Stud. Univ. Babe¸ s-Bolyai Math. 63(2018), No. 2, 175–188 DOI: 10.24193/subbmath.2018.2.02 Integral estimates for a class B n of operators Shah Lubna Wali and Abdul Liman Abstract. Let Pn be the class of polynomials of degree at most n. Rahman in- troduced a class Bn of operators B that map Pn into itself. In this paper, we establish certain integral estimates concerning B-operator, and thereby obtain generalizations as well as improvements of some well known inequalities for poly- nomials. Mathematics Subject Classification (2010): 26D10, 41A17. Keywords: B-Operator, polynomial inequalities, integral estimates. 1. Introduction and Statement of Results Let P n be the class of polynomials P (z) := n  j=0 a j z j of degree at most n with complex coefficients. For P ∈P n , define ‖P ‖ 0 := exp  1 2π 2π  0 log |P (e iθ )|dθ  , ‖P ‖ p :=  1 2π 2π  0 |P (e iθ )| p dθ  1 p 0 <p< ∞, ‖P ‖ ∞ := max |z|=1 |P (z)|. It is known that if P ∈P n , then ‖P ′ ‖ ∞ ≤ n‖P ‖ ∞ (1.1) and for R> 1, ‖P (R.)‖ ∞ ≤ R n ‖P ‖ ∞ . (1.2) Inequality (1.1) is an immediate consequence of a famous result due to Bernstein on the derivative of a trigonometric polynomial (for reference see [6], [14]), where as