Prime producing quadratic polynomials and class number one or two

Let d equivalent to 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field Q(root d). Let p be a divisor of d = pq and let f(p)( x) = vertical bar px(2) + px + p-q/4 vertical bar. Assume that f(p)( x) is prime or equal to 1 for all integers x with 0 <= x < W. Under the assumption that the Riemann hypothesis is true, we prove that if W = 1/2 (root d/5 - 1), then h(d) <= 2. Furthermore we show that h(d) <= 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If W = root d/4 - 1/2 then h <= 2 or h = 4. If W = 1/2 (root d/5 - 1), then h <= 2, h = 4 or h = 2(t-2), where t is the number of primes dividing d. In the case when h = 2(t-2) we have d = p(2)phi(2) +/- p, where phi = 2 or 4.