$k$-essence $f(R)$ Gravity Inflation
  In this work we study a modified version of $f(R)$ gravity in which higher
order kinetic terms of a scalar field are added in the action of vacuum $f(R)$
gravity. This type of theory is a type of $k$-essence $f(R)$ gravity, and it
belongs to the general class of $f(R,\phi ,X)$ theories of gravity, where
$\phi$ is a scalar field and
$X=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi$. We focus on the
inflationary phenomenology of the model, in the slow-roll approximation, and we
investigate whether viable inflationary evolutions can be realized in the
context of this theory. We use two approaches, firstly by imposing the
slow-roll conditions and by using a non-viable vacuum $f(R)$ gravity. As we
demonstrate, the spectral index of the primordial scalar perturbations and the
tensor-to-scalar ratio of the resulting theory can be compatible with the
latest observational data. In the second approach, we fix the functional form
of the Hubble rate as a function of the $e$-foldings number, and we modify
well-known vacuum $f(R)$ gravity reconstruction techniques, in order to find
the $k$-essence $f(R)$ gravity which can realize the given Hubble rate.
Accordingly, we calculate the slow-roll indices and the corresponding
observational indices, and we also provide general formulas of these quantities
in the slow-roll approximation. As we demonstrate, viability can be obtained in
this case too, however the result is strongly model dependent. In addition, we
discuss when ghosts can occur in the theory, and we investigate under which
conditions ghosts can be avoided by using a particular class of models.
Finally, we qualitatively discuss the existence of inflationary attractors for
the non-slow-roll theory, and we provide hints towards finding general de
Sitter attractors for the theory at hand.
