let , then see and .
another possible way: through Taylor's Mean Value Theorem. But I have not tried it.
Given two positive real numbers , prove that
One thing that strikes me first is that the middle term is actually the point in (a,b) which satisfies Lagrange's Mean Value Theorem for the function
But I can't think of any way of proving that the point lies between the geometric mean and the arithmetic mean of the numbers a,b.