  The relations between the resource theoretic measures of quantum coherence
are rigorously investigated for various Markovian and non-Markovian channels
for the two-qubit $X$ states with specific attention to the maximum and minimum
attainable coherence and usefulness of these states in performing quantum
teleportation in noisy environment. The investigation has revealed that under
both dephasing and dissipative type noises the maximally entangled mixed states
and Werner states lose their form and usefulness. However, maximally non-local
mixed states (MNMSs) lose their identity in dissipative noise only. Thus, MNMSs
are established to be useful in teleporting a qubit with fidelity greater than
the classical limit in the presence of dephasing noise. MNMSs also remain
useful for device independent quantum key distribution in this case as they
still violate Bell's inequality. In the presence of noise, coherence measured
by relative entropy of coherence is found to fall faster than the same measured
using $l_1$ norm of coherence. Further, information back-flow from the
environment to the system is observed over non-Markovian channels which leads
to revival in coherence. Additionally, sequential interaction of two qubits
with the same environment is found to result in correlated noise on both
qubits, and coherence is observed to be frozen in this case under dephasing
channel. Under the effect of Markovian and non-Markovian dephasing channels
studied here, we observed that MNMSs have maximum relative coherence, i.e.,
they have the maximum amount of $l_1$ norm of coherence among the states with
the same amount of relative entropy of coherence. However, this feature is not
visible in any $X$ state evolving over dissipative channels.
