The problem of finding bounds of time-averaged characteristics of dynamical systems, such as the bound on the mean energy dissipation rate in a turbulent flow governed by incompressible Navier–Stokes equations, is considered. It is shown that the direct method described by Seis (2015 J. Fluid Mech. 777 , 591–603. ( doi:10.1017/jfm.2015.384 )) and the auxiliary functional method by Chernyshenko et al. (2014 Phil. Trans. R. Soc. B 372 , 20130350. ( doi:10.1098/rsta.2013.0350 )) are related and can lead to the same bound. The well-known background flow method of Doering and Constantin is equivalent to the auxiliary functional method with a quadratic auxiliary functional. The known implementations of the direct method apparently also correspond to quadratic auxiliary functionals. The findings are illustrated by the analysis of the plane Couette flow. Three routes of further progress using non-quadratic auxiliary functionals and at the same time allowing to use the experience accumulated with the background flow method are proposed: making the balance parameter dependent on the energy, making the background flow time-dependent in a specific way and adding helicity to the auxiliary functional. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.