The germ of this post arises when I was trying to think what terms of the so-called exact form of the prime-counting function satisfy an inequality of the form $\text{term}(x+y)\leq \text{term}(x)+\text{term}(y)$, or of the form $\text{term}(x+y)\leq \text{term}(x)+\text{term}(y)$.

Is in the literature several problems related to the prime-counting function $\pi(x)$, one of the most famous is the Riemann hypothesis. Riemann provide us a formula, called the exact form, that you can see in the section Exact form of the Wikipedia Prime counting function. But is in the literature other unsolved problem, a less famous problem the Second Hardy–Littlewood conjecture, see the corresponding Wikipedia.

My belief is the following conjecture should be easy to get, since I believe that the mistery of these unsolved problems does not lie in the term that I evoke in the following conjecture, any case I believe that this question is interesting for this site, and I'm curious to know how you analyze the inequality.

Conjecture. For real numbers $x\geq 2$ and $y\geq 2$ the following inequality holds

$$\begin{multline} \frac{1}{\pi}\left(\arctan\left(\frac{\pi}{\log x}\right)+\arctan\left(\frac{\pi}{\log y}\right)-\arctan\left(\frac{\pi}{\log (x+y)}\right)\right) \\ \leq\frac{1}{\log x}+\frac{1}{\log y}-\frac{1}{\log (x+y)}. \end{multline}$$

Question. Can you prove previous conjecture? Many thanks.

I know methods to solve inequalities more easy than this.