On the Łojasiewicz exponent of the gradient of a holomorphic function

The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality ${\displaystyle |\nabla h(x,y)|\ge c{\left|\begin{array}{cc}x& y\end{array}\right|}^{\lambda }}$ holds near ${\displaystyle 0\in C^2}$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.