Sharpened versions of Mitrinović-Adamović, Lazarević and Wilkers inequalities for trigonometric and hyperbolic functions

Volume 9, Issue 5, pp 2688--2696 http://dx.doi.org/10.22436/jnsa.009.05.65

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Authors

Shan-He Wu - Department of Mathematics, Longyan University, Longyan, Fujian, 364012, P. R. China. Shu-Guang Li - Department of Mathematics, Longyan University, Longyan, Fujian, 364012, P. R. China. Mihály Bencze - Department of Mathematics, University of Craiova, Craiova, RO-200585, Romania.

Abstract

In this paper, we establish new sharpened versions of Mitrinović-Adamović and Lazarević's inequalities. Further, we provide an application of our results to the improvements of Wilker's inequality for trigonometric and hyperbolic functions. We show that the coefficient assigned to each of these sharpened inequalities is best possible.

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Keywords

• Mitrinović-Adamović's inequality
• Lazarević's inequality
• Wilker's inequality
• sharpening
• best possible coefficient
• trigonometric functions
• hyperbolic functions.

MSC

•  26D05
•  26D15
•  33B10

References

• [1] C. P. Chen, Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions, Integral Transforms Spec. Funct., 23 (2012), 865-873.


• [2] C. P. Chen, W. S. Cheung, Wilker and Huygens-type inequalities and solution to Oppenheim's problem, Integral Transforms Spec. Funct., 23 (2012), 325-336.


• [3] B. N. Guo, B. M. Qiao, F. Qi, W. Li , On new proofs of Wilker's inequalities involving trigonometric functions, Math. Inequal. Appl., 6 (2003), 19-22.


• [4] R. Klén, M. Visuri, M. Vuorinen, On Jordan type inequalities for hyperbolic functions, J. Inequal. Appl., 2010 (2010), 14 pages.


• [5] I. Lazarević, Certain inequalities with hyperbolic functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 159-170 (1966), 41-48.


• [6] D. S. Mitrinović, D. D. Adamović, Sur une inégalité élémentaire ou interviennent des fonctions trigonométriques, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 143-155 (1965), 23-34.


• [7] D. S. Mitrinović, P. M. Vasić, Analytic Inequalities, Springer-Verlag, New York (1970)


• [8] L. H. Montgomery, D. J. Vaaler, J. Delany, E. D. Knuth, D. Vialetto, M. S. Wilker, J. B. Wilker, Problems and Solutions: Elementary Problems, E3301-E3306, Amer. Math. Monthly, 96 (1989), 54-55.


• [9] C. Mortici , A subtly analysis of Wilker inequality, Appl. Math. Comput., 231 (2014), 516-520.


• [10] E. Neuman, On Wilker and Huygens type inequalities, Math. Inequal. Appl., 15 (2012), 271-279.


• [11] E. Neuman , Wilker and Huygens-type inequalities for Jacobian elliptic and theta functions, Integral Transforms Spec. Funct., 25 (2014), 240-248.


• [12] I. Pinelis, L' Hospital rules for monotonicity and the Wilker-Anglesio inequality, Amer. Math. Monthly, 111 (2004), 905-909.


• [13] J. S. Sumner, A. A. Jagers, M. Vowe, J. Anglesio, Inequalities involving trigonometric functions , Amer. Math. Monthly, 98 (1991), 264-267.


• [14] S. Wu , On extension and refinement of Wilker's inequality , Rocky Mountains J. Math., 39 (2009), 683-687.


• [15] S. Wu, A. Baricz, Generalizations of Mitrinović, Adamović and Lazarević's inequalities and their applications, Publ. Math. Debrecen, 75 (2009), 447-458.


• [16] S. Wu, H. M. Srivastava, A weighted and exponential generalization of Wilker's inequality and its applications , Integral Transform. Spec. Funct., 18 (2007), 529-535.


• [17] S. Wu, H. M. Srivastava, A further refinement of Wilker's inequality, Integral Transform. Spec. Funct., 19 (2008), 757-765.


• [18] S. H. Wu, H. P. Yue, Y. P. Deng, Y. M. Chu, Several improvements of Mitrinović-Adamović and Lazarević's inequalities with applications to the sharpening of Wilker-type inequalities, J. Nonlinear Sci. Appl., 9 (2016), 1755-1765.


• [19] Z. H. Yang, Renements of a two-sided inequality for trigonometric functions , J. Math. Inequal., 7 (2013), 601-615.


• [20] C. Y. Yang , Inequalities on generalized trigonometric and hyperbolic functions , J. Math. Anal. Appl., 419 (2014), 775-782.


• [21] Z. H. Yang, Y. M. Chu, A note on Jordan, Mitrinović-Adamović, and Cusa inequalities, Abstr. Appl. Anal., 2014 (2014), 12 pages.


• [22] L. Yin, L. Huang, F. Qi , Some inequalities for the generalized trigonometric and hyperbolic functions , Turkish J. Anal. Number theo., 2 (2014), 96-101.


• [23] L. Zhang, L. Zhu, A new elementary proof of Wilker's inequalities, Math. Inequal. Appl., 11 (2008), 149-151.


• [24] L. Zhu, A new simple proof of Wilker's inequality, Math. Inequal. Appl., 8 (2005), 749-750.


• [25] L. Zhu, On Wilker-type inequalities, Math. Inequal. Appl., 10 (2007), 727-731.


• [26] L. Zhu, Generalized Lazarević's inequality and its applications II, J. Inequal. Appl., 2009 (2009), 4 pages.