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The monotonicity of ratios involving arc tangent function with applications. (English) Zbl 1513.33002


MSC:

33B10 Exponential and trigonometric functions
26D05 Inequalities for trigonometric functions and polynomials

References:

[1] Shafer R.E., Elementary problems: E 1867, Amer. Math. Monthly, 1966, 73, 309.
[2] Shafer R.E., Grinstein L.S., Marsh D.C.B., Konhauser J.D.E., Problems and solutions: Solutions of elementary problems: E1867, Amer. Math. Monthly, 1967, 74, 726-727.
[3] Qi F., Zhang S.-Q., Guo B.-N., Sharpening and generalizations of Shafer’s inequality for the arc tangent function, J. Inequal. Appl., 2009, 2009:930294. · Zbl 1175.26049
[4] Chen C.-P., Cheung W.-S., Wang W., On Shafer and Carlson inequalities, J. Inequal. Appl., 2011, 2011:840206. · Zbl 1230.26008
[5] Mortici C., Srivastava H.M., Estimates for the arctangent function related to Shafer’s inequality, Colloq. Math., 2014, 136, 263-270. · Zbl 1305.26010
[6] Malešević B., Rašajski M., Lutovac T., Refined estimates and generalizations of inequalities related to the arctangent function and Shafer’s inequality, Math. Probl. Eng., 2018, 2018:4178629. · Zbl 1427.26004
[7] Shafer R.E., Analytic inequalities obtained by quadratic approximation, Publ. Elektroteh. Fak. Univ. Beogr. Ser. Mat. Fiz., 1977, 96-97, 577-598. · Zbl 0379.65012
[8] Zhu L., On a quadratic estimate of Shafer, J. Math. Inequal., 2008, 2, 571-574. · Zbl 1165.26314
[9] Alirezaei G., A sharp double inequality for the inverse tangent function, arXiv:1307.4983v1.
[10] Nishizawa Y., Refined quadratic estimations of Shafer’s inequality, J. Inequal. Appl., 2017, 2017(40). · Zbl 1357.26042
[11] Sun J.-L., Chen C.-P., Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions, J. Inequal. Appl., 2016, 2016:212. · Zbl 1346.26005
[12] Qiao Q.-X., Chen C.-P., Approximations to inverse tangent function, J. Inequal. Appl., 2018, 2018:141. · Zbl 1498.26028
[13] Gasull A., Utzet F., Approximating Mills ratio, J. Math. Anal. Appl., 2014, 420, 1832-1853. · Zbl 1415.60014
[14] Tian J.-F., Triple Diamond-Alpha integral and Hölder-type inequalities, J. Inequal. Appl., 2018, 2018:111. · Zbl 1497.26036
[15] Wang M.K., Zhang W., Chu Y.M., Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta Math. Sci. Ser. B (Engl. Ed.), 2019, 39, 1440-1450. · Zbl 1499.33072
[16] Tian J.-F., Zhu Y.-R., Cheung W.-S., N-tuple Diamond-Alpha integral and inequalities on time scales, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 2019, 113, 2189-2200. · Zbl 1429.26048
[17] Tian J.-F., Ha M.-H., Wang C., Improvements of generalized Hölder’s inequalities and their applications, J. Math. Inequal., 2018, 12, 459-471. · Zbl 1391.26060
[18] Yang Z.-H., Tian J.-F., A class of completely mixed monotonic functions involving the gamma function with applications, Proc. Amer. Math. Soc., 2018, 146, 4707-4721. · Zbl 1408.33006
[19] Yang Z.-H., A new way to prove L’Hospital monotone rules with applications, arXiv:1409.6408 [math.CA].
[20] Yang Z.-H., Chu Y.-M., A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 2017, 20, 729-735. · Zbl 1375.33026
[21] Yang Z.-H., Chu Y.-M., Wang M.-K., Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl., 2015, 428, 587-604. · Zbl 1321.26019
[22] Yang Z.-H., Zhang W., Chu Y.-M., Sharp Gautschi inequality for parameter 0 < p < 1 with applications, Math. Inequal. Appl., 2017, 20, 1107-1120. · Zbl 1386.33005
[23] Anderson G.D., Vamanamurthy M., Vuorinen M., Monotonicity rules in calculus, Amer. Math. Monthly, 2016, 113, 805-816. · Zbl 1167.26306
[24] Biernacki M., Krzyz J., On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sklodowska, 1995, 9, 135-145.
[25] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. · Zbl 0515.33001
[26] Mortici C., The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl., 2011, 14, 535-541. · Zbl 1222.26020
[27] Malešević B., Lutovac T., Rašajski M., Mortici C., Extensions of the natural approach to refinements, and generalizations of some trigonometric inequalities, Adv. Differ. Equ., 2018, 2018:90. · Zbl 1445.26015
[28] Nenezic M., Zhu L., Some improvements of Jordan-Steckin and Becker-Stark inequalities, Appl. Anal. Discr. Math., 2018, 12, 244-256. · Zbl 1499.41020
[29] Malesevic B., Makragic M., A method for proving some inequalities on mixed trigonometric polynomial functions, J. Math. Inequal., 2016, 10, 849-876. · Zbl 1351.26030
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