As has already been pointed out, Montgomery and Vaughn have proven that $$\pi(x+y) \leq \pi(x)+ 2 \frac{y}{\log(y)} $$ from which, with some care, one can derive that: $\pi(x+y) \leq \pi(x)+ 2\pi(y)$ (this is worked out in the original paper of Montgomery and Vaughn).

The natural question is thus to further refine the constant $2$. There seems to be no unrestricted result on this problem, however Friedlander and Iwaniec (in their recent book on sieve theory) have shown that

$$\pi(x+y) \leq \pi(x) + (2-\delta) \frac{y} {\log(y)} $$

holds for $x^{\theta} < y < x$ where $\delta:= \delta(\theta)$ is a function of $\theta$. In other words, one can improve the constant $2$ as long as $y$ isn't too small compared to $x$.

Recently Bourgain and Garaev have refined this result to give the quantitative relationship of $\delta \sim \theta^2$.

As has already been pointed out, Montgomery and Vaughan have proven that $$\pi(x+y) \leq \pi(x)+ 2 \frac{y}{\log(y)} $$ from which, with some care, one can derive that: $\pi(x+y) \leq \pi(x)+ 2\pi(y)$ (this is worked out in the original paper of Montgomery and Vaughan).

The natural question is thus to further refine the constant $2$. There seems to be no unrestricted result on this problem, however Friedlander and Iwaniec (in their recent book on sieve theory) have shown that

$$\pi(x+y) \leq \pi(x) + (2-\delta) \frac{y} {\log(y)} $$

holds for $x^{\theta} < y < x$ where $\delta:= \delta(\theta)$ is a function of $\theta$. In other words, one can improve the constant $2$ as long as $y$ isn't too small compared to $x$.

Recently Bourgain and Garaev have refined this result to give the quantitative relationship of $\delta \sim \theta^2$.