In this paper, we prove that if ${A}$ is a Banach $*$-subalgebra of $B(H)$, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$¦ and $\alpha:T\rightarrow [0,\infty)$ is a integrable function and $(A_t), (B_t)$ are appropriate integrable fields of operators in ${A}$ having the almost synchronous property for the Hadamard product, then $$ ıt_T\!lpha(s)d\mu(s)ıt_T\!lpha(t)\big(A_t\circ B_t\big) d\mu(t) \geq ıt_T\!lpha(t)A_td\mu(t)\circıt_T\!lpha(t)B_td\mu(t). $$ We also introduce a semi-inner product for square integrable fields of operators in a Hilbert space and using it, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product and the trace of operators.