Bertrand's Postulate for $4a+1$ type primes

Question

Bertrand's postulate proves $p_{n+1} < 2p_{n}$ where $p_n$ is the $n^\text{th}$ prime.

Let $q_m$ be the $m^\text{th}$ prime of form $4a+1$. Is this statement been proved/researched before?

$$q_{m+1}

for some constant $k$.

Answer 1

See this paper:

P. Moree, Bertrand's Postulate for primes in arithmetical progressions, Computers & Mathematics with Applications, volume 26, issue 5, september 1993, pages 35-43, DOI 10.1016/0898-1221(93)90071-3

Here is the abstract:

Bertrand's Postulate is the theorem that the interval $(x, 2x)$ contains at least one prime for $x >1$. We prove, building on work of Erdős, analogues of this result, in which the interval is of the form $(x, zx)$ and there are at least $m$ primes $\equiv a \bmod d$ required to be contained in this interval, and where $z$, $a$ and $d$ have to satisfy some conditions. For the case $m = $1 the results are worked out using a computer. They can be found in Table 1.