Let t n ( x) be any real trigonometric polynomial of degree n such that ∥ t n ∥ L ∞ ⩽ 1. Here we are concerned with obtaining the best possible upper estimate of ∝ 0 2π(t n (k)(x)) 2r + 2dx ∝ 0 2π(t n (k)(x)) 2rdx where r ⩾ 0 (integer). As a special case we obtain (1.2). Let ∥ t n ∥ L 2 and ∥ t n ( r) ∥ L 2 be given where t n is any real trigonometric polynomial. In Theorem 2 we shall obtain the estimate of ∥ t n ( j) ∥ L 2 in terms of ∥ t n ∥ L 2 and ∥ t n ( r) ∥ L 2 . The results are again the best possible.