Kazdan inequality

Let be a real -dimensional scalar inner product space (cf. also Inner product; Pre-Hilbert space), let be the space of linear operators of , and let be a given family of symmetric linear operators depending continuously on . For , denote by the solution of the initial value problem

Suppose that is invertible for all . Then for every positive -function on satisfying on , one has

with equality if and only if for all .

References

[a1]	J.L. Kazdan, "An inequality arising in geometry" A.L. Besse (ed.) , Manifolds all of whose Geodesics are Closed , Springer (1978) pp. 243–246; Appendix E
[a2]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)