A new proof and extension of the recent result of Iwasaki et al. (Systems Control Lett. 2005; 54(7):681-691) concerning equivalence of frequency domain inequality on finite frequency range and constrained dissipativity property for linear systems is given. The results of this paper employ an extended version of S-procedure based on duality in the space of positive definite matrices and introduction of matrix Lagrange multipliers, proposed by Yakubovich in 1992.