I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.

$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$

$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $

$c) (\log p_n)^{1/2} - (\log p_{n-1})^{1/2} < 1/4$

$d) (\log p_n)^{1/n} - (\log p_{n-1})^{1/n} < 1/n^X, n\geq7,X=2$

In $d)$ I tried to find a larger $X$, but I failed.

Maybe some will fail for a larger $n$. Are any of these know to be true? Also, what can be deduced from they?