L^q inequalities for the sth derivative of a polynomial

Let f (z) be an analytic function on the unit disk {z ɛ c, │z│≤ 1}, for each q > 0, the ││f ││ q is defined as follows││f ││q: ={ 1/2π ∫^2π _ 0 │∫ (e^ iө) │ q dө } ^ 1/q, 0 0,ip'i_q≤ n/ ik + ziq ipiq .in this paper, we shall present an interesting generalization and refinement of this result which include some previous results.