  This paper is concerned with the upper bound of the lifespan of solutions to
nonlinear Schr\"odinger equations with general homogeneous nonlinearity of the
critical order. In [8], Masaki and the first author obtain the upper bound of
the lifespan of solutions to our equation via a test function method introduced
by [16, 17]. Their nonlinearity contains a non-oscillating term $|u|^{1+2/d}$
which causes difficultly for constructing an even small data global solution.
The non-oscillating term corresponds to the $L^1$-scaling critical. In this
paper, it turns out that the upper bound can be refined by employing an unified
test function by Ikeda and the second author [5].
