symmetric. For \({\mathcal {F}}^{(n)}\), where \(n\) is odd positive integer, the following sets, \(\bigcap _{f\in {\mathcal {F}}^{(n)}} f(\Delta )\)—the Koebe set and \(\bigcup _{f\in {\mathcal {F}}^{(n)}} f(\Delta )\)—the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for \({\mathcal {F}}^{(n)}\).

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Covering Problems for Functions \(n\) -Fold Symmetric and Convex in the Direction of the Real Axis II

Bull. Malays. Math. Sci. Soc. Covering Problems for Functions n-Fold Symmetric and Convex in the Direction of the Real Axis II Leopold Koczan 0 Paweł Zaprawa 0 Mathematics Subject Classification 0 0 Primary 30C45 Let F denote the class of all functions univalent in the unit disk ≡ {ζ ∈ C : |ζ | < 1} and convex in the direction of the real axis. The paper deals with the subclass F (n) of these functions f which satisfy the property f (εz) = ε f (z) for all z ∈ , where ε = e2πi/n . The functions of this subclass are called n-fold symmetric. For F (n), where n is odd positive integer, the following sets, f ∈F (n) f ( )-the Koebe set and f ∈F (n) f ( )-the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for F (n). Communicated by Saminathan Ponusammy. Covering domain; Koebe domain; Convexity in one direction; n-fold symmetry 1 Introduction Let F denote the class of all functions f which are univalent in ≡ {ζ ∈ C : |ζ | < 1}, convex in the direction of the real axis and normalized by f (0) = f (0) − 1 = 0. Recall that an analytic function f is said to be convex in the direction of the real axis if the intersection of f ( ) with each horizontal line is either a connected set or empty. For a given subclass A of F , the following sets: f ∈A f ( ) and f ∈A f ( ) are called the Koebe set for A and the covering set for A. We denote them by K A and L A, respectively. The radius of the largest disk with center at the origin contained in K A is called the Koebe constant for A. Analogously, the radius of the smallest disk with center at the origin that contains L A is called the covering constant for A. In the class F , we consider functions which satisfy the property of n-fold symmetry: f (εz) = ε f (z) for all z ∈ , where ε = e2πi/n . The subclass of F consisting of n-fold symmetric functions is denoted by F (n). By the definition, for every f ∈ F (n) a set f ( ) is n-fold symmetric, which means that f ( ) = ε f ( ). In other words, f ( ) may be obtained as the union of rotations about a multiple of 2π/n from a set f ( ) ∩ {w : arg w ∈ [0, 2π/n]}. From this reason, the following notation is useful: 2π n 0 = , j = ε j 0 , j = 1, 2, . . . , n − 1 ∗ = . The main aim of the paper is to find the Koebe set and the covering set for the class F (n) when n is an odd positive integer. Similar problems in related classes were discussed, for instance, in [1,2,5] and in the papers of the authors [3,4]. At the beginning, let us consider the general properties of the Koebe sets and the covering sets for F (n). In [4], we proved that Theorem 1 The sets K (n) and LF(n) , for n ∈ N, are symmetric with respect to both F axes of the coordinate system. Theorem 2 The sets KF(n) and LF(n) , for n ∈ N, are n-fold symmetric. To prove both the theorems, it is enough to consider functions and and Obviously, g(z) = f (z) h(z) = − f (−z). f ∈ F (n) ⇔ g, h ∈ F (n). ( 1 ) ( 2 ) ( 3 ) Moreover, if D = f ( ) then g( ) = D, h( ) = −D. Taking ( 3 ) into account, it is clear that the coordinate axes are the lines of symmetry for both the sets KF(n) and LF(n) for all n ∈ N. Furthermore, Lemma 1 Each straight line ε j/4 · {ζ = t , t ∈ R}, j = 0, 1, . . . , 4n − 1 is the line of symmetry of KF(n) and LF(n) for every positive odd integer n. Proof Let n be a positive odd integer and let D be one of the two sets: K (n) or L (n) . F F Since D is symmetric with respect to the real axis and the positive real half-axis contains one side of the sector 0, each rotation of the real axis about a multiple of 2π/n is the line of symmetry of D. Because of the equality {ζ = t , t ∈ R} · ε1/2 = {ζ = t , t ∈ R} · ε(n+1)/2 , our claim is true for all even j , j = 0, 1, . . . , 4n − 1. Let n = 4k + 1, k ≥ 1. The bisector of (n−1)/4 divides this sector into two subsectors: {w : arg w ∈ [π/2 − π/2n, π/2 + π/2n]} and {w : arg w ∈ [π/2 + π/2n, π/2 + 3π/2n]}. Hence the imaginary axis is the bisector of the former. For this reason, each rotation of the imaginary axis about a multiple of 2π/n is the line of symmetry of D. Moreover, {ζ = i t , t ∈ R} · ε1/2 = {ζ = i t , t ∈ R} · ε(n+1)/2. Hence our claim is valid also for all odd j , j = 0, 1, . . . , 4n − 1. If n = 4k + 3, k ≥ 0, then the bisector of (n−3)/4 divides this sector into two subsectors: {w : arg w ∈ [π/2 − 3π/2n, π/2 − π/2n]} and {w : arg w ∈ [π/2 − π/2n, π/2 + π/2n]}. The imaginary axis is the bisector of the latter. Similar argument to the one for n = 4k + 1 completes the proof for this choice of n. Theorem 3 The sets KF(n) and LF(n) , for positive odd integers n, are 2n-fold symmetric. pPoroinotfbLeleotnDginbge otonethoef bthoeuntwdaorsyeotsf: DKFsu(nc) horthLaFta(nr)g. ϕL0et∈w[00,=π|/w20n|]e.iϕ0 be an arbitrary It is sufficient to apply Lemma 1. Firstly, the symmetric point to w0 with respect to the straight line ε1/4 · {ζ = t , t ∈ R} is w1 = |w0|ei(π/n−ϕ0). Secondly, the symmetric point to w0 with respect to the real axis is w2 = |w0|e−i (...truncated)


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Leopold Koczan, Paweł Zaprawa. Covering Problems for Functions \(n\) -Fold Symmetric and Convex in the Direction of the Real Axis II, Bulletin of the Malaysian Mathematical Sciences Society, 2015, pp. 1637-1655, Volume 38, Issue 4, DOI: 10.1007/s40840-014-0107-8