On an isoperimetric problem on the Lobachevsky hyperbolic plane with left-invariant Finsler structure

We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the left-invariant volume form. The optimal isoperimetric loops are found in terms of convex trigonometry functions. We also propose a generalization of the isoperimetric inequality in parametric form.