An L form of Bernstein's inequality

Suppose P is a pth degree real polynomial function in n variables and f=PS(n-1) is the restriction of P to the unit sphere S(n-1) in R(n). Bernstein's inequality asserts that ([unk](0) (k)f)(2) + p(2)([unk](0) (k-1)f)(2) /= 1 and differentiation is with respect to arc length theta along any geodesic in S(n-1). We find the constant corresponding to p(2k) when parallelf parallelinfinity is replaced by parallelf parallel(2). One application is a condition on the coefficients of the expansion in surface spherical harmonics of any g: S(n-1) --> R, which condition suffices to assure that g is k times differentiable.