Some Results on a Conjecture of Hardy and Littlewood

Let $m$ and $n$ be positive integers with $m,n\ge 2$. The second Hardy-Littlewood conjecture states that the number of primes in the interval $(m,m+n]$ is always less than or equal to the number of primes in the interval $[2,n]$. Based on new explicit estimates for the prime counting function $\pi (x)$, we give some new ranges in which this conjecture holds.