Qiu Ming LUO, Bai Ni GUO, Feng QI, Jiao Lian ZHAO

Remarks on inequalities for the tangent function

In the paper, the authors analyze and compare two double inequali- ties for bounding the tangent function, reorganize the proof in C.-P. Chen and F. Qi (A double inequality for remainder of power series of tangent function, Tamkang J. Math. 34 (4), 351–355, 2003) by using the usual definition of Bernoulli numbers, and correct some errors on page 6, (1.29) and (1.30) of F. Qi, D. -W. Niu, and B. -N. Guo (Re- finements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages, 2009). Moreover, the authors propose a sharp double inequality as a conjecture.

PDF

___

[1] Abramowitz, M. and Stegun, I.A. (Eds) Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables (National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970).
[2] Andrews, G. E., Askey, R. and Roy, R. Special Functions (Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999).
[3] Becker, M. and Strak, E. L. On a hierarchy of polynomial inequalities for tan x, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602-633, 133–138, 1978.
[4] Chen, C. -P. and Qi, F. A double inequality for remainder of power series of tangent function, Tamkang J. Math. 34 (4), 351–355, 2003; Available online at http://dx.doi.org//10.5556/j.tkjm.34.2003.351-355
[5] Guo, B. -N. and Qi, F. A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2), 21–30, 2010.
[6] Guo, B.-N. and Qi, F. Sharpening and generalizations of Carlson’s inequality for the arc cosine function, Hacet. J. Math. Stat. 39 (3), 403–409, 2010.
[7] Kuang, J. -C. Ch´angy`ong B`udˇengsh`ı (Applied Inequalities), 3rd ed. (Shandong Science and Technology Press, Ji’nan City, Shandong Province, China, 2004). (in Chinese)
[8] Mitrinovi´c, D. S. Analytic Inequalities (Springer-Verlag, New York, Heidelberg, Berlin, 1970).
[9] Qi, F. and Guo, B. -N. Sharpening and generalizations of Shafer’s inequality for the arc sine function, Integral Transforms Spec. Funct. 23 (2), 129–134, 2012; Available online at http://dx.doi.org/10.1080/10652469.2011.564578.
[10] Qi, F., Niu, D. -W. and Guo, B. -N. Refinements, generalizations, and applications of Jor- dan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages, 2009; Available online at http://dx.doi.org/10.1155/2009/271923.
[11] Qi, F., Zhang, S. -Q. and Guo, B. -N. Sharpening and generalizations of Shafer’s inequality for the arc tangent function, J. Inequal. Appl. 2009 (2009), Article ID 930294, 9 pages, 2009; Available online at http://dx.doi.org/10.1155/2009/930294.
[12] Steckin, S.B. Some remarks on trigonmetric polynomials, Uspehi Mat. Nauk. (N.S.) 10 1 (63), 159–166, 1955. (in Russian)
[13] Zhao, J. -L., Wei, C. -F., Guo, B. -N. and Qi, F. Sharpening and generalizations of Carlson’s double inequality for the arc cosine function, Hacet. J. Math. Stat. 41 (2), 201–209, 2012.
[14] Zhu, L. and Hua, J. -K. Sharpening the Becker-Stark inequalities, J. Inequal. Appl. 2010 (2010), Article ID 931275, 4 pages, 2010; Available online at http://dx.doi.org/10.1155/2010/931275.