It is well known that primes of form $4k+3$, call them $3=q_1 < q_2 < \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of Dusarttype (Strong Bertrand postulate) with concrete bounds for $N$ such that $q_{n+1} < 1.1 q_n$ provided $q_n > N$ or something like this (I am not sure that I need a multiple exactly $1.1$)
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Have a look at (Satz 9 and the preceeding calculations): 193510 P. Erdõs: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473491; Zentralblatt 10,293. Available at: http://www.renyi.hu/~p_erdos/Erdos.html 

