“…However, it only applies to those functions Φ having strictly positive second derivative and such that 1/Φ ′′ is concave, e.g., Φ(x) = |x| r , for r ∈ (1, 2], or Φ(x) = x log x. Moreover, the upper bounds on (3.1) for those Φ are linear in n as easily seen via, e.g., [6,Theorem 14.6], together with (2.1). In [6,Chapter 15], some further generalizations of Efron-stein inequality for Φ(x) = |x| r , r ≥ 2, are also obtained, which, together with (2.1), also imply upper bounds of order n r/2 on the r-th central moments (cf.…”