Mahler's inequality explained

In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means:

\prod ^{n} _{k =1} (x _{k} + y ^{1 /n} _{k)} \ge \prod ^{n} _{k =1} x ^{1 /n} _{k} + \prod ^{n} _{k =1} y ^{1 /n} _{k}

when x_k, y_k > 0 for all k.

Proof

By the inequality of arithmetic and geometric means, we have:

\prod ^{n} _{k =1} \left({x _{k} \over x _{k} + y ^{1 /n} _{k}\right)} \le { 1 \over n} \sum ^{n} _{k =1} {x _{k} \over x _{k} + y _{k},}

and

\prod ^{n} _{k =1} \left({y _{k} \over x _{k} + y ^{1 /n} _{k}\right)} \le { 1 \over n} \sum ^{n} _{k =1} {y _{k} \over x _{k} + y _{k}.}

Hence,

\prod ^{n} _{k =1} \left({x _{k} \over x _{k} + y ^{1 /n} _{k}\right)} + \prod ^{n} _{k =1} \left({y _{k} \over x _{k} + y ^{1 /n} _{k}\right)} \le { 1 \over n} n = 1 .

Clearing denominators then gives the desired result.

See also

• Minkowski inequality

References

• Minkowski inequality in the Encyclopedia of Mathematics