L’Hôpital’s monotone rule, Gromov’s theorem, and operations that preserve the monotonicity of quotients. (English) Zbl 1474.26036
Summary: We study several operators \(T\) that when applied to both the numerator and denominator of an increasing, or decreasing, function \(u/v\) produce another increasing, or decreasing, function \(T(u)/T(v)\). We also give new proofs of the monotone form of L’Hôpital’s rule and of Gromov’s theorem.
MSC:
| 26A48 | Monotonic functions, generalizations |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
References:
| [1] | G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen,Inequalities for quasiconformal mappings in space, Pac. J. Math.192(1993), 1-18. · Zbl 0793.30014 |
| [2] | R. G. Bartle,A Modern Theory of Integration, Amer. Math. Soc., Providence, 2001. · Zbl 0968.26001 |
| [3] | M. Biernacki, J. Krzyż,On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska2(1955), 134-145. |
| [4] | E. Cesàro,Sur la convergence des séries, Nouv. Ann. (3) VII (1888), 49-59. · JFM 20.0242.01 |
| [5] | J. Cheeger, M. Gromov, M. Taylor,Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemann manifolds, J. Differ. Geom. 17(1982), 15-53. · Zbl 0493.53035 |
| [6] | R. Estrada, J. Vindas,A general integral, Diss. Math.483(2012), 49 pages. · Zbl 1245.26006 |
| [7] | R. A. Gordon,The integrals of Lebesgue, Denjoy, Perron, and Henstock, Am. Math. Soc., Providence, 1994. · Zbl 0807.26004 |
| [8] | G. H. Hardy,The mean value of the modulus of an analytic function, Proc. Lond. Math. Soc. (2)14(1915), 269-277. · JFM 45.1331.03 |
| [9] | V. Heikkala, M. K. Vamanamurthy, M. Vuorinen,Generalized elliptic integrals, Comput. Methods Funct. Theory9(2009), 75-109. · Zbl 1198.33002 |
| [10] | M. Pavlović,A simple proof of the monotone form of L’Hospital’s rule. Gromov’s monotonicity theorem and a generalization, Research Gate, Technical Report, 2015. |
| [11] | I. Pinelis,L’Hospital rules for oscillation, with applications, JIPAM, J. Inequal. Pure Appl. Math.3(2002), Article 33, 24 pp., (electronic). |
| [12] | O. Stolz,Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten, Teubners, Leipzig, 1885. · JFM 17.0116.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
