On the convexity of lemniscates

Abstract.Sharp bounds are derived for the curvature of level curves of analytic functions in the complex plane whose logarithmic derivative has the representation c/(w-g(w)), where g(w) is analytic for |w|>a and \g(w)\fía, c real. These results are applied in particular to lemniscates and sharpened for the level curves of lacunary polynomials. Extensions to the level curves of Green's function and rational functions are indicated.1. Introduction. In this paper sharp bounds are derived for the curvature of various classes of level curves of analytic functions in the complex plane. In § §2 and 3 a considerably simplified proof is given for the estimates of the curvature of lemniscates, the level curves of Green's function and rational functions derived by the author in previous publications [4], [5]. In §4 new estimates for the curvature of the level lines of lacunary polynomials and their orthogonal trajectories are derived. The methods used combine the author's previous calculations with an application of a new coincidence lemma due to Rubinstein and Walsh [3]. These estimates are sharp. Applications to the special lemniscate of radius one are indicated in the final section.

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On the convexity of lemniscates

Abstract: Let Li denote the lemniscate | n?-i(z-f»)| =L Assume the poles f, are inscribed in the disc \z\ Sa. Let Zo = w~'/ .?_if" Conditions for the convexity of ¿i are established in terms of a and Zo. Sharp bounds are derived for real f».

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Lemniscates and equipotentials

Lemniscates and equipotentials

The curvature of level curves

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