Abstract
In this paper, we derive an improved sharp version of a reverse isoperimetric inequality for convex planar curves of Pan and Zhang (Beiträge Algebra Geom 48:303–308, 2007), with a simpler Fourier series proof. Moreover our result also confirm a conjecture by Pan et al. (J Math Inequal (preprint), 2010). Furthermore we also present a stability property of our reverse isoperimetric inequality (near equality implies curve nearly circular).
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References
Pan S.L., Zhang H.: A reverse isoperimetric inequality for convex plane curves. Beiträge Algebra Geom. 48, 303–308 (2007)
Pan, S.L., Tang, X.Y., Wang, X.Y.: A refined reverse isoperimetric inequality in the plane. J. Math. Inequal (2010) (preprint)
Pan S.L., Xu H.P.: Stability of a reverse isoperimetric inequality. J. Math. Anal. Appl. 350, 348–353 (2009)
Groemer, H.: Geometric applications of Fourier series and spherical harmonics, Encyclopedia of Mathematics and its Applications, vol. 61. Cambridge University Press, Cambridge (1996)
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Gao, X. A Note on the Reverse Isoperimetric Inequality. Results. Math. 59, 83–90 (2011). https://doi.org/10.1007/s00025-010-0056-y
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DOI: https://doi.org/10.1007/s00025-010-0056-y