Harmonic Mean Inequalities for Hyperbolic Functions
Abstract
Inequalities involving hyperbolic functions have been the subject of intense discussion in recent times. In this work, we establish harmonic mean inequalities for these functions. This complements the results known in the literature. The techniques adopted in proving our results are analytical in nature.
References
H. Alzer, A harmonic mean inequality for the gamma function, J. Comp. Appl. Math. 87 (1997), 195-198. https://doi.org/10.1016/S0377-0427(96)00181-1
H. Alzer, Inequalities for the gamma function, Proc. Amer. Math. Soc. 128 (1999), 141-147. https://doi.org/10.1090/S0002-9939-99-04993-X
H. Alzer, On a gamma function inequality of Gautschi, Proc. Edinburgh Math. Soc. 45 (2002), 589-600. https://doi.org/10.1017/S0013091501000943
H. Alzer, On Gautschi’s harmonic mean inequality for the gamma function, J. Comp. Appl. Math. 157 (2003), 243-249. https://doi.org/10.1016/S0377-0427(03)00456-4
H. Alzer, Inequalities involving Г(x) and Г(1/x), J. Comp. Appl. Math. 192 (2006), 460-480. https://doi.org/10.1016/j.cam.2005.04.063
H. Alzer, Gamma function inequalities, Numer. Algor. 49 (2008), 53-84. https://doi.org/10.1007/s11075-008-9160-4
H. Alzer and G. Jameson, A harmonic mean inequality for the digamma function and related results, Rend. Sem. Mat. Univ. Padova. 137 (2017), 203-209. https://doi.org/10.4171/RSMUP/137-10
M. Bouali, A harmonic mean inequality for the q-gamma and q-digamma functions, arXiv:2005.08945 [math.CA].
W. Gautschi, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278-281. https://doi.org/10.1137/0505030
C. Giordano and A. Laforgia, Inequalities and monotonicity properties for the gamma function, J. Comp. Appl. Math. 133 (2001), 387-396. https://doi.org/10.1016/S0377-0427(00)00659-2
G. J. O. Jameson and T. P. Jameson, An inequality for the gamma function conjectured by D. Kershaw, J. Math. Ineq. 6 (2012), 175-181. https://doi.org/10.7153/jmi-06-18
L. Matejicka, Proof of a conjecture On Nielsen’s β-function, Probl. Anal. Issues Anal. 8(26) (2019), 105-111. https://doi.org/10.15393/j3.art.2019.6810
K. Nantomah, On some properties and inequalities for the Nielsen’s β-function, Sci. Ser. A Math. Sci. (N.S.) 28 (2017-2018), 43-54.
K. Nantomah, Certain properties of the Nielsen’s β-function, Bull. Int. Math. Virtual Inst. 9 (2019), 263-269.
E. Yildirim, Monotonicity properties on k-digamma function and its related inequalities, J. Math. Inequal. 14(1) (2020), 161-173. https://doi.org/10.7153/jmi-2020-14-12
L. Yin, L-G. Huang, X-L. Lin and Y-L. Wang, Monotonicity, concavity, and inequalities related to the generalized digamma function, Adv. Difference Equ. (2018) 2018:246. https://doi.org/10.1186/s13662-018-1695-7
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
