Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
I have finally found the following papers and results, which predate Nagura's paper of 1952. I cite them from newest to oldest:
- (Molsen, 1941):
- For $n\geq 118$ there are primes in $(n,\frac43n)$ congruent to 1,5,7,11 modulo 12.
- For $n\geq 199$ there are primes in $(n,\frac87n)$ congruent to 1,2 modulo 3. This result implies that of Nagura.
K. Molsen, Zur Verallgemeinerung des Bertrandschen Postulates, Deutsche Math. 6 (1941), 248-256. MR0017770
- (Breusch, 1932):
- For $n\geq 7$ there are primes in $(n,2n)$ congruent to 1,2 modulo 3 and to 1,3 modulo 4.
- For $n\geq 48$ there is a prime in $(n,\frac98n)$. This result implies those of Nagura and Molsen (but not the congruences part).
R. Breusch, Zur Verallgemeinerung des Bertrandschen Postulates, dass zwischen x und 2x stets Primzahlen liegen, Math. Z. 34 (1932), 505–526. MR1545270
- (Schur, 1929, according to Breusch in the previous paper):
- For $n\geq 24$ there is a prime in $(n,\frac54n)$. This result already implies those of Hanson and El Bachraoui.
I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen I, Sitzungsberichte der preuss. Akad. d. Wissensch., phys.-math. Klasse 1929, S.128.
EDIT: Jose Brox, in another answer, had provided references which not noly predate Nagura's result, but they are also stronger.
J. Nagura, already in 1952, proved that the interval $(x,\frac{6}{5}x)$ contains a prime for any $x \ge 25$. The proof appears in the paper "On the interval containing at least one prime number" published in Proc. Japan Acad. Volume 28, Number 4 (1952), 177-181 (see this link). From Nagura's theorem one obtains the results mentioned in your post by choosing $x=2n$ or $x=3n$, and checking what happens for small $n$.