Min-entropy ($H_{\infty }(X)$)

For a discrete random variable with largest point probability $p_{\max }$, its min-entropy is

For a discrete random variable, $H(X)\ge H_{\infty }(X)$. Indeed, every point probability satisfies $p_x\le p_{\max }$, so averaging $-\log p_x\ge -\log p_{\max }$ proves the inequality.