Abstract
We present various properties of the Erlang loss function
Among other results, we prove:
-
(1)
The function \(x\mapsto B(x,a)^{\lambda}\) is convex on \([0,\infty)\) for every \(a>0\) if and only if \(\lambda\leq 0\) or \(\lambda \geq 1\).
-
(2)
The function \(x\mapsto ({1-B(1/x,a)})^{-1} \) is strictly convex on \((0,\infty)\). This leads to the functional inequality
$$\frac{2}{1-B( H(x,y) ,a)}< \frac{1}{1-B(x,a)} +\frac{1}{1-B(y,a)}\quad{(x,y>0,\ x\neq y)}, $$where \(H(x,y)=2xy/(x+y)\) denotes the harmonic mean of x and y.
-
(3)
Let \(a>0\). The inequality \(B(x,a) + B(1/x,a)\leq 1\) holds for all \(x>0\) if and only if \(a\leq 1\).
Similar content being viewed by others
References
Brenner, J.L.: Analytic inequalities with applications to special functions. J. Math. Anal. Appl. 106, 427–442 (1985)
Gautschi, W.: A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5, 278–281 (1974)
G. Giambene, Queuing Theory and Telecommunications, Springer (New York, 2014)
Jagerman, D.L.: Some properties of the Erlang loss function. Bell System Tech. J. 53, 525–551 (1974)
Jagers, A.A., van Doorn, E.A.: On the continued Erlang loss function. Oper. Res. Letters 5, 43–46 (1986)
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Birkhäuser (Basel, 2009)
Medhi, J.: The evergreen Erlang loss function. Opsearch 43, 309–319 (2006)
D. S. Mitrinović, Analytic Inequalities, Springer (New York, 1970)
Montel, P.: Sur les fonctions convexes et les fonctions sousharmoniques. J. Math. Pure Appl. 7, 29–60 (1928)
Petrović, M.: Sur une équation fonctionelle. Publ. Math. Univ. Belgrade 1, 149–156 (1932)
Acknowledgement
We thank the referee for the careful reading of our manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alzer, H., Kwong, M.K. On the Erlang loss function. Acta Math. Hungar. 162, 14–31 (2020). https://doi.org/10.1007/s10474-020-01046-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01046-1