Optimal bounds for the generalized Euler-Mascheroni constant. (English) Zbl 1497.11292
Summary: We provide several sharp upper and lower bounds for the generalized Euler-Mascheroni constant. As consequences, some previous bounds for the Euler-Mascheroni constant are improved.
MSC:
| 11Y60 | Evaluation of number-theoretic constants |
| 40A05 | Convergence and divergence of series and sequences |
| 33B15 | Gamma, beta and polygamma functions |
References:
| [1] | Knopp, K.: Theory and Applications of Infinite Series. Dover Publications, New York (1990) |
| [2] | Chen, C.-P., Qi, F.: The best bounds of the n-th harmonic number. Glob. J. Appl. Math. Math. Sci. 1(1), 41-49 (2008) |
| [3] | Niu, D.-W., Zhang, Y.-J., Qi, F.: A double inequality for the harmonic number in terms of the hyperbolic cosine. Turk. J. Anal. Number Theory 2(6), 223-225 (2014) · doi:10.12691/tjant-2-6-6 |
| [4] | Wang, M.-K., Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 521-537 (2018) · Zbl 1384.33034 |
| [5] | Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch-Pfluger distortion function. Math. Inequal. Appl. 21(3), 629-648 (2018) · Zbl 1402.30022 |
| [6] | Alzer, H.: Inequalities for the gamma and polygamma functions. Abh. Math. Semin. Univ. Hamb. 68, 363-372 (1998) · Zbl 0945.33003 · doi:10.1007/BF02942573 |
| [7] | Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23(1), 97-100 (2010) · Zbl 1183.33003 · doi:10.1016/j.aml.2009.08.012 |
| [8] | Mortici, C.: Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant. Anal. Appl. 8(1), 99-107 (2010) · Zbl 1185.26047 · doi:10.1142/S0219530510001539 |
| [9] | Mortici, C.: Improved convergence towards generalized Euler-Mascheroni constant. Appl. Math. Comput. 215(9), 3443-3448 (2010) · Zbl 1186.11076 |
| [10] | Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 24(7), 1239-1244 (2011) · Zbl 1216.33005 · doi:10.1016/j.aml.2011.02.015 |
| [11] | Negoi, T.: A faster convergence to the constant of Euler. Gaz. Mat., Ser. A 15, 111-113 (1997) |
| [12] | Qiu, S.-L., Vuorinen, M.: Some properties of the gamma and psi functions, with applications. Math. Comput. 74(250), 723-742 (2005) · Zbl 1060.33006 · doi:10.1090/S0025-5718-04-01675-8 |
| [13] | DeTemple, D.W.: A geometric look at sequences that converge to Euler’s constant. Coll. Math. J. 37(2), 128-131 (2006) · doi:10.2307/27646302 |
| [14] | Chen, C.-P.: The best bounds in Vernescu’s inequalities for the Euler’s constant. RGMIA Res. Rep. Collect. 12(3), Article ID 11 (2009) |
| [15] | Sîntămărian, A.: A generalization of Euler’s constant. Numer. Algorithms 46(2), 141-151 (2007) · Zbl 1130.11075 · doi:10.1007/s11075-007-9132-0 |
| [16] | Berinde, V., Mortici, C.: New sharp estimates of the generalized Euler-Mascheroni constant. Math. Inequal. Appl. 16(1), 279-288 (2013) · Zbl 1259.33004 |
| [17] | Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012). https://doi.org/10.1007/s00025-010-0090-9 · Zbl 1259.26035 · doi:10.1007/s00025-010-0090-9 |
| [18] | Guo, B.-N., Qi, F.: Sharp inequalities for the psi function and harmonic numbers. Analysis 34(2), 201-208 (2014) · Zbl 1294.33003 |
| [19] | Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729-735 (2017) · Zbl 1375.33026 |
| [20] | Yang, Z.-H., Chu, Y.-M., Zhang, X.-H.: Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind. J. Nonlinear Sci. Appl. 10(3), 929-936 (2017) · Zbl 1412.33033 · doi:10.22436/jnsa.010.03.06 |
| [21] | Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautsch inequality for parameter 0<p<\(10< p<1\) with applications. Math. Inequal. Appl. 20(4), 1107-1120 (2017) · Zbl 1386.33005 |
| [22] | Guo, B.-N., Qi, F.: Sharp bounds for harmonic numbers. Appl. Math. Comput. 218(3), 991-995 (2011) · Zbl 1229.11027 |
| [23] | Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21(2), 469-479 (2018) · Zbl 1384.33008 |
| [24] | Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714-1726 (2018) · Zbl 1387.33029 · doi:10.1016/j.jmaa.2018.03.005 |
| [25] | Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U. S. Government Printing Office, Washington (1964) · Zbl 0171.38503 |
| [26] | Chen, C.-P.: Inequalities for the Euler-Mascheroni constant. Appl. Math. Lett. 23(2), 161-164 (2010) · Zbl 1203.26025 · doi:10.1016/j.aml.2009.09.005 |
| [27] | Chen, C.-P.: Inequalities and monotonicity properties for some special functions. J. Math. Inequal. 3(1), 79-91 (2009) · Zbl 1177.33005 · doi:10.7153/jmi-03-07 |
| [28] | Chen, C.-P., Mortici, C.: Limits and inequalities associated with the Euler-Mascheroni constant. Appl. Math. Comput. 219(18), 9755-9761 (2013). https://doi.org/10.1016/j.amc.2013.03.089 · Zbl 1297.33002 · doi:10.1016/j.amc.2013.03.089 |
| [29] | Qiu, S.-L., Zhao, X.: Some properties of the psi function and evaluations of γ. Appl. Math. J. Chin. Univ. Ser. A 31(1), 103-111 (2016). https://doi.org/10.1007/s11766-016-3272-8 · Zbl 1363.33004 · doi:10.1007/s11766-016-3272-8 |
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