Piyavskii's algorithm maximizes a univariate function f satisfying a Lipschitz condition. We compare the numbers of iterations needed to obtain a bound within ε of the (unknown) globally optimal value with a best possible algorithm (n_B) and with Piyavskii's algorithm (n_P). The main result if that: n_P ≤ 2n_B + 1 and that this bound is sharp. We also show that the number of iterations needed by Piyavskii's algorithm to obtain a proved globally ε-optimal value together with a corresponding point (n_PY) satisfies n_PY ≤ 4n_B + 1 and that this bound is again sharp. Lower and upper bounds on n_B are obtained as functions of f(x), ε, L₀ and L₁, where L₀ is the smallest valid Lipschitz constant and L₁ the Lipschitz constant used. Several properties of n_B and of the distribution of evaluation poins are deduced from these bounds.