Sharp Shafer-Fink type inequalities for Gauss lemniscate functions. (English) Zbl 1332.26029
Summary: In this paper, we establish sharp Shafer-Fink type inequalities for Gauss lemniscate functions.
MSC:
| 26D07 | Inequalities involving other types of functions |
References:
| [1] | Borwein JM, Borwein PB: Pi and the AGM: A Study in the Analytic Number Theory and Computational Complexity. Wiley, New York; 1987. · Zbl 0611.10001 |
| [2] | Carlson BC: Algorithms involving arithmetic and geometric means.Am. Math. Mon. 1971, 78:496-505. · Zbl 0218.65035 · doi:10.2307/2317754 |
| [3] | Neuman E: On Gauss lemniscate functions and lemniscatic mean.Math. Pannon. 2007, 18:77-94. · Zbl 1164.33007 |
| [4] | Neuman E: Two-sided inequalities for the lemniscate functions.J. Inequal. Spec. Funct. 2010, 1:1-7. · Zbl 1312.33055 |
| [5] | Neuman E: On Gauss lemniscate functions and lemniscatic mean II.Math. Pannon. 2012, 23:65-73. · Zbl 1289.33013 |
| [6] | Neuman E: Inequalities for Jacobian elliptic functions and Gauss lemniscate functions.Appl. Math. Comput. 2012, 218:7774-7782. · Zbl 1241.33018 · doi:10.1016/j.amc.2012.01.041 |
| [7] | Neuman E: On lemniscate functions.Integral Transforms Spec. Funct. 2013, 24:164-171. · Zbl 1273.26020 · doi:10.1080/10652469.2012.684054 |
| [8] | Siegel, CL, 1 (1969), New York |
| [9] | Mitrinović DS, Vasić PM: Analytic Inequalities. Springer, New York; 1970. · Zbl 0199.38101 · doi:10.1007/978-3-642-99970-3 |
| [10] | Fink AM: Two inequalities.Publ. Elektroteh. Fak. Univ. Beogr., Mat. 1995, 6:48-49. · Zbl 0841.26009 |
| [11] | Malešević BJ: Application ofλ-method on Shafer-Fink’s inequality.Publ. Elektroteh. Fak. Univ. Beogr., Mat. 1997, 8:103-105. · Zbl 0886.26009 |
| [12] | Malešević, BJ, One method for proving inequalities by computer, No. 2007 (2007) · Zbl 1133.26320 |
| [13] | Malešević BJ: An application ofλ-method on inequalities of Shafer-Fink’s type.Math. Inequal. Appl. 2007, 10:529-534. · Zbl 1127.26010 |
| [14] | Zhu L: On Shafer-Fink inequalities.Math. Inequal. Appl. 2005, 8:571-574. · Zbl 1084.26007 |
| [15] | Zhu, L., On Shafer-Fink-type inequality, No. 2007 (2007) · Zbl 1133.26303 |
| [16] | Pan, W.; Zhu, L., Generalizations of Shafer-Fink-type inequalities for the arc sine function, No. 2009 (2009) · Zbl 1175.26046 |
| [17] | Zhu L: A solution of a problem of Oppenheim.Math. Inequal. Appl. 2007, 10:57-61. · Zbl 1115.26009 |
| [18] | Oppenheim A: E1277.Am. Math. Mon. 1957, 64:504. · doi:10.2307/2308467 |
| [19] | Chen CP, Cheung WS: Wilker- and Huygens-type inequalities and solution to Oppenheim’s problem.Integral Transforms Spec. Funct. 2012, 23:325-336. · Zbl 1254.26022 · doi:10.1080/10652469.2011.586637 |
| [20] | Qi F, Guo BN: Sharpening and generalizations of Shafer’s inequality for the arc sine function.Integral Transforms Spec. Funct. 2012, 23:129-134. · Zbl 1247.33002 · doi:10.1080/10652469.2011.564578 |
| [21] | Qi F, Guo BN: Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function.Filomat 2013, 27:261-265. · Zbl 1324.33004 · doi:10.2298/FIL1304601Q |
| [22] | Shafer RE: Problem E1867.Am. Math. Mon. 1966, 73:309. · doi:10.2307/2315358 |
| [23] | Shafer RE: Problems E1867 (solution).Am. Math. Mon. 1967, 74:726-727. · doi:10.2307/2314285 |
| [24] | Kuang JC: Applied Inequalities. 3rd edition. Shangdong Science and Technology Press, Jinan; 2004. |
| [25] | Qi, F.; Zhang, SQ; Guo, BN, Sharpening and generalizations of Shafer’s inequality for the arc tangent function, No. 2009 (2009) · Zbl 1175.26049 |
| [26] | Chen, CP; Cheung, WS; Wang, W., On Shafer and Carlson inequalities, No. 2011 (2011) · Zbl 1230.26008 |
| [27] | Zhu, L., New inequalities of Shafer-Fink type for arc hyperbolic sine, No. 2008 (2008) · Zbl 1165.26350 |
| [28] | Chen CP: Wilker and Huygens type inequalities for the lemniscate functions.J. Math. Inequal. 2012, 6:673-684. · Zbl 1268.26025 · doi:10.7153/jmi-06-65 |
| [29] | Chen CP: Wilker and Huygens type inequalities for the lemniscate functions, II.Math. Inequal. Appl. 2013, 16:577-586. · Zbl 1259.33031 |
| [30] | Anderson GD, Qiu SL, Vamanamurthy MK, Vuorinen M: Generalized elliptic integral and modular equations.Pac. J. Math. 2000, 192:1-37. · Zbl 0951.33012 · doi:10.2140/pjm.2000.192.1 |
| [31] | Anderson GD, Vamanamurthy MK, Vuorinen M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York; 1997. · Zbl 0885.30012 |
| [32] | Anderson GD, Vamanamurthy MK, Vuorinen M: Monotonicity rules in calculus.Am. Math. Mon. 2006, 113:805-816. · Zbl 1167.26306 · doi:10.2307/27642062 |
| [33] | Lindqvist P: Some remarkable sine and cosine functions.Ric. Mat. 1995, 44:269-290. · Zbl 0944.33002 |
| [34] | Biezuner RJ, Ercole G, Martins EM: Computing the first eigenvalue of thep-Laplacian via the inverse power method.J. Funct. Anal. 2009, 257:243-270. · Zbl 1172.35047 · doi:10.1016/j.jfa.2009.01.023 |
| [35] | Bushell PJ, Edmunds DE: Remarks on generalised trigonometric functions.Rocky Mt. J. Math. 2012, 42:25-57. · Zbl 1246.33001 · doi:10.1216/RMJ-2012-42-1-25 |
| [36] | Drábek P, Manásevich R: On the closed solution to somep-Laplacian nonhomogeneous eigenvalue problems.Differ. Integral Equ. 1999, 12:773-788. · Zbl 1015.34071 |
| [37] | Lang, J.; Edmunds, DE, Lecture Notes in Mathematics 2016 (2011), Berlin · Zbl 1220.47001 · doi:10.1007/978-3-642-18429-1 |
| [38] | Takeuchi S: Generalized Jacobian elliptic functions and their application to bifurcation problems associated withp-Laplacian.J. Math. Anal. Appl. 2012, 385:24-35. · Zbl 1232.34029 · doi:10.1016/j.jmaa.2011.06.063 |
| [39] | Edmunds DE, Gurka P, Lang J: Properties of generalized trigonometric functions.J. Approx. Theory 2012, 164:47-56. · Zbl 1241.42019 · doi:10.1016/j.jat.2011.09.004 |
| [40] | Baricz, A.; Bhayo, BA; Klén, R., Convexity properties of generalized trigonometric and hyperbolic functions (2013) |
| [41] | Bhayo BA, Vuorinen M: On generalized trigonometric functions with two parameters.J. Approx. Theory 2012, 164:1415-1426. · Zbl 1257.33052 · doi:10.1016/j.jat.2012.06.003 |
| [42] | Klén R, Vuorinen M, Zhang X: Inequalities for the generalized trigonometric and hyperbolic functions.J. Math. Anal. Appl. 2014, 409:521-529. · Zbl 1306.33013 · doi:10.1016/j.jmaa.2013.07.021 |
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.
