Are there any well-known conditions that guarantee that a probability distribution isn’t too “spiky”? I ask because I am interested in families of probability distributions $ f(x)$ on the unit interval such that the following criterion holds: there exists a measurable subset $ S\subset[0,1]$ such that $ $ 4\left(\int_{S}f(x)\,dx\right)^{2}\geq\lambda(S)\int_{0}^{1}f(x)^{2}\,dx$ $ where $ \lambda(S)$ denotes the Lebesgue measure of $ S$ , i.e. the sum of the intervals that comprise it. This looks like a reversed Jensen’s inequality, except for the fact that we’re taking integrals over two separate domains. Is there a well-known sufficient condition that would cause this to hold? How about if I increase the coefficient $ 4$ ?