Given an integer $n>1$ with two factorization $n=\prod\limits_{i=1}^rp_i$ and $n=\prod\limits_{i=1}^tq_i$ where the $p_i$'s are primes(not necessarily distinct) and the $q_i$'s are arbitrary integers $>1$. If $\alpha\ge1$, then we can prove that $$\sum_{i=1}^rp_i^\alpha\le\sum_{i=1}^tq_i^\alpha$$
The problem 4.17.b in Apostol's analytic number theory says:
Obtain a corresponding inequality relating these sums if $0\le\alpha<1$.
Let $f(\alpha)=\sum\limits_{i=1}^rp_i^\alpha-\sum\limits_{i=1}^tq_i^\alpha$. Clearly $r\ge t$. If $r=t$, then $f(\alpha)=0$. Also $r>t$ gives $f(0)>0$ and $f(1)<0$. This means there is a unique $\alpha_0$ such that $f(\alpha)>0$ for $\alpha<\alpha_0$ and $f(\alpha)<0$ if $\alpha>\alpha_0$ and $f(\alpha_0)=0$.
Now my question is if this is the answer that is required by the author or yet we should improve the solution to for example find a formula for $\alpha_0$ and other, as the author says "Obtain a corresponding inequality"? Thanks.