Numerical Radius Inequalities for Sums and Products of Operators

A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.