Title: Absolute Monotonicity of Functions Related To Estimates of First Eigenvalue of Laplace Operator on Riemannian Manifolds
Author(s): Feng Qi, Miao-Miao Zheng
Pages: 123-131
Cite as:
Feng Qi, Miao-Miao Zheng, Absolute Monotonicity of Functions Related To Estimates of First Eigenvalue of Laplace Operator on Riemannian Manifolds, Int. J. Anal. Appl., 6 (2) (2014), 123-131.

Abstract


The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.

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References


  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, 9th printing, Washington, 1970. Google Scholar

  2. H. Alzer, Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel) 74 (2000), 207-211. Google Scholar

  3. H.-F. Ge, New sharp bounds for the Bernoulli numbers and refinement of Becker-Stark inequalities, J. Appl. Math. 2012 (2012), Article ID 137507, 7 pages. Google Scholar

  4. B.-N. Guo, Q.-M. Luo, and F. Qi, Monotonicity results and inequalities for the inverse hyperbolic sine function, J. Inequal. Appl. 2013 (2013), Article ID 536, 6 pages. Google Scholar

  5. B.-N. Guo and F. Qi, Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function, Filomat 27 (2013), no. 2, 261–265. Google Scholar

  6. B.-N. Guo and F. Qi, 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 (2010), no. 2, 21–30. Google Scholar

  7. B.-N. Guo and F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat. 26 (2015), in press. Google Scholar

  8. F.-B. Hang and X.-D. Wang, A remark on Zhong-Yang’s eigenvalue estimate. Int. Math. Res. Not. IMRN 2007, no. 18, Article ID rnm064, 9 pages. Google Scholar

  9. Q.-D. Hao and B.-N. Guo, A method of finding extremums of composite functions of trigonometric functions, Ku`ang Y`e (Mining) (1993), no. 4, 80–83. (Chinese) Google Scholar

  10. Z.-H. Huo, F. Qi, and B.-N. Guo, Laplace operator ∆ and its representations, Zh`engzh¯ou Fˇangzh¯i G¯ongx´uey`uan X´ueb`ao (Journal of Zhengzhou Textile Institute) 4 (1993), no. 2, 52–57. (Chinese) Google Scholar

  11. P. Li. Poincar´e inequalities on Riemannian manifolds, Seminar on Differential Geometry (Ann. of Math. Stud. 102), 73–83, Princeton University Press, 1982. Google Scholar

  12. P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), 205–239, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. Google Scholar

  13. J. Ling, The first eigenvalue of a closed manifold with positive Ricci curvature, Proc. Amer. Math. Soc. 134 (2006), no. 10, 3071–3079. Google Scholar

  14. D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993. Google Scholar

  15. F. Qi, A double inequality for ratios of Bernoulli numbers, ResearchGate Dataset, available online at http://dx.doi.org/10.13140/2.1.2367.2962. Google Scholar

  16. F. Qi, A double inequality for ratios of Bernoulli numbers, RGMIA Res. Rep. Coll. 17 (2014), Article 103, 4 pages. Google Scholar

  17. F. Qi, An estimate of the gap of the first two eigenvalues in the Schr¨odinger operator, Ji¯aozu`o Ku`angy`e Xu´eyu`an Xu´eb`ao (Journal of Jiaozuo Mining Institute) 12 (1993), no. 2, 108–112. Google Scholar

  18. F. Qi, Estimates of the Gap of Two Eigenvalues in the Schr¨odinger Operator and the First Eigenvalue of Laplace Operator, Thesis supervised by Professor Yi-Pei Chen and submitted for the Master Degree of Sceince in Mathematics at Xiamen University by Feng Qi in April 1989. (Chinese) Google Scholar

  19. F. Qi, Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function, J. Comput. Appl. Math. 268 (2014), 155–167. Google Scholar

  20. F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl. (2015), in press. Google Scholar

  21. F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), no. 4, 1685–1696. Google Scholar

  22. F. Qi, P. Cerone, and S. S. Dragomir, Complete monotonicity of a function involving the divided difference of psi functions, Bull. Aust. Math. Soc. 88 (2013), no. 2, 309–319. Google Scholar

  23. F. Qi and B.-N. Guo, Lower bound of the first eigenvalue for the Laplace operator on compact Riemannian manifold, Chinese Quart. J. Math. 8 (1993), no. 2, 40–49. Google Scholar

  24. F. Qi and B.-N. Guo, Sharpening and generalizations of Shafer’s inequality for the arc sine function, Integral Transforms Spec. Funct. 23 (2012), no. 2, 129–134. Google Scholar

  25. F. Qi, B.-N. Guo, and R.-Q. Cui, Estimates of the upper bound of the difference of two arbitrary neighboring eigenvalues of the Schr¨odinger operator, J. Math. (Wuhan) 16 (1996), no. 1, 81–86. (Chinese) Google Scholar

  26. F. Qi, B.-N. Guo, and Q.-D. Hao, Estimate of the lower bound for the gap between the first two eigenvalues of Laplace operator, Ku`ang Y`e (Mining) (1994), no. 2, 86–93. (Chinese) Google Scholar

  27. F. Qi, H.-C. Li, B.-N. Guo and Q.-M. Luo, Inequalities and estimates of the eigenvalue for Laplace operator, Ji¯aozu`o Ku`angy`e Xu´eyu`an Xu´eb`ao (Journal of Jiaozuo Mining Institute) 13 (1994), no. 3, 89–95. (Chinese) Google Scholar

  28. F. Qi, D.-W. Niu, and B.-N. Guo, Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages. Google Scholar

  29. F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal. 2 (2014), no. 3, 91–97. Google Scholar

  30. F. Qi, L.-Q. Yu, and Q.-M. Luo, Estimates for the upper bounds of the first eigenvalue on submanifolds, Chinese Quart. J. Math. 9 (1994), no. 2, 40–43. Google Scholar

  31. R. L. Schilling, R. Song, and Z. Vondraˇcek, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012. Google Scholar

  32. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, Wiley 1996. Google Scholar

  33. D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946. Google Scholar

  34. H. C. Yang, Estimate of the first eigenvalue of Laplace operator on Riemannian manifolds whose Ricci curvature has a negative lower bound, Sci. Sinica Ser. A 32 (1989), no. 7, 689–700. (Chinese) Google Scholar

  35. H. C. Yang, Estimates of the first eigenvalue of compact Riemannian manifolds with boundary with Dirichlet boundary conditions Acta Math. Sinica 34 (1991), no. 3, 329–342. (Chinese) Google Scholar

  36. D.-G. Yang, Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature, Pacific J. Math. 190 (1999), no. 2, 383–398. Google Scholar

  37. Q. H. Yu and J. Q. Zhong, Lower bounds of the gap between the first and second eigenvalues of the Schr¨odinger operator. Trans. Amer. Math. Soc. 294 (1986), no. 1, 341–349. Google Scholar

  38. J. Q. Zhong and H. C. Yang, Estimates of the first eigenvalue of Laplace operator on compact Riemannian manifolds, Sci. Sinica Ser. A 26 (1983), no. 9, 812–820. (Chinese) Google Scholar

  39. J. Q. Zhong and H. C. Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. Google Scholar

  40. L. Zhu and J.-K. Hua, Sharpening the Becker-Stark inequalities, J. Inequal. Appl. 2010 (2010), Article ID 931275, 4 pages. Google Scholar


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