Bounds of ideal class numbers of real quadratic function fields

The theory of continued fractions of functions $\sqrt D$ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function fields $K$, the bounds of $h(D)$ are given more explicitly, e.g., if $D=F^2+c,$ \mbox{}\hspace{0.1cm} then $h(D)\geq {deg}F /{deg} P;$ \hspace{0.1cm} if $D=(SG)^2+cS,$ then $h(D)\geq {deg}S / {deg} P;$ if $D=(A^m+a)^2+A,$ then $h(D)\geq {deg}A / {deg} P,$ where $P$ is irreducible polynomial splitting in $K, c\in {\bf F}_q$ is any constant. In addition, six types of quadratic function fields are found to have ideal class numbers bounded and bigger than one. {\bf keywords:} quadratic function field, ideal class number, continued fractions of functions