On the existence of twin prime in an interval

Abstract: Let $S_{(x,y]} = \left{\frac{p_n}{p_{n+1}-2} :~ n\in I \right}$, where $I = \left{n :~ x&lt;p_n \le y \right\}$, $p_n$ is the $n$-th prime and $$x, y \in \mathbb{R}_{\&gt;0}$$. If $M_\alpha(x,y)$ denotes the $\alpha$-power mean of the elements of $S_{(x,y]}$, it is shown that the existence of a twin prime pair in $(x,y]$ is implied if $\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,y) &gt; 1 - 2/y + O(y<sup>{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,x^{\beta)&gt;1-c/x^{\beta+O(x^{{-\beta}\log^{-1}}}} x)$, where the constant $c&gt;0$ and $\beta = 1+c/\log² x$ or equivalently, $$x^{\beta=x+cx/\log} x+O(x/\log² x)$$. With $c&lt;2$, the lower bound for $$\displaystyle \lim_{\alpha \rightarrow \infty}M_{\alpha}(x,x<sup>\beta)$$ satisfies the inequality on the existence of a twin prime in the interval $(x,x<sup>\beta]$.