On Banach-Mazur distance between planar convex bodies

Upper estimates of the diameter and the radius of the family of all planar convex bodies with respect to the Banach-Mazur distance are obtained. Namely, it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx 2.614$,whichimprovesthepreviouslyknownboundof$3$, and that the radius does not exceed $\frac{117}{70}\approx 1.671$.