An Entire Holomorphic Function Associated to an Entire Harmonic Function

Let h be a harmonic function on ^N. Then there exists a holomorphic function f on such that f(t)=h(t, 0, …, 0) for all real t. Precise inequalities relating the growth rate of f to that of h are proved. These results are applied to deduce uniqueness theorems for harmonic functions of sufficiently slow growth that vanish at certain lattice points. Another application concerns the rate at which a harmonic function of finite order can decay along a ray.