An Inequality with Determinants

Statement

Inequality With Determinants

Solution

It could be seen that the determinant $\Delta\;$ in the left-hand side of the required inequality equals $\Delta = abcd+acd+abd+abc+bcd\;$ which is evaluated via the AM-GM inequality:

$\displaystyle\begin{align} \Delta &= abcd+acd+abd+abc+bcd\\ &=\frac{1}{2}abcd+\frac{1}{2}abcd+acd+abd+abc+bcd\\ &\ge 6[\frac{1}{4}(abcd)^2(acd)(abd)(abc)(bcd)]^{1/6}\\ &=6\left(\frac{1}{4}\right)^{1/6}(abcd)^{5/6}\\ &=3\cdot 4^{1/3}(abcd)^{5/6}. \end{align}$

The equality holds when $\displaystyle\frac{1}{2}abcd=abc=bcd=\ldots,\;$ i.e., when $a=b=c=d=2.$

Acknowledgment

The inequality from the book Math Accent has been posted at the CutTheKnotMath facebook page by Dan Sitaru along with a solution by Ravi Prakash.

 

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