Hardy's inequality

Hardy's inequality is a mathematical statement that relates the size of a function to the size of its average values, especially near zero. It shows that, under certain conditions, the integral (sum) of a function divided by its distance from zero is controlled by the integral of its derivative (how quickly it changes). Essentially, it provides a way to estimate how a function behaves near a point by examining how rapidly it varies, ensuring that functions that don’t change too wildly near zero have controlled overall behavior. This is useful in analysis and solving differential equations.