  This work tests the no-hair conjecture in $f(R)$ gravity models. No-hair
conjecture asserts that all black holes in General Relativity coupled to any
matter must be Kerr-Newman type. However, the conjecture fails in some cases
with non-linear matter sources. Here, we address this by explicitly
constructing multiple slow-rotating black hole solutions, up to second order in
rotational parameter, for a class of $f(R)$ models ( $f(R) =(\alpha_{0} +
\alpha_{1}\,R)^{p}, p > 1$). Such an $f(R)$ includes all higher-powers of $R$.
We analytically show that multiple vacuum solutions satisfy the field equations
up to the second order in the rotational parameter. In other words, we show
that the multiple vacuum solutions depend on arbitrary constants, which depend
on the coupling parameters of the model. Hence, our results indicate that the
no-hair theorem for modified gravity theories merits extending to include the
coupling constants. The uniqueness of our result stems from the fact that these
are obtained directly from metric formalism without conformal transformation.
We discuss the kinematical properties of these black hole solutions and compare
them with slow-rotating Kerr. Specifically, we show that the circular orbits
for the black holes in $f(R)$ are smaller than that of Kerr. This implies that
the inner-most stable circular orbit for black holes in $f(R)$ is smaller than
Kerr's; hence, the shadow radius might also be smaller. Finally, we discuss the
implications of our results for future observations.
