It it well-known that the prime-counting function $\pi(x)$ satisfies the prime number theorem and that were in the literature two related conjectures to this arithmetic function, these are: the Riemann hypothesis, I mean in this ocassion the equivalence due to von Koch ([1]), and the second Hardy–Littlewood conjecture (see the corresponding Wikipedia or [2])
I would like to know if it is in the literature a continuous function $f(t)$ on some interval $(L,\infty)$, for some constant $L>0$, satisfying the following conditions $$\pi(x+y)\leq \pi(x)+f(y)\tag{C1}$$ for all positive real numbers $x$ and $y$ such that $x\leq y$ where $L<x$. And also that satisfies $$|f(x)-\pi(x)|=O\left(\sqrt{x}\log x\right)\tag{C2}$$ as $x\to\infty$, and also satisfying the asymptotic equivalence $$\pi(x)\sim f(x)\tag{C3}$$ as $x\to\infty$.
Question*. Do you know any continuous function $f(x)$, on our interval $(L,\infty)$ being $L$ a positive constant, satisfying previous conditions $\text{(C1)}$, $\text{(C2)}$ and $\text{(C3)}$? Alternatively, is it possible to prove the existence of such continuous function? If it is in the literature please refer it answering as a reference request and I try to search and read the statement from the literature. Many thanks.
If isn't in the literature I'm asking about what work can be done to elucidate the problem in my Question, that is at least feedback about the existence of such function: I don't know if it's easy to prove that such a continuous function (and its corresponding interval $(L,\infty)$) exists. If you can to prove the existence and you want to add more remarks about the function(s) $f(x)$ feel free to do it.
References:
[1] Helge von Koch , Sur la distribution des nombres premiers, Acta Mathematica volume 24, Article number: 159 (1901).
[2] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’ III: On the expression of a number as a sum of primes, Acta Math. (44): 1–70 (1923).
*An hour ago I've asked a similar question without a good mathematical content (thanks for the user in comments), I've updated it in a new post.