  For a noninteracting Bose gas with a fixed one-body Hamiltonian H^0
independent of the number of particles we derive the inequalities <N_i>_N <
<N_i>_{N+1}, <N_i N_j>_N < <N_i>_N <N_j>_N for i\neq j, \partial
<N_0>_N/\partial \beta >0 and <N_i>^+_N < <N_i>_N. Here N_i is the occupation
number of the ith eigenstate of H^0, \beta is the inverse temperature and the
superscript + refers to adding an extra level to those of H^0. The results
follow from the convexity of the N-particle free energy as a function of N.
