Abstract This article is about transient information regarding the functions that satisfy the conditions of monotonicity and convexity on a set, onset of matrices of particular patterns with finite or infinite order called operator monotone or operator convex functions. Data is gathered by investigating numerous existing articles, review papers, publications that show wide scattering in multiple dimensions according to the behavior and prolific results. This work emphasizes not only the origin, key results, and applications of convexity and monotonicity but also their extensions in different directions. It also takes us to the current research scenario in relating fields and may provide an opportunity to express some new ideas, methods, or concepts.