  We establish exact inequalities for the structure-function scaling exponents
of a passively advected scalar in both the inertial-convective and
viscous-convective ranges. These inequalities involve the scaling exponents of
the velocity structure functions and, in a refined form, an intermittency
exponent of the convective-range scalar flux. They are valid for 3D
Navier-Stokes turbulence and satisfied within errors by present experimental
data. The inequalities also hold for any ``synthetic'' turbulent velocity
statistics with a finite correlation in time. We show that for time-correlation
exponents of the velocity smaller than the ``local turnover'' exponent, the
scalar spectral exponent is strictly less than that in Kraichnan's soluble
``rapid-change'' model with velocity delta-correlated in time. Our results
include as a special case an exponent-inequality derived previously by
Constantin \& Procaccia [Nonlinearity {\bf 7} 1045 (1994)], but with a more
direct proof. The inequalities in their simplest form follow from a
Kolmogorov-type relation for the turbulent passive scalar valid in each space
dimension $d.$ Our improved inequalities are based upon a rigorous version of
the refined similarity hypothesis for passive scalars. These are compared with
the relations implied by ``fusion rules'' hypothesized for scalar gradients.
