It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem of convergence and control the occurrence of real poles.In this paper,we establish a family of linear Hermite barycentric rational interpolants r that has no real poles on any interval and in the case k =0,1,2,the function r(k)(x) converges to f(k)(x) at the rate of O(h3d+3-k) as h → 0 on any real interpolation interval,regardless of the distribution of the interpolation points.Also,the function r(x) is linear in data.