A Remark on "Two-Sided" Monotonicity Condition: An Application to   $L^{p}$ Convergence
  To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the $L^{p}$ convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial. 
Rui-Jun Le and Song-Ping Zhou