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The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n_0(X)$.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows for $p_n>7$ that $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

Regarding d), the Prime Number Theorem gives that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{p_n-p_{n-1}}{n p_n\log p_n}\sim\frac{p_n-p_{n-1}}{n^2\log^2 n}.$$ Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) restricted to large $n$ is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$ and $n>n_0(X)$.

The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n_0(X)$.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows for $p_n>7$ that $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

Regarding d), the Prime Number Theorem gives that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{p_n-p_{n-1}}{n p_n\log p_n}\sim\frac{p_n-p_{n-1}}{n^2\log^2 n}.$$ Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) restricted to large $n$ is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$ and $n>n_0(X)$.

The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n_0(X)$.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows that for $p_n>7$ we have $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

Regarding d), we have by standard estimates that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{p_n-p_{n-1}}{n p_n\log p_n}\sim\frac{p_n-p_{n-1}}{n^2\log^2 n}.$$ Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) (for large $n$) is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$.

The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n_0(X)$.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows for $p_n>7$ that $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

Regarding d), the Prime Number Theorem gives that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{p_n-p_{n-1}}{n p_n\log p_n}\sim\frac{p_n-p_{n-1}}{n^2\log^2 n}.$$ Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) restricted to large $n$ is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$ and $n>n_0(X)$.

The first three statements are true for $n$ sufficiently large.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows that for $p_n>7$ we have $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n_0(X)$.

Regarding a), a classical result of Ingham (1937) shows that $$p_n^{1/3} - p_{n-1}^{1/3}<\frac{p_n-p_{n-1}}{3p_{n-1}^{2/3}}=o(1).$$

Regarding b), the Prime Number Theorem gives that $$p_n^{1/n} - p_{n-1}^{1/n}<\frac{p_n-p_{n-1}}{np_{n-1}^{(n-1)/n}}\sim\frac{p_n-p_{n-1}}{n p_{n-1}}=o\left(\frac{1}{n}\right).$$

Regarding c), Bertrand's postulate shows that for $p_n>7$ we have $$(\log p_n)^{1/2}-(\log p_{n-1})^{1/2}<\frac{\log p_n-\log p_{n-1}}{2(\log p_{n-1})^{1/2}}<\frac{\log 2}{2(\log p_{n-1})^{1/2}}<\frac{1}{4}.$$ In fact the left-hand side tends to zero (as $n\to\infty$).

Regarding d), we have by standard estimates that $$(\log p_n)^{1/n}-(\log p_{n-1})^{1/n}\sim\frac{p_n-p_{n-1}}{n p_n\log p_n}\sim\frac{p_n-p_{n-1}}{n^2\log^2 n}.$$ Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) (for large $n$) is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$.