Evolution of geometric constant evolving along extended Ricci flow

We investigate the behavior of the lowest geometric constant, lambda(c)(a,b) (g), along the extended Ricci flow such that there exist positive solutions to the following partial differential equation: -Delta u + au log u + bS(c)u = lambda(c)(a, b)(g) u with integral(M) u(2)d mu = 1, where a, b and c are real constants. We drive the evolution formula for the geometric constant lambda(c)(a,b) (g) along the unnormalized and normalized extended Ricci flow. Moreover, we give some monotonic quantities involving lambda(c)(a,b) (g) along the extended Ricci flow by imposing some geometric conditions.