Let P={P1,P2,…,Pn} be a set of n points in Rd. For every 1≤i≤n, define the star rooted at Pi as the union of all straight line segments joining Pi to all the other points in the set P. A Steiner star is the union of all straight line segments connecting some point in Rd to each point of P. The length of a star is defined as the total Euclidean length of its edges. We consider the problem of estimating the supremum of the ratio between the rooted star of minimal length and the Steiner star of minimal length, taken over all n point configurations in Rd. This is known as the Steiner ratio in Rd. It is conjectured that this ratio is 4/π when d=2 and 4/3 when d=3. Fekete and Meijer proved that for every d, this ratio is bounded from above by 2. Very recently, Dumitrescu, Tóth and Xu proved better upper bounds: 1.3631 for d=2 and 1.3833 for d=3. By a refinement of their approach we further improve these bounds to 1.3546 in the plane and 1.3801 in 3-space. These estimates yield improved upper bounds on the maximum ratio between the minimum star and the maximum matching in two and three dimensions.