Summary: Let \(P: X\to Y\) be a homogeneous polynomial of degree \(\leq m\) between the normed spaces \(X\) and \(Y\). In 1997 L. A. Harris proved that the Fréchet derivative of \(P\) satisfies the Markov inequality \(\|\widehat D^k P\|\leq c_{m,k}\| P\|\), where the best constant \(c_{m,k}\) can be obtained as a solution of an external problem for polynomials on the real line. He also gave upper and lower estimates to \(c_{m,k}\) and computed exact values up to \(m= 20\). Here we obtain improved estimates and thus we find that the exact order of magnitude of \(c_{m,k}\) is the \((m\log m)^k\) order of the upper estimate of Harris for at least \(k= 1,2\). Our method relies on the technique of potential theory with external fields, what we apply with varying weights and at a border-case situation.