  We consider inflationary models with the inflaton coupled to the Gauss-Bonnet
term assuming a special relation $\delta_1=2\lambda\epsilon_1$ between the two
slow-roll parameters $\delta_1$ and $\epsilon_1$. For the slow-roll inflation,
the assumed relation leads to the reciprocal relation between the Gauss-Bonnet
coupling function $\xi(\phi)$ and the potential $V(\phi)$, and it leads to the
relation $r=16(1-\lambda)\epsilon_1$ that reduces the tensor-to-scalar ratio
$r$ by a factor of $1-\lambda$. For the constant-roll inflation, we derive the
analytical expressions for the scalar and tensor power spectra, the scalar and
tensor spectral tilts, and the tensor-to-scalar ratio to the first order of
$\epsilon_1$ by using the method of Bessel function approximation. The
tensor-to-scalar ratio is reduced by a factor of $1-\lambda+\lambda\tilde
\eta$. Comparing the derived $n_s$-$r$ with the observations, we obtain the
constraints on the model parameters $\tilde\eta$ and $\lambda$.
