Define a function for by
- Prove that is strictly increasing on the nonnegative real axis.
- If denotes the inverse of prove that is proportional to and find the constant of proportionality.
- Proof. To show is strictly increasing we take the derivative,
Since for all we have that is strictly increasing on the nonnegative real axis
- Proof. If is the inverse of then we know (Theorem 6.7 on page 252 of Apostol)
But, we have defined to the function such that . Therefore,
Therefore, we have that
Taking another derivative of with respect to we then have