  In this work we generalize the constant-roll condition for minimally coupled
canonical scalar field inflation. Particularly, we shall assume that the scalar
field satisfies the condition $\ddot{\phi}=\alpha (\phi) V'(\phi)$, and we
derive the field equations under this assumption. We call the framework
extended constant-roll framework. Accordingly we calculate the inflationary
indices and the corresponding observational indices of inflation. In order to
demonstrate the inflationary viability, we choose three potentials that are
problematic in the context of slow-roll dynamics, namely chaotic, linear
power-law and exponential inflation, and by choosing a simple power-law form
for the smooth function $\alpha (\phi)$, we show that in the extended
constant-roll framework, the models are compatible with the latest 2018 Planck
constraints on inflation. We also justify appropriately why we called this new
framework extended constant-roll framework, and we show that the condition
$\ddot{\phi}=\alpha (\phi) V'(\phi)$ is equivalent to the condition
$\ddot{\phi}=\beta (\phi) H \dot{\phi}$, with the latter condition being a
simple generalization of the constant-roll condition. Finally, we examine an
interesting physical situation, in which a general extended constant-roll
scalar field model is required to satisfy the cosmological tracker condition
used in quintessence models. In contrast to the slow-roll and ordinary
constant-roll cases, in which case the tracker condition is not compatible with
neither the slow-roll or the ordinary constant-roll conditions, the extended
constant-roll condition can be compatible with the tracker condition. This
feature leads to a new inflationary phenomenological framework, the essential
features of which we develop in brief.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory