I need help in proving this inequality->
Assuming q to be any positive integer prove that ( 1+ log(q) ) d(q) $\phi^{-2} (q) $ $\leq$ $ q^{-5/3}$ . where d(q) means number of divsor of q and $\phi(q) $ = number of integer less than q which are co-prime to q.
I am unable to deduce (39) on page 6 of research paper and this inequality will be used there. The motive of asking the question and putting the bounty is to prove (39) .I hope that I am not wrong in thinking that the above mentioned inequality will lead to proving (39) ?
The inequality asked for is equivalent to $\phi(n) \geq cn^{5/6}\sqrt{d(n)\log n}$ and this follows from known estimates on $\phi(n)$ and $d(n)$. For example, $\phi(n)$ is asymptotically at least $\dfrac{n}{\log \log n}$, whereas the divisor function is (asymptotically) upper-bounded by $n^{\epsilon}$ for every $\epsilon>0$. The required bound also follows from $\varphi(n)d(n) \geq cn^2$.
For proofs of the three bounds stated above, see: Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$? and https://terrytao.wordpress.com/2008/09/23/the-divisor-bound/ and Lower bound of Euler phi function times sum of divisors