I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function. 

>**Result 1.** For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for all $x\ge x^k_0$ the following inequality holds, $$\pi(kx)+\pi(x)>2\pi\left(\left(\dfrac{k+1}{2}\right)x\right)$$
>**Result 2.** For all real $k\ge 2$ and $\varepsilon>0$ there exists a prime between $((1+\varepsilon)n+1)^k$ and $n^k$ for all sufficiently large $n$.

I have searched the internet for references on these results. But the closest I found via searching the first was something called the Second Hardy-Littlewood Conjecture. The third seemed to be very close to [Legendre's Conjecture](https://en.wikipedia.org/wiki/Legendre%27s_conjecture). In fact, when these results were shown to an expert, she said that these results are all deducible by elementary methods. That's why I would be glad if someone could point me to some references from which these results can be deduced.