I was thinking what should be a good way (a reasoning), if there is one, to combine the convexity of the function $$-\operatorname{Li}(x)=-\int_2^x\frac{dt}{\log(t)}$$ on the interval $[2,x]$ with the prime-counting function $\pi(x)$ in the context of the Second Hardy–Littlewood conjecture to state some claims or an interesting open question.
Attempt. My idea is to combine with Jensen's inequality, see this Wikipedia. I say maybe for $\varphi(x)=-\int_2^x\frac{dt}{\log(t)}$ and taking $f(x)=\pi(x)$ with the purpose to write an interesting fact or open question in the context the Second Hardy–Littlewood conjecture. I don't know if such idea was in the literature or well if this idea is feasible.
Question. What are your ideas to state an interesting claim or an interesting open question, if is it feasible, combining the convexity of $-\int_2^x\frac{dt}{\log(t)}$ and propositions/questions or heuristics involving the prime-counting function $\pi(x)$ in the context of Second Hardy–Littlewood conjecture? Many thanks.
If you think that combine with Jensen's inequality is a good way feel free to do it, obviously my reasoning is unfinished. If my question was in the literature please refers those papers and I try find and read those papers.
References:
[1] Arguments for the second Hardy–Littlewood conjecture being false?, from MathOverflow.
I believe that the original conjecture due to Hardy and Littlewood was in next reference:
[2] Hardy and Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, (1923).
The problem here and in the ref [1] you mentioned is that $Li(x)$ is very smooth increasing function with $\frac{1}{\ln x}$ is decreasing, so to put it in words , the function $Li(x)$ is from the heaven of functions on the other side $\pi(x)$ is not-smooth nor decreasing or increasing and the most difficulty with $\pi(x)$ is that for infinitely many times $\pi(x) >> Li(x)$ and for infinitely many times $\pi(x) << Li(x)$.
Also its hard work and if its going to tell you something, i bet its about $Li(x)$ (which we do know a lot about) and not $\pi(x)$ (which we don't).