We propose a new focus function Λ that, like many of the existing focus functions, consists of a convex function and an image enhancement filter. Λ is rather flexible because for any convex function and image enhancement filter, it is a focus function. We proved that Λ is a focus function using a model and Jensen's inequality. Furthermore, we generated random Λs and experimentally applied them to simulated and real blurred images, finding that 98% and 99% of the random Λs, respectively, have a maximum value at the best-focused image and most of them decrease as the defocus increases. We also applied random Λs to motion-blurred images, blurred images in different-sized windows, and blurred images with different types of noise. We found that Λ can be applied to motion blur and is robust to different-sized windows and different noise types.