Some comparison theorems in Finsler–Hadamard manifolds

We give upper and lower bounds for the ratio of the volume of metric ball to the area of metric sphere in Finsler–Hadamard manifolds with the pinched S-curvature. We apply these estimates to find the limit at infinity for this ratio. The estimates derived are the generalization of the result well known in Riemannian geometry. We also estimate the volume growth entropy for the balls in these manifolds.