In a Minkowski inner product space \((M, [\cdot,\cdot]_J)\), for timelike \(x\) and arbitary \(y\), we have the reverse Schwarz inequality: \[ |[x,y]_J|^2\geq [x,x]_J [y,y]_J. \] In this paper, the authors generalize it to several types of inequalities. The first two are Theorem 3.1 (reverse determinant Hadamard theorem). Let \(x_1, \dots, x_n\in M\) and \(x_1\) be timelike. (i) If \(x_1,\ldots, x_n\) are linearly independent, then \[ \det G_J(x_1,\dots, x_n)<0 \] where \(G_j=([x_i,j_j]_J)_{i,j=1}^n\) is the \(J\)-Gramian matrix. (ii) If furthermore \(x_1\) is \(J\)-orthogonal to \(x_2,\dots,x_n\) then \[ \begin{multlined} [x_1,x_1]_J[x_2,x_2]_J\cdots[x_n,x_n]_J \leq \det G_j(x_1,\dots,x_k) \det G_j(x+1,\dots,x_n)\leq \det G_J(x_1,\dots,x_n)\end{multlined} \] for \(k=1,\dots,n-1\). Theorem 3.2 (reversed mixed Schwarz inequality). Let \(T\) be a \(J\)-contraction and \(\alpha\in [0,1]\). Then \[ |[Tx,y]_J|^2 \geq [|T|_J^{2\alpha}x,x]{}_J[|T^{|*|}|_J^{2(1-\alpha)}y,y]{}_J \] for any \(x,y\) with \(x\) or \(y\) timelike. Here \(T^{|*|}_J\) is the \(J\)-adjoint of \(T\). From these two inequalities, they further deduce a reverse Heinz-Kato-Furuta inequality Theorem 4.1 Let \(A,B\) be \(J\)-Hermitian and \(\alpha,\beta\in [0,1]\) such that \(\alpha+\beta\geq 1\). If \(T\) satisfies \(I_n\geq^J A^2 \geq^J T^{|*|}T\) and \(I_n\geq^J B^2 \geq^J TT^{|*|}\) , then \[ \left|[T|T|_J^{\alpha+\beta-1}x,y]_J\right|\geq \|A^\alpha x\|_J\|B^\beta y\|_J \] for any timelike \(x\) and \(y\). A much generalised version of the reverse Heinz-Kato-Furuta inequality is proved in Theorem 5.1, and then two some-what less generalized versions are given in Theorem 5.2 and Theorem 5.3. A Bernstein type inequality is proved in Theorem 6.1. Finally, a sharpened reverse Heinz-Kato-Furuta inequality is presented in Theorem 7.1.