The study of operator Holder functions has a long history and plays an important role in the perturbation theory of linear operators on a Hilbert space. The starting point is the so-called Powers--Strømer inequality [Comm. Math. Phys. 1970] (and its generalization, the so-called Birman--Koplienko--Solomjak inequality [IVUZM, 1975]). Over the past decades, many mathematicians enlarged the classes of functions or the quasi-norms for which the inequality holds. Aleksandrov and Peller [JFA, 2010] showed Holder functions are not necessarily operator-Holder. However, it was unknown whether the simplest example of Holder functions, the fractional power functions, are operator-Holder for every Lp-norm or not before being proved by E. Ricard recently [Adv. Math. 2018]. After reviewing the background material, I will sketch Ricard's proof and present the joint work with F. Sukochev and D. Zanin, which extends several existing results.