Abstract
The Bhatia—Li mean \(\mathcal {B}_{p}\left( x,y\right) \) of positive numbers x and y is defined as
where \(B\left( \cdot ,\cdot \right) \) is the Beta function. This new family of means includes the famous logarithmic mean, the Gaussian arithmetic-geometric mean etc. In 2012, Bhatia and Li conjectured that \(\mathcal {B}_{p}\left( x,y\right) \) is an increasing function of the parameter p on \(\left[ 0,\infty \right] \). In this paper, we give a positive answer to this conjecture. Moreover, the mean \(\mathcal {B} _{p}\left( x,y\right) \) is generalized to an multivariate mean and its elementary properties are investigated.
Similar content being viewed by others
References
Carlson, B.C.: A hypergeometric mean value. Proc. Am. Math. Soc. 16(4), 759–766 (1965)
Carlson, B.C.: Lauricella’s hypergeometric function \(F_{D}\). J. Math. Anal. Appl. 7, 452–470 (1963)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Carlson, B.C.: Special Functions of Applied Mathematics. Academic Press, New York (1977)
Carlson, B.C.: Some inequalities for hypergeometric functions. Proc. Am. Math. Soc. 17(1), 32–39 (1966)
Stolarsky, K.B.: Generalizations of the logarithmic mean. Math. Mag. 48, 87–92 (1975)
Yang, Z.-H.: On the log-convexity of two-parameter homogeneous functions. Math. Inequal. Appl. 10(3), 499–516 (2007)
Yang, Z.-H.: On the monotonicity and log-convexity of a four-parameter homogeneous mean. J. Inequal. Appl. 149286, 12 (2008)
Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, Hoboken (1987)
Bhatia, R., Li, R.: An interpolating family of means. Commun. Stoch. Anal. 6(1), 15–31 (2012)
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing. Dover Publications, New York and Washington (1972)
Batir, N.: Inequalities for the gamma function. Arch. Math. 91(6), 54–63 (2008)
Yang, Z.-H., Chu, Y.-M., Zhang, X.-H.: Sharp bounds for psi function. Appl. Math. Comput. 268, 1055–1063 (2015)
Yang, Z.-H., Chu, Y.-M., Tao, X.-J.: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 702718, 9 (2014)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, 210 (2017)
Yang, Z.-H., Tian, J.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1–22 (2018)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462, 1714–1726 (2018)
Yang, Z.-H.: Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means. J. Math. Anal. Appl. 467, 446–461 (2018)
Yang, Z., Tian, J.-F.: Monotonicity rules for the ratio of two Laplace transforms with applications. J. Math. Anal. Appl. 470, 821–845 (2019)
Yang, Z.-H., Tian, J.: Convesity and monotonicity for elliptic integrals of the first kind and applications. Appl. Anal. Discrete Math. 13, 240–260 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the 60th anniversary of Zhejiang Electric Power Company Research Institute
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yang, ZH., Tian, JF. & Wang, MK. A positive answer to Bhatia—Li conjecture on the monotonicity for a new mean in its parameter. RACSAM 114, 126 (2020). https://doi.org/10.1007/s13398-020-00856-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00856-w