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Let $d$ be a square-free positive integer and $h\left(d\right)$ be the class number of the real quadratic field $\mathbb{Q}\left(\sqrt{d}\right).$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g\mid h\left(d\right)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb{Q}\left(\sqrt{n^{2g}+4}\right)$. As another application, we deduce some criteria for a class group of prime power order to be cyclic.