×

Monotonicity properties and inequalities related to generalized Grötzsch ring functions. (English) Zbl 1427.33012

Summary: In the paper, the authors present some monotonicity properties and some sharp inequalities for the generalized Grötzsch ring function and related elementary functions. Consequently, the authors obtain new bounds for solutions of the Ramanujan generalized modular equation.

MSC:

33E05 Elliptic functions and integrals
26A48 Monotonic functions, generalizations
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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. · Zbl 0543.33001
[2] Alzer H., Qiu S.-L., Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 2004, 172(2), 289-312; Available online at doi:10.1016/j.cam.2004.02.009. · Zbl 1059.33029 · doi:10.1016/j.cam.2004.02.009
[3] Alzer H., Richards K., On the modulus of the Grötzsch ring, J. Math. Anal. Appl., 2015, 432(1), 134-141; Available online at doi:10.1016/j.jmaa.2015.06.057. · Zbl 1327.30021 · doi:10.1016/j.jmaa.2015.06.057
[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., 1995, 347(5), 1713-1723; Available online at doi:10.2307/2154966. · Zbl 0826.33003 · doi:10.2307/2154966
[5] Anderson G.-D., Qiu S.-L., Vamanamurthy M.-K., Elliptic integral inequalities, with applications, Constr. Approx., 1998, 14(2), 195-207; Available online at doi:10.1007/s003659900070. · Zbl 0901.33011 · doi:10.1007/s003659900070
[6] Anderson G.-D., Qiu S.-L., Vamanamurthy M.-K., Vuorinen M., Generalized elliptic integrals and modular equations, Pacific J. Math., 2000, 192(1), 1-37; Available online at doi:10.2140/pjm.2000.192.1. · Zbl 0951.33012 · doi:10.2140/pjm.2000.192.1
[7] Anderson G.-D., Vamanamurthy M.-K., Vuorinen M., Conformal Invariants, Inequalities, and Quasiconformal Maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1997. · Zbl 0885.30012
[8] Anderson G.-D., Vamanamurthy M.-K., Vuorinen M., Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM J. Math. Anal., 1992, 23(2), 512-524; Available online at doi:10.1137/0523025. · Zbl 0764.33009 · doi:10.1137/0523025
[9] András S., Baricz Á., Bounds for complete elliptic integral of the first kind, Expo. Math., 2010, 28(4), 357-364; Available online at doi:10.1016/j.exmath.2009.12.005. · Zbl 1204.33004 · doi:10.1016/j.exmath.2009.12.005
[10] Baricz Á., Turán type inequalities for generalized complete elliptic integrals, Math. Z., 2007, 256(4), 895-911; Available online at doi:10.1007/s00209-007-0111-x. · Zbl 1125.26022 · doi:10.1007/s00209-007-0111-x
[11] Byrd P.F., Friedman M.D., Handbook of Elliptic Integrals for Engineers and Scientists, Second edition, revised, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg 1971. · Zbl 0213.16602
[12] Chen C.-P., Qi F., The best bounds of the n-th harmonic number, Glob. J. Appl. Math. Math. Sci., 2008, 1(1), 41-49.
[13] Guo B.-N., Qi F., Sharp bounds for harmonic numbers, Appl. Math. Comput., 2011, 218(3), 991-995; Available online at doi:10.1016/j.amc.2011.01.089. · Zbl 1229.11027 · doi:10.1016/j.amc.2011.01.089
[14] Guo B.-N., Qi F., Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin), 2014, 34(2), 201-208; Available online at doi:10.1515/anly-2014-0001. · Zbl 1294.33003 · doi:10.1515/anly-2014-0001
[15] Guo B.-N., Qi F., Some bounds for the complete elliptic integrals of the first and second kind, Math. Inequal. Appl., 2011, 14(2), 323-334; Available online at doi:10.7153/mia-14-26. · Zbl 1217.26042 · doi:10.7153/mia-14-26
[16] Ma X.-Y., Chu Y.-M., Wang F., Monotonicity and inequalities for the generalized distortion function, Acta Math. Sci. Ser. B Engl. Ed., 2013, 33(6), 1759-1766; Available online at doi:10.1016/S0252-9602(13)60121-6. · Zbl 1313.33004 · doi:10.1016/S0252-9602(13)60121-6
[17] Neuman E., Inequalities and bounds for generalized complete elliptic integrals, J. Math. Anal. Appl., 2011, 373(1), 203-213; Available online at doi:10.1016/j.jmaa.2010.06.060. · Zbl 1206.33020 · doi:10.1016/j.jmaa.2010.06.060
[18] Niu D.-W., Zhang Y.-J., Qi F., A double inequality for the harmonic number in terms of the hyperbolic cosine, Turkish J. Anal. Number Theory, 2014, 2(6), 223-225; Available online at doi:10.12691/tjant-2-6-6. · doi:10.12691/tjant-2-6-6
[19] Qi F., Akkurt A., Yildirim H., Catalan numbers, k-gamma and k-beta functions, and parametric integrals, J. Comput. Anal. Appl., 2018, 25(6), 1036-1042.
[20] Qi F., Guo B.-N., Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 2017, 111(2), 425-434; Available online at doi:10.1007/s13398-016-0302-6. · Zbl 1360.33004 · doi:10.1007/s13398-016-0302-6
[21] Qi F., Huang Z., Inequalities of the complete elliptic integrals, Tamkang J. Math., 1998, 29(3), 165-169. · Zbl 0912.33012
[22] Qi F., Mahmoud M., Bounding the gamma function in terms of the trigonometric and exponential functions, Acta Sci. Math. (Szeged), 2017, 83(1-2), 125-141; Available online at doi:10.14232/actasm-016-813-x. · Zbl 1399.33002 · doi:10.14232/actasm-016-813-x
[23] Qi F., Shi X.-T., Liu F.-F., Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ. Sapientiae Math., 2016, 8(2), 282-297; Available online at doi:10.1515/ausm-2016-0019. · Zbl 1360.11052 · doi:10.1515/ausm-2016-0019
[24] Qiu S.-L., Grötzsch ring and Ramanujan’s modular equations, Acta Math. Sinica (Chin. Ser.), 2000, 43(2), 283-290. (Chinese) · Zbl 1005.11013
[25] Qiu S.-L., Monotonicity of ĽHôpitaľs rule with applications, J. Hangzhou Dianzi Univ., 1995, 15(4), 23-30. (Chinese)
[26] Qiu S.-L., Singular values, quasiconformal maps and the Schottky upper bound, Sci. China Ser. A, 1998, 41(12), 1241-1247; Available online at doi:10.1007/BF02882264. · Zbl 0972.30011 · doi:10.1007/BF02882264
[27] Qiu S.-L., Ma X.-Y., Huang T.R., Some properties of the difference between the Ramanujan constant and beta function, J. Math. Anal. Appl., 2017, 446(1), 114-129; Available online at doi:10.1016/j.jmaa.2016.08.043. · Zbl 1350.26024 · doi:10.1016/j.jmaa.2016.08.043
[28] Qiu S.-L., Vuorinen M., Infinite products and normalized quotients of hypergeometric functions, SIAM J. Math. Anal., 1999, 30(5), 1057-1075; Available online at doi:10.1137/S0036141097326805. · Zbl 0931.30011 · doi:10.1137/S0036141097326805
[29] Qiu S.-L., Vuorinen M., Special functions in geometric function theory, Handbook of Complex Analysis: Geometric Function Theory., Vol. 2, 621-659, Elsevier Sci. B. V., Amsterdam, 2005; Available online at doi:10.1016/S1874-5709(05)80018-6. · Zbl 1073.30007 · doi:10.1016/S1874-5709(05)80018-6
[30] Vuorinen M., Singular values, Ramanujan modular equations, and Landen transformations, Studia Math., 1996, 121(3), 221-230; Available online at doi:10.4064/sm-121-3-221-230. · Zbl 0872.30010 · doi:10.4064/sm-121-3-221-230
[31] Wang F., Ma X.-Y., Zhou P.-G., Monotonicity and inequalities for the generalized complete elliptic integrals, College Math., 2016, 3, 77-82. (Chinese)
[32] Wang M.-K., Qiu S.-L., Chu Y.-M., Jiang Y.-P., Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221-229; Available online at doi:10.1016/j.jmaa.2011.06.039 · Zbl 1229.33024 · doi:10.1016/j.jmaa.2011.06.039
[33] Wang G.-D., Zhang X.-H., Jiang Y. P., Concavity with respect to Hölder means involving the generalized Grötzsch function, J. Math. Anal. Appl., 2011, 379(1), 200-204; Available online at doi:10.1016/j.jmaa.2010.12.055. · Zbl 1218.30067 · doi:10.1016/j.jmaa.2010.12.055
[34] Wang G.-D., Zhang X.-H., Qiu S.-L., Chu Y.-M., The bounds of the solutions to generalized modular equations, J. Math. Anal. Appl., 2006, 321(2), 589-594; Available online at doi:10.1016/j.jmaa.2005.08.064. · Zbl 1098.33015 · doi:10.1016/j.jmaa.2005.08.064
[35] Yin L., Qi F., Some inequalities for complete elliptic integrals, Appl. Math. E-Notes, 2014, 14, 192-199. · Zbl 1321.33019
[36] Zhang X.-H., On the generalized modulus, J. Ramanujan., 2017, 43(2), 403-415; Available online at doi:10.1007/s11139-015-9746-0. · Zbl 1375.33009 · doi:10.1007/s11139-015-9746-0
[37] Zhang X.-H., Wang G.-D., Chu Y.-M., Some inequalities for the generalized Grötzsch functions, In: Proc. Edinb. Math. Soc., 2008, 51(1), 265-272; Available online at doi:10.1017/S001309150500132X. · Zbl 1135.33009 · doi:10.1017/S001309150500132X
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.