An Extension of the Turan Inequality in L_p-space for 0

1. Introduction Denote by H_n. the set of n-th algebraic polynomials all whose roots lie inside -1,1],R_n is the class of algebraic polynomials of degree n which have only real roots, and R_n~* is the class of trigonometric polynomials of degree n having only real zeros. Let C be a positive absolute constant, which may be different in different places.