1. ## Lagrange's Theorem(?)

Given two positive real numbers $a,b (a, prove that

$\displaystyle \sqrt{ab}<[(a-b)/(\ln(a)-\ln(b))]<(a+b)/2$

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 $\displaystyle \ln(x)$

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.

Any suggestions??

2. ## maybe you can consider this

let $\displaystyle f(x)=\frac{x-a}{lnx-lna}-\sqrt{ax}$, then see $\displaystyle f'(x)$ and $\displaystyle f''(x)$.

another possible way: through Taylor's Mean Value Theorem. But I have not tried it.