An inequality for coefficients of the real-rooted polynomials

Boas, 1954 Boros, 2004 Brändén, 2011, Iterated sequences and the geometry of zeros, J. Reine Angew. Math., 658, 115 Chen, 2019, Higher order Turán inequalities for the partition function, Trans. Am. Math. Soc., 372, 2143, 10.1090/tran/7707 Chen, 2016, Finite differences of the logarithm of the partition function, Math. Comput., 85, 825, 10.1090/mcom/2999 Craven, 2002, Iterated Laguerre and Turán inequalities, JIPAM. J. Inequal. Pure Appl. Math., 3 Csordas, 1986, The Riemann hypothesis and the Turán inequalities, Trans. Am. Math. Soc., 296, 521 Dimitrov, 1998, Higher order Turán inequalities, Proc. Am. Math. Soc., 126, 2033, 10.1090/S0002-9939-98-04438-4 Dimitrov, 2011, Higher order Turán inequalities for the Riemann ξ-function, Proc. Am. Math. Soc., 139, 1013, 10.1090/S0002-9939-2010-10515-4 Fisk Honsberger, 1991, 237 Hou, 2019, r-log-Concavity of partition functions, Ramanujan J., 48, 117, 10.1007/s11139-017-9975-5 Levin, 1980, Distribution of Zeros of Entire Functions, vol. 5 Mar̆ík, 1964, On polynomials with all real zeros, Čas. Pěst. Mat., 89, 5, 10.21136/CPM.1964.117490 McNamara, 2010, Infinite log-concavity: developments and conjectures, Adv. Appl. Math., 44, 1, 10.1016/j.aam.2009.03.001 Pólya, 1927, Über die algebraisch-funktionentheoretischen Untersuchungen vonj. L. W. V. Jensen, K. Dan. Vidensk. Selsk. Math.-Fys. Medd., 7, 3 Pólya, 1914, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., 144, 89 Rahman, 2002 R.P. Stanley, Personal communication, May 2008.