  The recent IceCube observation of ultra-high-energy astrophysical neutrinos
has begun the era of neutrino astronomy. In this work, using the unitarity of
leptonic mixing matrix, we derive nontrivial unitarity constraints on the
flavor composition of astrophysical neutrinos detected by IceCube. Applying
leptonic unitarity triangles, we deduce these unitarity bounds from geometrical
conditions, such as triangular inequalities. These new bounds generally hold
for three flavor neutrinos, and are independent of any experimental input or
the pattern of leptonic mixing. We apply our unitarity bounds to derive general
constraints on the flavor compositions for three types of astrophysical
neutrino sources (and their general mixture), and compare them with the IceCube
measurements. Furthermore, we prove that for any sources without $\nu_\tau$
neutrinos, a detected $\nu_\mu$ flux ratio $< 1/4$ will require the initial
flavor composition with more $\nu_e$ neutrinos than $\nu_\mu$ neutrinos.
