How Broad is the Class of Normal Scale Mixtures?

We study the class of scale mixtures of normal distributions with mean zero. Given that the cdf F(x) of such a mixture is fixed at two points, say F(x₁) = α₁, F(x₂) = α₂, we answer the question of how widely F(x₃) can vary at some third point x₃. A brief final section mentions extensions of our theorem.