Abstract: Let H_n be the set of real algebraic polynomials of degree n, whose zeros all lie in the interval [-1,1]. The well known Turán type inequalities tell us that for f(x)∈H_n, it holds ‖f′‖≥C√n‖f‖. This note deals with the weighted Turán type inequalities with the weights having inner singularities under L^p norm for 0<p≤∞. Our results essentially extend the result of Wang and Zhou (2002), and the method used in this paper is simpler and more direct than that of Wang and Zhou (2002). The results and methods have their own values in approximation theory and computation.

Key words: Turán type inequality, Doubling weights, A^* weights

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