Geometric and arithmetic mean of real numbers in (0,1]

Question

Assume two sequences $x_1$ and $x_2$, each of $n$ numbers is in $(0,1]$. Now take the geometric($g$) and arithmetic($a$) mean of $x_1$ and $x_2$.

Is it possible that if $a_1 > a_2$ that $g_1 < g_2$ ??

Thanks

Answer 1

Let's think about what we would need in order to have such sequences. We know that $g_1\leq a_1$ and $g_2\leq a_2$, with equality iff the sequences are constant. So in order to have $a_1>a_2$ and $g_1<g_2$, we would need to have $$g_1<g_2\leq a_2<a_1.$$ If we first pick $x_1$ to be any nonconstant sequence, that will guarantee $g_1<a_1$. We then just need to pick $x_2$ such that both $g_2$ and $a_2$ are between $g_1$ and $a_1$. We can do this by simply picking $x_2$ such that all of its terms are between $g_1$ and $a_1$, since $g_2$ and $a_2$ must both be between the smallest term of $x_2$ and the largest term.

So to find an example, let $x_1$ be any nonconstant sequence, with arithmetic mean $a_1$ and geometric mean $g_1$, and let $x_2$ be any sequence all of whose terms are between $g_1$ and $a_1$.