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In this paper, we prove an upper bound for the first Robin eigenvalue of the$p$-Laplacian with a positive boundary parameter and a quantitative version ofthe reverse Faber-Krahn type inequality for the first Robin eigenvalue of the$p$-Laplacian with negative boundary parameter, among convex sets withprescribed perimeter.  The proofs are based on a comparison argument obtained by means of innersets, introduced by Payne, Weimberger and Polya.

Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter