On the existence of twin prime in an interval

Let $S_{(x,y]} = \left\{\frac{p_n}{p_{n+1}-2} :~ n\in I \right\}$, where $I = \left\{n :~ x<p_n \le y \right\}$, $p_n$ is the $n$-th prime and $x, y \in \mathbb{R}_{>0}$. If $M_α(x,y)$ denotes the $α$-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 $\displaystyle \lim_{α\rightarrow \infty}M_α(x,y) > 1 - 2/y + O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)>1-c/x^β+O(x^{-β}\log^{-1} x)$, where the constant $c>0$ and $β= 1+c/\log^2 x$ or equivalently, $x^β=x+cx/\log x+O(x/\log^2 x)$. With $c<2$, the lower bound for $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)$ satisfies the inequality on the existence of a twin prime in the interval $(x,x^β]$.

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