Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence $\{\sqrt[n]{f_n}\}_{n\ge 1}$ by using the aforementioned bounds.

[1] M. E. Dasef and S. M. Kautz, Some sums of some importance, College Math. J., 28 (1997) 52–55.

[2] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009.

[3] T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015.

[4] A. Dil and V. Kurt, Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices, J. Integer Seq., 14 (2011) pp. 12.

[5] N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Math. Mag., 83 (2010) 331–346.

[6] O. A. Gross, Preferential Arrangements, Amer. Math. Monthly, 69 (1962) 4–8.

[7] Z. W. Sun, Conjectures Involving Arithmetical Sequences, Proceedings of the 6th China-Japan Seminar, S. Kane-mitsu, H. Li and J. Liu eds., World Scientific, Singapore, 2013 244–258.

[8] W. Y. C. Chen and E. X. W. Xia, 2-log-concavity of the Boros-Moll Polynomials, Proc. Edinb. Math. Soc., 56 (2013) 701–722.

[9] W. Y. C. Chen, J. J. F. Guo and L. X. W. Wang, Infinitely Log-monotonic Combinatorial Sequences, Adv. in Appl. Math., 52 (2014) 99–120.

[10] E. H. Liu and L. J. Jin, The Ratio Log-concavity of the Cohen Numbers, J. Inequal. Appl., 2016 (2016) pp. 9.

[11] F.-Z. Zhao, The Log-convexity of Genocchi Numbers and the Monotonicity of Some Sequences Related to Genocchi Numbers, J. Math. Inequal., 10 (2016) 541–550.

[12] J. P. Barthelemy, An Asymptotic Equivalent for the Number of Total Preorders on a Finite Set, Discrete Math., 29 (1980) 311–313.

[13] M. Aigner, Combinatorial Theory, Grundlehren der Mathematischen Wissenschaften, 234, Springer, Berlin, 1979.

[14] A. Knopfmacher and M. E. Mays, A Survey of Factorization Counting Functions, Int. J. Number Theory, 1 (2005) 563–581.

[15] Y. Wang and B.-X. Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial se-quences, Science China Math., 57 no. 11 (2014) 2429–2435.