  In this paper we show a new inequality which generalizes to the unit sphere
the Lebedev-Milin inequality of the exponentiation of functions on the unit
circle. It may also be regarded as the counterpart on the sphere of the second
inequality in the Szeg\"o limit theorem on the Toeplitz determinants on the
circle. On the other hand, this inequality is also a variant of several
classical inequalities of Moser-Trudinger type on the sphere. The inequality
incorporates the deviation of the center of mass from the origin into the
optimal inequality of Aubin for functions with mass centered at the origin, and
improves Onofri's inequality with the contribution of the shifting of the mass
center explicitly expressed.
