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The arc length of the lemniscate \(\left|w^n\!+\!c\right|=1\). (English) Zbl 1134.31003

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)].

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
26D05 Inequalities for trigonometric functions and polynomials

Citations:

Zbl 0088.25302
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References:

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