Abstract
This paper presents an interpolation-based method for bounding some smooth functions, including the arctangent functions related to Shafer’s inequality. Given the form of new bounding functions, the interpolation technique is also utilized for determining the corresponding unknown coefficients, and the resulting functions bound the given function very well under some preset condition. Two applications are shown, one is to refine Shafer’s inequality, and the other is to approximate the definite integrals of some special functions; both of them have wide applications in computer science, mathematics, physical sciences and engineering. Experimental results show that the new bounds achieve much better bounds than those of prevailing methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrievskii, V., Blatt, H.P.: On approximation of continuous functions by trigonometric polynomials. J. Approx. Theory 163(2), 249–266 (2011)
Abdel-Raouf, O., Abdel-Baset, M., El-Henawy, I.: Chaotic firefly algorithm for solving definite integrals. Int. J. Inf. Technol. Comput. Sci. 6(6), 19–24 (2014)
Atluri, S.N., Nakagaki, M.: J-integral estimates for strain-hardening materials in ductile fracture problems. AIAA J. 15(7), 923–931 (2015)
Bhayo, B.A., Ozsef, J., Andor, S.: On certain old and new trigonometric and hyperbolic inequalities. Anal. Math. 41(1–2), 3–15 (2015)
Chen, C.-P., Cheung W.-S., Wang, W.-S.: On Shafer and Carlson inequalities, J. Inequal. Appl., Article ID 840206, 110 (2011)
Davis, P.J.: Interpolation and Approximation. Dover Publications, New York (1975)
Guo, B.-N., Luo, Q.-M., Qi, F.: Sharpening and generalizations of Shafer-Finks double inequality for the arc sine function. Filomat 27, 261–265 (2013)
Hasegawa, T., Sugiura, H.: A user-friendly method for computing indefinite integrals of oscillatory functions. J. Comput. Appl. Math. 315, 126–141 (2017)
Kuang, J.-C.: Applied Inequalities, 3rd edn. Shangdong Science and Technology Press, Ji’nan City (2004). (in Chinese)
Malešević, B.J.: An application of \(\lambda \)-method on inequalities of ShaferFinks type. Math. Inequal. Appl. 10, 529534 (2007)
Malešević, B. J.: One method for proving inequalities by computer. J. Inequal. Appl., Article ID 78691, 1–8 (2007)
Maleknejad, K., Saeedipoor, E.: An efficient method based on hybrid functions for Fredholm integral equation of the first kind with convergence analysis. Appl. Math. Comput. 304, 93–102 (2017)
Mortici, C., Srivastava, H.M.: Estimates for the arctangent function related to Shafer’s inequality. Colloq. Math. 136(2), 263–270 (2014)
Nenezic, M., Malesevic, B., Mortici, C.: New approximations of some expressions involving trigonometric functions. Appl. Math. Comput. 283, 299–315 (2016)
Qi, F., Guo, B.-N.: Sharpening and generalizations of Shafer’s inequality for the arc sine function. Integral Transforms Spec. Funct. 23, 129–134 (2012)
Qi, F., Zhang, S.-Q., Guo, B.-N.: Sharpening and generalizations of Shafers inequality for the arc tangent function. J. Inequal. Appl., Article ID 930294, 1–10 (2009)
Shafer, R.E., Problem, E.: Am. Math. Month. 73(1966), 309–309 (1867)
Shadimetov, K., Hayotov, A.R.: Optimal quadrature formulas with positive coefficients in \(L^{(m)}_2 (0, 1)\) space. J. Comput. Appl. Math. 235, 1114–1128 (2011)
Yang, C.Y.: Inequalities on generalized trigonometric and hyperbolic functions. J. Math. Anal. Appl. 419(2), 775–782 (2014)
Zhu, L.: On Shafer-Fink type inequality. J. Inequal. Appl., Article ID 67430, 1–4 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, XD., Jin, S., Li-Geng, C. et al. A New Method for Refining the Shafer’s Equality and Bounding the Definite Integrals. Results Math 73, 78 (2018). https://doi.org/10.1007/s00025-018-0836-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0836-3