Araştırma Makalesi
BibTex RIS Kaynak Göster

Yıl 2021, Cilt: 50 Sayı: 1, 46 - 62, 04.02.2021

Ling Zhu , Branko Malesevic

https://doi.org/10.15672/hujms.550184
Cited By: 1

Öz

Kaynakça

  • [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, Dover Publications, 1972.
  • [2] B. Banjac, M. Makragić, and B. Malešević, Some notes on a method for proving inequalities by computer, Results Math. 69, 161–176, 2016.
  • [3] A. Baricz and J. Sandor, Extensions of the generalized Wilker inequality to Bessel functions, J. Math. Inequal. 2 (3), 397–406, 2008.
  • [4] F. Cajori, A History of Mathematics, 2nd ed., New York, 1929.
  • [5] F.T. Campan, The Story of Number π, Ed. Albatros, Romania, 1977.
  • [6] C.-P. Chen and W.-S. Cheung, Wilker- and Huygens-type inequalities and solution to Oppenheim’s problem, Int. Trans. Spec. Funct. 23 (5), 325–336, 2012.
  • [7] B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On new proofs of Wilker inequalities involving trigonometric functions, Math. Inequal. Appl. 6 (1), 19–22, 2003.
  • [8] C. Huygens, Oeuvres Completes, Publiees Par la Societe Hollandaise des Science, Haga, 20 volumes, 1888–1940.
  • [9] A.P. Iuskevici, History of Mathematics in 16th and 17th Centuries, Moskva, 1961.
  • [10] A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 3rd ed., Elsevier Acad. Press, San Diego, CA, 2004.
  • [11] J.-C. Kuang, Applied Inequalities, 3rd ed. (in Chinese), Shangdong Science and Technology Press, Jinan City, Shangdong Province, China, 2004.
  • [12] J.-L. Li, An identity related to Jordan’s inequality, Int. J. Math. Math. Sci. 6, Article ID 76782, 2016.
  • [13] T. Lutovac, B. Malešević, and C. Mortici, The natural algorithmic approach of mixed trigonometric-polynomial problems, J. Inequal. Appl. 2017, Article No: 116, 2017.
  • [14] T. Lutovac, B. Malešević, and M. Rašajski, A new method for proving some inequalities related to several special functions, Results Math. 73, Article No: 100, 2018.
  • [15] B.J. Malešević, Application of λ-method on Shafer–Fink’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8, 103–105, 1997.
  • [16] B.J. Malešević, An application of λ-method on inequalities of Shafer–Fink’s type, Math. Inequal. Appl. 10, 529–534, 2007.
  • [17] B. Malešević, T. Lutovac, M. Rašajski, and C. Mortici, Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities, Adv. Difference Equ. 2018, Article No: 90, 2018.
  • [18] B. Malešević, M. Nenezić, L. Zhu, B. Banjac and M. Petrovic, Some new estimates of precision of Cusa-Huygens and Huygens approximations, accepted in Appl. Anal. Discrete Math., 2020.
  • [19] B. Malešević, M. Rašajski, and T. Lutovac, Refinements and generalizations of some inequalities of Shafer-Fink’s type for the inverse sine function, J. Inequal. Appl. 2017, Article No: 275, 2017.
  • [20] B. Malešević, M. Rašajski, and T. Lutovac, Double-sided Taylor’s approximations and their applications in Theory of analytic inequalities. in: Differential and Integral Inequalities, Th.M. Rassias, D. Andrica (eds.), Optimization and Its Applications, vol. 151, 569-582, 2019.
  • [21] D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, New York, Berlin 1970.
  • [22] M. Nenezić and L. Zhu, Some improvements of Jordan-Steckin and Becker-Stark inequalities, Appl. Anal. Discrete Math. 12, 244–256, 2018.
  • [23] E. Neuman, Wilker and Huygens-type inequalities for Jacobian elliptic and theta functions, Int. Trans. Spec. Funct. 25 (3), 240–248, 2014.
  • [24] E. Neuman and J. Sandor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl. 13 (4), 715–723, 2010.
  • [25] F. Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish J. Anal. Number Theory 6 (5), 129–131, 2018.
  • [26] F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math. 351, 1–5, 2019.
  • [27] M. Rašajski, T. Lutovac, and B. Malešević, About some exponential inequalities related to the sinc function, J. Inequal. Appl. 2018, Article No: 150, 2018.
  • [28] J. Sandor and M. Bencze, On Huygens’s trigonometric inequality, RGMIA Research Report Collection 8 (3), Art. 14, 2005.
  • [29] J.S. Sumner, A.A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, Amer. Math. Monthly 98, 264–267, 1991.
  • [30] J.B. Wilker, Problem E 3306, Amer. Math. Monthly 96, 55, 1989.
  • [31] S.-H. Wu, L. Debnath, A generalization of L’Hôspital-type rules for monotonicity and its application, Appl. Math. Lett. 22 (2), 284-290, 2009.
  • [32] S.-H. Wu, H.M. Srivastava, A weighted and exponential generalization of Wilker’s inequality and its applications, Int. Trans. Spec. Funct. 18 (8), 529–535, 2008.
  • [33] Z.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the arithmeticgeometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl. 462, 1714–1726, 2018.
  • [34] Z.-H. Yang and J.-F. Tian, Convexity and monotonicity for the elliptic integrals of the first kind and applications, Appl. Anal. Discrete Math. 13 (1), 240-260, 2019.
  • [35] Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math. 364, 112359, 14 pages, 2020.
  • [36] L. Zhu, On Shafer-Fink inequalities, Math. Inequal. Appl. 8 (4), 571–574, 2005.
  • [37] L. Zhu, A new simple proof of Wilker’s inequality, Math. Inequal. Appl. 8 (4), 749– 750, 2005.
  • [38] L. Zhu, A solution of a problem of Oppenheim, Math. Inequal. Appl. 10, 57–61, 2007.
  • [39] L. Zhu, On Wilker-type inequalities, Math. Inequal. Appl. 10 (4), 727–731, 2007.
  • [40] L. Zhu, New inequalities of Shafer–Fink type for arc hyperbolic sine, J. Inequal. Appl. 2008, Article ID 368275, 2008.
  • [41] L. Zhu, Some new inequalities of the Huygens type, Comput. Math. Appl. 58, 1180– 1182, 2009.
  • [42] L. Zhu, Some new Wilker-type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal. 2009, Article ID 485842, 9 pages, 2009.
  • [43] L. Zhu, A source of inequalities for circular functions, Comput. Math. Appl. 58, 1998–2004, 2009.
  • [44] L. Zhu, Inequalities for hyperbolic functions and their applications, J. Inequal. Appl. 2010, Article ID 130821, 10 pages, 2010.
  • [45] L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 2020.
  • [46] D. Zwillinger, CRC Standard Mathematical Tables and Formulae, CRC Press, 1996.

New Wilker-type and Huygens-type inequalities

Yıl 2021, Cilt: 50 Sayı: 1, 46 - 62, 04.02.2021

Ling Zhu , Branko Malesevic

https://doi.org/10.15672/hujms.550184
Cited By: 1

Öz

In this paper, we first determine the relationships between the first Wilker's inequality, the second Wilker's inequality, the first Huygens inequality, and the second Huygens inequality for circular functions and for hyperbolic functions, respectively. Then, we establish new Wilker-type inequalities and Huygens-type inequalities for two function pairs, $x/\sin^{-1}x$ and $x/\tan ^{-1}x$, $x/\sinh ^{-1}x$ and $x/\tanh ^{-1}x$. Finally, we obtain some more general conclusions than the first work of this paper, which reveal the absolute monotonicity of four functions involving the four inequalities mentioned above.

Anahtar Kelimeler

Wilker-type inequalities Huygens-type inequalities circular functions hyperbolic functions inverse circular functions inverse hyperbolic functions

Kaynakça

  • [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, Dover Publications, 1972.
  • [2] B. Banjac, M. Makragić, and B. Malešević, Some notes on a method for proving inequalities by computer, Results Math. 69, 161–176, 2016.
  • [3] A. Baricz and J. Sandor, Extensions of the generalized Wilker inequality to Bessel functions, J. Math. Inequal. 2 (3), 397–406, 2008.
  • [4] F. Cajori, A History of Mathematics, 2nd ed., New York, 1929.
  • [5] F.T. Campan, The Story of Number π, Ed. Albatros, Romania, 1977.
  • [6] C.-P. Chen and W.-S. Cheung, Wilker- and Huygens-type inequalities and solution to Oppenheim’s problem, Int. Trans. Spec. Funct. 23 (5), 325–336, 2012.
  • [7] B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On new proofs of Wilker inequalities involving trigonometric functions, Math. Inequal. Appl. 6 (1), 19–22, 2003.
  • [8] C. Huygens, Oeuvres Completes, Publiees Par la Societe Hollandaise des Science, Haga, 20 volumes, 1888–1940.
  • [9] A.P. Iuskevici, History of Mathematics in 16th and 17th Centuries, Moskva, 1961.
  • [10] A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 3rd ed., Elsevier Acad. Press, San Diego, CA, 2004.
  • [11] J.-C. Kuang, Applied Inequalities, 3rd ed. (in Chinese), Shangdong Science and Technology Press, Jinan City, Shangdong Province, China, 2004.
  • [12] J.-L. Li, An identity related to Jordan’s inequality, Int. J. Math. Math. Sci. 6, Article ID 76782, 2016.
  • [13] T. Lutovac, B. Malešević, and C. Mortici, The natural algorithmic approach of mixed trigonometric-polynomial problems, J. Inequal. Appl. 2017, Article No: 116, 2017.
  • [14] T. Lutovac, B. Malešević, and M. Rašajski, A new method for proving some inequalities related to several special functions, Results Math. 73, Article No: 100, 2018.
  • [15] B.J. Malešević, Application of λ-method on Shafer–Fink’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8, 103–105, 1997.
  • [16] B.J. Malešević, An application of λ-method on inequalities of Shafer–Fink’s type, Math. Inequal. Appl. 10, 529–534, 2007.
  • [17] B. Malešević, T. Lutovac, M. Rašajski, and C. Mortici, Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities, Adv. Difference Equ. 2018, Article No: 90, 2018.
  • [18] B. Malešević, M. Nenezić, L. Zhu, B. Banjac and M. Petrovic, Some new estimates of precision of Cusa-Huygens and Huygens approximations, accepted in Appl. Anal. Discrete Math., 2020.
  • [19] B. Malešević, M. Rašajski, and T. Lutovac, Refinements and generalizations of some inequalities of Shafer-Fink’s type for the inverse sine function, J. Inequal. Appl. 2017, Article No: 275, 2017.
  • [20] B. Malešević, M. Rašajski, and T. Lutovac, Double-sided Taylor’s approximations and their applications in Theory of analytic inequalities. in: Differential and Integral Inequalities, Th.M. Rassias, D. Andrica (eds.), Optimization and Its Applications, vol. 151, 569-582, 2019.
  • [21] D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, New York, Berlin 1970.
  • [22] M. Nenezić and L. Zhu, Some improvements of Jordan-Steckin and Becker-Stark inequalities, Appl. Anal. Discrete Math. 12, 244–256, 2018.
  • [23] E. Neuman, Wilker and Huygens-type inequalities for Jacobian elliptic and theta functions, Int. Trans. Spec. Funct. 25 (3), 240–248, 2014.
  • [24] E. Neuman and J. Sandor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl. 13 (4), 715–723, 2010.
  • [25] F. Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish J. Anal. Number Theory 6 (5), 129–131, 2018.
  • [26] F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math. 351, 1–5, 2019.
  • [27] M. Rašajski, T. Lutovac, and B. Malešević, About some exponential inequalities related to the sinc function, J. Inequal. Appl. 2018, Article No: 150, 2018.
  • [28] J. Sandor and M. Bencze, On Huygens’s trigonometric inequality, RGMIA Research Report Collection 8 (3), Art. 14, 2005.
  • [29] J.S. Sumner, A.A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, Amer. Math. Monthly 98, 264–267, 1991.
  • [30] J.B. Wilker, Problem E 3306, Amer. Math. Monthly 96, 55, 1989.
  • [31] S.-H. Wu, L. Debnath, A generalization of L’Hôspital-type rules for monotonicity and its application, Appl. Math. Lett. 22 (2), 284-290, 2009.
  • [32] S.-H. Wu, H.M. Srivastava, A weighted and exponential generalization of Wilker’s inequality and its applications, Int. Trans. Spec. Funct. 18 (8), 529–535, 2008.
  • [33] Z.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the arithmeticgeometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl. 462, 1714–1726, 2018.
  • [34] Z.-H. Yang and J.-F. Tian, Convexity and monotonicity for the elliptic integrals of the first kind and applications, Appl. Anal. Discrete Math. 13 (1), 240-260, 2019.
  • [35] Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math. 364, 112359, 14 pages, 2020.
  • [36] L. Zhu, On Shafer-Fink inequalities, Math. Inequal. Appl. 8 (4), 571–574, 2005.
  • [37] L. Zhu, A new simple proof of Wilker’s inequality, Math. Inequal. Appl. 8 (4), 749– 750, 2005.
  • [38] L. Zhu, A solution of a problem of Oppenheim, Math. Inequal. Appl. 10, 57–61, 2007.
  • [39] L. Zhu, On Wilker-type inequalities, Math. Inequal. Appl. 10 (4), 727–731, 2007.
  • [40] L. Zhu, New inequalities of Shafer–Fink type for arc hyperbolic sine, J. Inequal. Appl. 2008, Article ID 368275, 2008.
  • [41] L. Zhu, Some new inequalities of the Huygens type, Comput. Math. Appl. 58, 1180– 1182, 2009.
  • [42] L. Zhu, Some new Wilker-type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal. 2009, Article ID 485842, 9 pages, 2009.
  • [43] L. Zhu, A source of inequalities for circular functions, Comput. Math. Appl. 58, 1998–2004, 2009.
  • [44] L. Zhu, Inequalities for hyperbolic functions and their applications, J. Inequal. Appl. 2010, Article ID 130821, 10 pages, 2010.
  • [45] L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 2020.
  • [46] D. Zwillinger, CRC Standard Mathematical Tables and Formulae, CRC Press, 1996.
Toplam 46 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Ling Zhu 0000-0002-9059-3983

Branko Malesevic 0000-0002-4963-4149

Yayımlanma Tarihi 4 Şubat 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 50 Sayı: 1

Kaynak Göster

APA Zhu, L., & Malesevic, B. (2021). New Wilker-type and Huygens-type inequalities. Hacettepe Journal of Mathematics and Statistics, 50(1), 46-62. https://doi.org/10.15672/hujms.550184
AMA Zhu L, Malesevic B. New Wilker-type and Huygens-type inequalities. Hacettepe Journal of Mathematics and Statistics. Şubat 2021;50(1):46-62. doi:10.15672/hujms.550184
Chicago Zhu, Ling, ve Branko Malesevic. “New Wilker-Type and Huygens-Type Inequalities”. Hacettepe Journal of Mathematics and Statistics 50, sy. 1 (Şubat 2021): 46-62. https://doi.org/10.15672/hujms.550184.
EndNote Zhu L, Malesevic B (01 Şubat 2021) New Wilker-type and Huygens-type inequalities. Hacettepe Journal of Mathematics and Statistics 50 1 46–62.
IEEE L. Zhu ve B. Malesevic, “New Wilker-type and Huygens-type inequalities”, Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 1, ss. 46–62, 2021, doi: 10.15672/hujms.550184.
ISNAD Zhu, Ling - Malesevic, Branko. “New Wilker-Type and Huygens-Type Inequalities”. Hacettepe Journal of Mathematics and Statistics 50/1 (Şubat 2021), 46-62. https://doi.org/10.15672/hujms.550184.
JAMA Zhu L, Malesevic B. New Wilker-type and Huygens-type inequalities. Hacettepe Journal of Mathematics and Statistics. 2021;50:46–62.
MLA Zhu, Ling ve Branko Malesevic. “New Wilker-Type and Huygens-Type Inequalities”. Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 1, 2021, ss. 46-62, doi:10.15672/hujms.550184.
Vancouver Zhu L, Malesevic B. New Wilker-type and Huygens-type inequalities. Hacettepe Journal of Mathematics and Statistics. 2021;50(1):46-62.

Cited By

Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions
Axioms
https://doi.org/10.3390/axioms10040308

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms