Abstract
Let \(\overline{p}(n)\) denote the overpartition function. Engel showed that for \(n\ge 2\), \(\overline{p}(n)\) satisfy the Turán inequalities, that is, \(\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0\) for \(n\ge 2\). In this paper, we prove several inequalities for \(\overline{p}(n)\). Moreover, motivated by the work of Chen, Jia and Wang, we find that the higher order Turán inequalities of \(\overline{p}(n)\) can also be determined.
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Ahlgren, S., Andersen, N.: Algebraic and transcendental formulas for the smallest parts function. Adv. Math. 289, 411–437 (2016)
Bessenrodt, C., Ono, K.: Maximal multiplicative properties of partitions. Ann. Comb. 20(1), 59–64 (2016)
Chen, W.Y.C.: Recent developments on log-concavity and \(q\)-log-concavity of combinatorial polynomials. In: FPSAC 2010 Conference Talk Slides. http://www.billchen.org/talks/2010-FPSAC (2010)
Chen, W.Y.C.: The spt-function of Andrews. Surv. Comb. 2017(440), 141 (2017)
Chen, W.Y.C., Wang, L.X.W., Xie, G.Y.B.: Finite differences of the logarithm of the partition function. Math. Comput. 85(298), 825–847 (2016)
Chen, W.Y.C., Jia, D.X.Q., Wang, L.X.W.: Higher order Turán inequalities for the partition function. Trans. Am. Math. Soc. 372, 2143–2165 (2019)
Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)
Dawsey, M.L., Masri, R.: Effective bounds for the Andrews spt-function. Forum Math. 31(3), 743–767 (2019)
DeSalvo, S., Pak, I.: Log-concavity of the partition function. Ramanujan J. 38(1), 61–73 (2015)
Dimitrov, D.K.: Higher order Turán inequalities. Proc. Am. Math. Soc. 126(7), 2033–2037 (1998)
Engel, B.: Log-concavity of the overpartition function. Ramanujan J. 43(2), 229–241 (2017)
Griffin, M., Ono, K., Rolen, L., Zagier, D.: Jensen polynomials for the Riemann zeta function and other sequences. Proc. Natl. Acad. Sci. USA 116(23), 11103–11110 (2019)
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. (2) 17, 75–115 (1918)
Jensen, J.L.W.V.: Recherches sur la théorie des équations. Acta Math. 36, 181–195 (1913)
Larson, H., Wagner, I.: Hyperbolicity of the partition Jensen polynomials. Res. Number Theory 5, 19 (2019)
Pólya, G.: Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen. Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7, 3–33 (1927)
Pólya, G., Schur, J.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)
Wang, L.X.W., Xie, G.Y.B., Zhang, A.Q.: Finite difference of the overpartition function. Adv. Appl. Math. 92, 51–72 (2018)
Zuckerman, H.S.: On the coefficients of certain modular forms belonging to subgroups of the modular group. Trans. Am. Math. Soc. 45(2), 298–321 (1939)
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Liu, E.Y.S., Zhang, H.W.J. Inequalities for the overpartition function. Ramanujan J 54, 485–509 (2021). https://doi.org/10.1007/s11139-019-00227-z
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DOI: https://doi.org/10.1007/s11139-019-00227-z