A HYPERGEOMETRIC INEQUALITY ATUL DIXIT, VICTOR H. MOLL, AND VERONIKA PILLWEIN Abstract. A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. The original proof is based on a inequality for hypergeometric functions. A generalization is presented. 1. Introduction A sequence of numbers {a k :0 ≤ k ≤ n} is called unimodal if there is an index k ∗ such that a k−1 ≤ a k for 1 ≤ k ≤ k ∗ and a k−1 ≥ a k for k ∗ +1 ≤ k ≤ n. The prototypical example of unimodal sequences is a k = ( n k ) . A polynomial P (x) is called unimodal if its sequence of coefficients is unimodal. A simple criteria for unimodality of a polynomial was established in [4]: Theorem 1.1. If P (x) is a polynomial with positive nondecreasing coefficients, then P (x + 1) is unimodal. The original motivation for this result was the question of unimodality of the polynomial (1.1) P m (a)= m  ℓ=0 d ℓ (m)a ℓ with (1.2) d ℓ (m)=2 −2m m  k=ℓ 2 k  2m - 2k m - k  m + k m  k ℓ  . This example appeared in the evaluation of the formula (1.3)  ∞ 0 dx (x 4 +2ax 2 + 1) m+1 = π 2 P m (a) [2(a + 1)] m+1/2 given in [5]. A variety of proofs of (1.3) can be found in [3] and properties of the coefficients {d ℓ (m)} have been reviewed in [26]. A property stronger than unimodality is that of logconcavity : a sequence of positive numbers {a k : 0 ≤ k ≤ n} is called logconcave if a 2 k ≥ a k−1 a k+1 for 1 ≤ k ≤ n - 1. As before, a polynomial is called logconcave if its sequence of coefficients is logconcave. The logconcavity of P m (a) was first established by M. Kauers and P. Paule in [23] using computer algebra, in particular algorithms for automatically deriving recurrences for multiple sums and Cylindrical Algebraic Decomposition (CAD), and by W. Y. Chen et al in [10, 11, 12, 13] by a variety of classical techniques. Date : March 18, 2014. 2010 Mathematics Subject Classification. Primary 33C05. Key words and phrases. Hypergeometric function, unimodal polynomials, monotonicity. 1