An inequality on entropy

The entropy H(X) of a discrete random variable X of alphabet size m is always non-negative and upper-bounded by log m. In this paper, we present a theorem which gives a non-trivial lower bound for H(X). We show that for any discrete random variable X with range R={x₀,…,x_m-1}, if p_i=Pr{X=x_i} and p₀⩾p₁⩾…p_m-1, then H(X)⩾(2logm)/(m-1)Σ_i=0^m-1ip_i, with equality iff (i) X is uniformly distributed, i.e., p_i=1/m for all i, or trivially (ii) p₀=1, and p _i=0 for 1⩽i⩽m-1