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We establish two main inequalities; one for the norm of the secondfundamental form and the other for the matrix of the shape operator. Theresults obtained are for cosymplectic manifolds and, for these, we show thatthe contact warped product submanifolds naturally possess a geometric property;namely $\mathcal{D}_1$-minimality which, by means of the Gauss equation, allowsus to obtain an optimal general inequality. For sake of generalization, westate our hypotheses for nearly cosymplectic manifolds, then we obtain them asparticular cases for cosymplectic manifolds. For the other part of the paper,we derived some inequalities and applied them to construct and introduce ashape operator inequality for cosimpleptic manifolds involving the harmonicseries. As further research directions, we have addressed a couple of openproblems arose naturally during this work and which depend on its results.

General inequalities and new shape operator inequality for contact CR-warped product submanifolds in cosymplectic space form