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It is well known Cauchy's inequality is implied by Lagarange's Lagrange's identity. Bohr's inequality $|a -b|^2 \le p|a|^2 +q|b|^2$, where $\frac{1}{p}+\frac{1}{q}=1$, is implied by $|a -b|^2 +|\sqrt{p/q}a+\sqrt{q/p}b|^2= p|a|^2 +q|b|^2$. L.K Hua's determinant inequality $det(I-A^*A)\cdot \det(I-A^*A)\cdot \det(I-B^*B)\le |det(I-A^*B)|^2$ \det(I-A^*B)|^2$ is implied by Hua's matrix equality $(I-B^*B)-(I-B^*A)(I-A^*A)^{-1}(I-A^B)=-(A-B)^(I-AA^*)(A-B)$. (I-B^*B)-(I-B^*A)(I-A^*A)^{-1}(I-A^*B)=-(A-B)^*(I-AA^*)(A-B)$. What other examples can be found?
It is well known Cauchy's inequality is implied by Lagarange's identity. Bohr's inequality $|a -b|^2 \le p|a|^2 +q|b|^2$, where $\frac{1}{p}+\frac{1}{q}=1$, is implied by $|a -b|^2 +|\sqrt{p/q}a+\sqrt{q/p}b|^2= p|a|^2 +q|b|^2$. L.K Hua's determinant inequality $det(I-A^*A)\cdot det(I-B^*B)\le |det(I-A^*B)|^2$ is implied by Hua's matrix equality $(I-B^*B)-(I-B^*A)(I-A^*A)^{-1}(I-A^B)=-(A-B)^(I-AA^*)(A-B)$. What other examples can be found?