An isoperimetric inequality in the universal cover of the punctured plane Original Research Article

We find the largest image (approximately 1.71579) for which any simple closed path image in the universal cover image of image, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies image, where image is the length of image and image is the area enclosed by image. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in image.