  Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the
number of preimages of Euler's totient function $\phi$ and the sum-of-divisors
functions $\sigma$. In this paper, we will extend the upper bound to the number
of preimages of iterates of $\phi$ and $\sigma$. Using these new asymptotic
upper bounds, a conjecuture in Konick and Katai's paper, "On the uniform
distribution of certain sequences involving the Euler totient function and the
sum of divisors function" is now proven and many corollaries follow from their
proven conjecture.
