A note on the trade-off relationships of steering and Bell inequalities and the maximal mean values of Sliwa inequalities

The detection of quantum correlations and the trade-off relationships of correlation resources in multi-partite systems are vitally important in quantum information theory. We present the trade-off relation of any quantum state with respect to the steering inequality and the trade-off relation of generalized Greenberger-Horne-Zeilinger state with respect to the Mermin inequality. We also provide a quantitative analysis of the allocation of associated resources in multi-partite quantum systems. Computable formula of the maximal violation of the Sliwa inequality for any three-qubit state is presented.