Is it possible to show that, $$\int_{-1/2}^{1/2}\frac{\sin^4(2^n\pi \,f)}{\lvert2\sin(\pi \,f)\rvert^k} \leq \int_{-1/2}^{1/2}\frac{\sin^4(2^{n+1}\pi \,f)}{\lvert2\sin(\pi \,f)\rvert^k}$$ for $1<k<4$?
Since $\sin^4(2^{n+1}\pi \,f) = \sin^4(2(2^{n})\pi \,f) = 2^4\sin^4(2^{n}\pi \,f)\cos^4(2^{n}\pi \,f)$, we have
$$\int_{-1/2}^{1/2}\frac{2^4\sin^4(2^{n}\pi \,f)\cos^4(2^{n}\pi \,f)}{\lvert2\sin(\pi \,f)\rvert^k}$$
Which looks a bit more like the LHS of the first inequality, but I'm not sure how to proceed from here. Any help?
The problem I'm having is that neither expression has a closed from solution (that I'm aware of).