Monotonicity criterion for the quotient of power series with applications. (English) Zbl 1321.26019
Summary: In this paper, we present the necessary and sufficient condition for the monotonicity of the quotient of power series. As applications, some gaps and misquotations in certain published articles are pointed out and corrected, and some known results involving the Landen inequalities for zero-balanced hypergeometric functions are improved.
MSC:
| 26A48 | Monotonic functions, generalizations |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
quotient of power series; piecewise monotonicity; hypergeometric function; Landen inequalityReferences:
| [1] | Almkvist, G.; Berndt, B., Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, \(π\), and the Ladies diary, Amer. Math. Monthly, 95, 7, 585-608 (1988) · Zbl 0665.26007 |
| [2] | Alzer, H.; Qiu, S.-L., Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 172, 2, 289-312 (2004) · Zbl 1059.33029 |
| [3] | Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M. K., Hypergeometric functions and elliptic integrals, (Srivastava, H. M.; Owa, S., Current Topics in Analytic Function Theory (1992), World Scientific Publ. Co.: World Scientific Publ. Co. Singapore-London), 48-85 · Zbl 0984.33502 |
| [4] | Anderson, G. D.; Barnard, R. W.; Richards, K. C.; Vamanamurthy, M. K.; Vuorinen, M., Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc., 347, 5, 1713-1723 (1995) · Zbl 0826.33003 |
| [5] | Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M., Generalized convexity and inequalities, J. Math. Anal. Appl., 335, 2, 1294-1308 (2007) · Zbl 1125.26017 |
| [6] | Anderson, G. D.; Vuorinen, M.; Zhang, X.-H., Topics in special functions III, (Analytic Number Theory, Approximation Theory, and Special Functions (2014)), 297-345 · Zbl 1323.33001 |
| [7] | Balasubramanian, R.; Ponnusamy, S.; Vuorinen, M., Functional inequalities for the quotients of hypergeometric functions, J. Math. Anal. Appl., 218, 1, 256-268 (1998) · Zbl 0890.33004 |
| [8] | Baricz, Á., Landen-type inequality for Bessel functions, Comput. Methods Funct. Theory, 5, 2, 373-379 (2005) · Zbl 1104.33002 |
| [9] | Baricz, Á., Landen inequalities for special functions, Proc. Amer. Math. Soc., 142, 9, 3059-3066 (2014) · Zbl 1309.26020 |
| [10] | Baricz, Á.; Vesti, J.; Vuorinen, M., On Kaluza’s sign criterion for reciprocal power series, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 65, 2, 1-16 (2011) · Zbl 1260.30002 |
| [11] | Belzunce, F.; Ortega, E.-M.; Ruiz, J. M., On non-monotonic ageing properties from the Laplace transform, with actuarial applications, Insurance Math. Econom., 40, 1, 1-14 (2007) · Zbl 1273.91231 |
| [12] | Biernacki, M.; Krzyż, J., On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 9, 135-147 (1955) · Zbl 0078.26402 |
| [13] | Editorial Committee of Mathematics Handbook, Mathematics Handbook (1979), Peoples’ Education Press: Peoples’ Education Press Beijing, China, (Chinese) |
| [14] | Evans, R. J., Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related functions, (Ramanujan Revisited. Ramanujan Revisited, Urbana-Champaign, Ill., 1987 (1988), Academic Press: Academic Press Boston, MA), 537-560 · Zbl 0646.33003 |
| [15] | Heikkala, V.; Vamanamurthy, M. K.; Vuorinen, M., Generalized elliptic integrals, Comput. Methods Funct. Theory, 9, 1, 75-109 (2009) · Zbl 1198.33002 |
| [16] | Ponnusamy, S.; Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44, 2, 278-301 (1997) · Zbl 0897.33001 |
| [17] | Qian, W.-M.; Chu, Y.-M., On certain inequalities for Neuman-Sándor mean, Abstr. Appl. Anal., 2013 (2013), Article ID 790783, 6 pp · Zbl 1276.26060 |
| [18] | Qiu, S.-L.; Vuorinen, M., Landen inequalities for hypergeometric functions, Nagoya Math. J., 154, 31-56 (1999) · Zbl 0944.33004 |
| [19] | Simić, S.; Vuorinen, M., Landen inequalities for zero-balanced hypergeometric functions, Abstr. Appl. Anal., 2012 (2012), Article ID 932061, 11 pp · Zbl 1241.26021 |
| [20] | Sun, H.; Zhao, T.-H.; Chu, Y.-M.; Liu, B.-Y., A note on the Neuman-Sándor mean, J. Math. Inequal., 8, 2, 287-297 (2014) · Zbl 1297.26069 |
| [21] | Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1962), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 0105.26901 |
| [22] | Yang, Zh.-H., New sharp bounds for identric mean in terms of logarithmic mean and arithmetic mean, J. Math. Inequal., 6, 4, 533-543 (2012) · Zbl 1257.26032 |
| [23] | Yang, Zh.-H., A new way to prove L’Hospital Monotone Rules with applications, available online at |
| [24] | Yang, Zh.-H.; Jiang, Y.-L.; Song, Y.-Q.; Chu, Y.-M., Sharp inequalities for trigonometric functions, Abstr. Appl. Anal., 2014 (2014), Article ID 601839, 18 pp · Zbl 1474.26058 |
| [25] | Zhang, F.; Chu, Y.-M.; Qian, W.-M., Bounds for the arithmetic mean in terms of the Neuman-Sándor and other bivariate means, J. Appl. Math., 2013 (2013), Article ID 582504, 7 pp · Zbl 1397.26017 |
| [26] | Zhao, T.-H.; Chu, Y.-M.; Liu, B.-Y., Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means, Abstr. Appl. Anal., 2012 (2012), Article ID 302635, 9 pp · Zbl 1256.26018 |
| [27] | Zhu, L., Sharpening Jordan’s inequality and Yang Le inequality, Appl. Math. Lett., 19, 2, 240-243 (2006) · Zbl 1097.26012 |
| [28] | Zhu, L., New inequalities for hyperbolic functions and their applications, J. Inequal. Appl., 2012, 303 (2012), 9 pp · Zbl 1279.26067 |
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