  We study the quantum capacitance in a topological insulator thin film system
magnetized in the in-plane direction in the presence of an out-of-plane
magnetic field and hexagonal warping. To first order, the modification in
quantum capacitance due to hexagonal warping compared to the clean case, where
both the in-plane magnetization and hexagonal warping are absent, is always
negative, and increases in magnitude monotonically with the energy difference
from the charge neutrality point. In contrast, the change in the quantum
capacitance due to in-plane magnetization oscillates with the energy in
general, except when a certain relation between the inter-surface coupling, out
of plane Zeeman energy splitting and magnetic field strength is satisfied. In
this special case, the quantum capacitance remains unchanged by the in-plane
magnetization for all energies.
