In this article we present the polar decomposition of a bounded linear operator defined on an infinite-dimensional complex Hilbert space with respect to a conjugation. As a result, the polar decomposition for C-symmetric, C-skew-symmetric, and C-normal operators are obtained. As an another consequence the Godič –Lucenko theorem is obtained. We give two applications of the polar decomposition theorem. First, we obtain the polar decomposition of bounded antilinear operators and discuss various cases. Second, we derive an inequality for the bounded linear operator based on the Hölder-McCarthy’s inequality and derive an inequality for C-normal operators. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.