  We consider a subclass of Horndeski theories for studying cosmic inflation.
In particular, we investigate models of inflation in which the derivative
self-interaction of the scalar field and the non-minimal derivative coupling to
gravity are present in the action and equally important during inflation. In
order to control contributions of each term as well as to approach the
single-term limit, we introduce a special relation between the derivative
interaction and the coupling to gravity. By calculating observable quantities
including the power spectra and spectral tilts of scalar and tensor
perturbation modes, and the tensor-to-scalar ratio, we found that the
tensor-to-scalar ratio is suppressed by a factor of $(1+1/\gamma)$, where
$\gamma$ reflects the strength of the derivative self-interaction of the
inflaton field with respect to the derivative coupling gravity. We placed
observational constraints on the chaotic and natural inflation models and
showed that the models are consistent with the current observational data
mainly due to the suppressed tensor-to-scalar ratio.
