A Generalization of Jordan's Inequality and an Application  ABSTRACT  |  FULL TEXT

Yıl 2011, Cilt: 40 Sayı: 1, 53 - 61, 01.01.2011

Zh.-h. Huo D.-w. Niu J. Cao  f. Qi Feng Qi

Öz

In this article, a new generalization of Jordan’s inequality
Xn
k=1
µk

θ
t − x
t
k ≤
sin x
x
−
sin θ
θ
≤
Xn
k=1
ωk

θ
t − x
t
k
for t ≥ 2, n ∈ N and θ ∈ (0, π] is established, where the coefficients µk
and ωk are defined by recursion formulas, and are the best possible. As
an application, Yang’s inequality is refine

Anahtar Kelimeler

ds: Jordan’s inequali, Yang’s inequali

Kaynakça

• Abel, U. and Caccia, D. A sharpening of Jordan’s inequality, Amer. Math. Monthly 93 (7), –569, 1986.
• Abramowitz, M. and Stegun, I. A. (Eds), Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables(4th printing, with corrections, Applied Mathematics Series 55, National Bureau of Standards, Washington, 1965).
• Bullen, P. S. A Dictionary of Inequalities (Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, Harlow/Essex, 1998).
• Debnath, L. and Zhao, Ch. -J. New strengthened Jordan’s inequality and its applications, Appl. Math. Lett. 16 (4), 557–560, 2003.
• Feng, Y. -F. Proof without words: Jordan’s inequality x π ≤sin x ≤ x, 0 ≤ x ≤2, Math. π, Math. Mag. 69, 126, 1996. Jiang, W. -D. and Hua,
• Y. Sharpening of Jordan’s inequality and its applica- tions, J. Inequal. Pure Appl. Math. 7 (3), Art. 102, 2006; http://www.emis.de/journals/JIPAM/article719.html?sid=719. Available online at
• Kuang, J. -Ch. Ch´angy`ong B`udˇengsh`ı(Applied Inequalities) 3rd ed., Sh¯and¯ong K¯exu´e J`ısh`u Ch¯ubˇan Sh`e (Shandong Science and Technology Press, Ji’nan City, Shandong Province, China, 2004). (Chinese)
• Luo, Q. -M., Wei, Z. -L. and Qi, F. Lower and upper bounds of ζ(3), Adv. Stud. Contemp. Math. (Kyungshang) 6 (1), 47–51, 2003.
• Mercer, A. McD. Problem E 2952, Amer. Math. Monthly 89 (6), 424, 1982.
• Mitrinovi´c, D. S. Analytic Inequalities (Springer-Verlag, Berlin/Heildelberg/New York, ). Niu, D. -W. Generalizations of Jordan’s Inequality and Applications, Thesis supervised by Professor Feng Qi and submitted for the Master Degree at Henan Polytechnic University in June 2007. (Chinese)
• Niu, D. -W., Cao, J. and Qi, F. Generalizations of Jordan’s inequality and concerned rela- tions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (3), 85–98, 2010.
• Niu, D. -W., Huo, Zh. -H., Cao, J. and Qi, F. A general refinement of Jordan’s inequality and a refinement of L. Yang’s inequality, Integral Transforms Spec. Funct. 19 (3), 157–164, ¨Ozban, A. Y. A new refined form of Jordan’s inequality and its applications, Appl. Math. Lett. 19 (2), 155–160, 2006.
• Qi, F. Extensions and sharpenings of Jordan’s and Kober’s inequality, G¯ongk¯e Sh`uxu´e (Journal of Mathematics for Technology) 12 (4), 98–102, 1996. (Chinese)
• Qi, F., Cui, L. -H. and Xu, S. -L. Some inequalities constructed by Tchebysheff ’s integral inequality, Math. Inequal. Appl. 2 (4), 517–528, 1999.
• Qi, F. and Guo, B. -N. Extensions and sharpenings of the noted Kober’s inequality, Ji¯aozu`o Ku`angy`e Xu´eyu`an Xu´eb`ao (Journal of Jiaozuo Mining Institute) 12 (4), 101–103, 1993. (Chinese)
• Qi, F. and Guo, B. -N. On generalizations of Jordan’s inequality, M´eit`an G¯aodˇeng Ji`aoy`u (Coal Higher Education), Suppl., November/1993, 32–33, 1993. (Chinese)
• Qi, F. and Hao, Q. -D. Refinements and sharpenings of Jordan’s and Kober’s inequality, Mathematics and Informatics Quarterly 8 (3), 116–120, 1998.
• 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, Article ID 271923, 52 pages, ; Available online at http://dx.doi.org/10.1155/2009/271923.
• Redheffer, R. Correction, Amer. Math. Monthly 76 (4), 422, 1969.
• Redheffer, R. Problem 5642, Amer. Math. Monthly 75 (10), 1125, 1968.
• Vamanamurthy, M. K. and Vuorinen, M. Inequalities for means, J. Math. Anal. Appl. 183, –166, 1994.
• Williams, J. P. A delightful inequality, Amer. Math. Monthly 76 (10), 1153–1154, 1969.
• Wu, Sh. -H. On generalizations and refinements of Jordan type inequality, Octogon Math. Mag. 12 (1), 267–272, 2004.
• Wu, Sh. -H. Sharpness and generalization of Jordan’s inequality and its application, Tai- wanese J. Math. 12 (2), 325–336, 2008.
• Wu, Sh. -H. and Debnath, L. A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, Appl. Math. Lett. 19 (12), –1384, 2006.
• Wu, Sh. -H. and Debnath, L. A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, II, Appl. Math. Lett. 20, –538, 2007.
• Yang, L. Zh´ı F¯enb`u Lˇil`un J´ıq´ı X¯in Y´anji¯u(The Theory of Distribution of Values of Func- tions and Recent Researches), K¯exu´e Ch¯ubˇan Sh`e (Science Press, Beijing, 1982). (Chinese)
• Zhang, X. -H., Wang, G. -D. and Chu, Y. -M. Extensions and sharpenings of Jordan’s and Kober’s inequalities, J. Inequal. Pure Appl. Math. 7 (2), Art. 63, 2006; Avaialble online at http://www.emis.de/journals/JIPAM/article680.html?sid=680.
• Zhao, Ch. -J. The extension and strength of Yang Le inequality, Sh`uxu´e de Sh´ıji`an yˇu R`ensh´ı (Math. Practice Theory) 30 (4), 493–497, 2000. (Chinese) Zhao, ity, http://www.emis.de/journals/JIPAM/article208.html?sid=208. On generalizations Art. 56, ; Available online at
• J. Inequal. Pure Appl. Math. 3 (4), Zhu, L. Sharpening of Jordan’s inequalities and its applications, Math. Inequal. Appl. 9 (1), –106, 2006.
• Zhu, L. Sharpening Jordan’s inequality and the Yang Le inequality, Appl. Math. Lett. 19 (3), –243, 2006.
• Zhu, L. Sharpening Jordan’s inequality and the Yang Le inequality, II, Appl. Math. Lett. (9), 990–994, 2006.