(A) is incorrect: ep>1 (in absolute value) is the definition of price elastic demand, not inelastic — the statement has the classification backwards.
(B) is incorrect: the standard classification is the opposite — luxury goods tend to be more price elastic (since they're discretionary, consumers cut back sharply when prices rise), while necessities tend to be more price inelastic (people keep buying them regardless of price changes).
(C) is correct: luxury goods are precisely defined by having income elasticity eI>1 — demand for them rises more than proportionally as income rises, which is the standard textbook definition distinguishing luxuries from necessities (0<eI<1) and inferior goods (eI<0).
(D) is incorrect: 0<ep<1 is the definition of price inelastic demand, not elastic — again backwards from the correct classification.
Therefore only (C) is correct.