Wang, Chunjie; Peng, Lizhong The arc length of the lemniscate \(\left|w^n\!+\!c\right|=1\). (English) Zbl 1134.31003 Rocky Mt. J. Math. 36, No. 1, 337-347 (2006). Summary: Let \(s_n(c)\) be the arc length of the lemniscate \(|w^n + c| = 1\), \(c\in [0,\infty)\). We obtain some properties of the function \(s_n(c)\). In particular, we prove that \(s_n(c)\leq s_n(1)\), \(c\in [0,\infty)\). We also give a sharp bound for \(s_n(1)-2n\), that is, \(4 \log 2 <s_n(1) - 2n \leq 2(\pi - 1)\).We discuss the relations between these estimates a problem posed by P. Erdös, F. Herzog, and C. Piranian [J. Anal. Math. 6, 125–148 (1958; Zbl 0088.25302)]. Cited in 1 ReviewCited in 1 Document MSC: 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 26D05 Inequalities for trigonometric functions and polynomials Keywords:arc length; lemniscate Citations:Zbl 0088.25302 PDFBibTeX XMLCite \textit{C. Wang} and \textit{L. Peng}, Rocky Mt. J. Math. 36, No. 1, 337--347 (2006; Zbl 1134.31003) Full Text: DOI Euclid References: [1] P. Borwein, The arc length of the lemniscate \(\|p(z)|=1\$, Proc. Amer. Math. Soc. 123 (1995), 797-799. JSTOR:\) · Zbl 0821.31002 [2] P. Erdös, Extremal problems on polynomials , in Approximation theory II, Academic Press, New York, 1976, pp. 347-355. [3] P. Erdös, F. Herzog and G. Piranian, Metric properties of polynomials , J. Analyse Math. 6 (1958), 125-148. · Zbl 0088.25302 [4] A. Eremenko and W. Hayman, On the length of lemniscates , Michigan Math. J. 46 (1999), 409-415. · Zbl 0958.30002 [5] G.H. Hardy, J. E. Littlewood and G. Pólya, Inequalities , Cambridge Univ. Press, Cambridge, 1952. [6] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces , Springer-Verlag, New York, 2000. · Zbl 0955.32003 [7] E. Hille, Analytic function theory , Vol. II, Ginn and Co., Boston, 1962. · Zbl 0102.29401 [8] S. Lang, Complex analysis , 2 -1985. [9] Ch. Pommerenke, On some problems of Erdös, Herzog and Piranian , Michigan Math. J. 6 (1959), 221-225. · Zbl 0090.01601 [10] ——–, On some metric properties of polynomials with real zeros , Michigan Math. J. 6 (1959), 377-384. · Zbl 0122.25202 [11] ——–, On some metric properties of polynomials II, Michigan Math. J. 8 (1961), 49-54. [12] ——–, On metric properties of complex polynomials , Michigan Math. J. 8 (1961), 97-115. · Zbl 0100.25503 [13] Chunjie Wang, The arc length of the lemniscate \(\left|w^2+c\right|=1\), Acta Math. Sci. 18 (1998), 297-301 (in Chinese). · Zbl 0926.31001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.