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We take a pragmatic, model independent approach to single field slow-rollcanonical inflation by imposing conditions, not on the potential, but on theslow-roll parameter $\epsilon(\phi)$ and its derivatives $\epsilon^{\prime}(\phi)$ and $\epsilon^{\prime\prime }(\phi)$, thereby extracting generalconditions on the tensor-to-scalar ratio $r$ and the running $n_{sk}$ at$\phi_{H}$ where the perturbations are produced, some $50$ $-$ $60$ $e$-foldsbefore the end of inflation. We find quite generally that for models where$\epsilon(\phi)$ develops a maximum, a relatively large $r$ is most likelyaccompanied by a positive running while a negligible tensor-to-scalar ratioimplies negative running. The definitive answer, however, is given in terms ofthe slow-roll parameter $\xi_2(\phi)$. To accommodate a large tensor-to-scalarratio that meets the limiting values allowed by the Planck data, we study anon-monotonic $\epsilon(\phi)$ decreasing during most part of inflation. Sinceat $\phi_{H}$ the slow-roll parameter $\epsilon(\phi)$ is increasing, we thusrequire that $\epsilon(\phi)$ develops a maximum for $\phi > \phi_{H}$ afterwhich $\epsilon(\phi)$ decrease to small values where most $e$-folds areproduced. The end of inflation might occur trough a hybrid mechanism and asmall field excursion $\Delta\phi_e\equiv |\phi_H-\phi_e |$ is obtained with asufficiently thin profile for $\epsilon(\phi)$ which, however, should notconflict with the second slow-roll parameter $\eta(\phi)$. As a consequence ofthis analysis we find bounds for $\Delta \phi_e$, $r_H$ and for the scalarspectral index $n_{sH}$. Finally we provide examples where these considerationsare explicitly realised.

Canonical single field slow-roll inflation with a non-monotonic tensor-to-scalar ratio