A Short Note on Gaps between Powers of Consecutive Primes

Let $\alpha ,\beta \ge 0$ and $\alpha +\beta <1$. In this short note, we show that ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{p}_n^{\beta }(p_{n+1}^{\alpha }-p_n^{\alpha })=0$, where $p_n$ is the $n$th prime. This notes an improvement over results of S'{a}ndor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.