  Let $d$ be a square-free positive integer and $h(d)$ the class number of the
real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ In this paper we give an
explicit lower bound for $h(n^2+r)$, where $r=1,4$, and also establish an
equivalent criteria to attain this lower bound in terms of special value of
Dedekind zeta function. Our bounds enable us to reduce the real quadratic
families considered in Chowla and Yokoi's conjecture to comparatively small
subfamily. We also give an equivalent criteria for having an alternate proof of
both the conjectures. Also applying our results, we obtain some criteria for
class group of prime power order to be cyclic.
