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Nicolas [8] and DeSalvo and Pak [3] proved that the partition function p(n) is log concave for n ≥ 25 . Chen et al. [2] proved that p(n) satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for n ≥ 95 . Recently, Griffin et al. [5] proved more generally that for all d, the degree d Jensen polynomials associated to p(n) are hyperbolic for sufficiently large n. In this paper, we prove that the same result holds for the k-regular partition function p k (n) for k ≥ 2 . In particular, for any positive integers d and k, the order d Turán inequalities hold for p k (n) for sufficiently large n. The case when d = k = 2 proves a conjecture by Neil Sloane that p 2 (n) is log concave. [ABSTRACT FROM AUTHOR]