Variation of the centered maximal operator

Let M denote the centered Hardy--Littlewood operator on R. We prove that \[ { Var} (Mf)≤ { Var} (f) - 12| |f(∞)|-|f(-∞)|| \] for piecewise constant functions f with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt proved for indicator functions of bounded variation vanishing at ±∞. We conjecture that the inequality holds for all functions of bounded variation, which is a stronger version of the known conjecture Var (Mf)≤ Var (f). We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.