The extended reverse ultra log-concavity of transposed Boros-Moll sequences

The Boros-Moll sequences \d_ (m) \=₀ᵐ arise in the study of evaluation of a quartic integral. After the infinite log-concavity conjecture of the sequence \d_ (m) \=₀ᵐ was proposed by Boros and Moll, a lot of interesting inequalities on d_ (m) were obtained, although the conjecture is still open. Since d_ (m) has two parameters, it is natural to consider the properties for the sequences \d_ (m) \₌, which are called the transposed Boros-Moll sequences here. In this paper, we mainly prove the extended reverse ultra log-concavity of the transposed Boros-Moll sequences \d_ (m) \₌, and hence give an upper bound for the ratio d_² (m) / (d_ (m-1) d_ (m+1) ). A lower bound for this ratio is also established which implies a result stronger than the log-concavity of the sequences \d_ (m) \₌. As a consequence, we also show that the transposed Boros-Moll sequences possess a stronger log-concave property than the Boros-Moll sequences do. At last, we propose some conjectures on the Boros-Moll sequences and their transposes.