Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers. (English) Zbl 1272.11038
The authors obtain Turán type inequalities for the partial sums in the Maclaurin expansion of the generating functions of Bernoulli numbers, Euler numbers of the first and second kind and weighted Bernoulli numbers. First they prove general relations between Turán type inequalities for partial sums and remainders in any Maclaurin expansion and then they apply them to the above generating functions. Also, results concerning the complete monotonicity of the remainders in asymptotic expansions of the \(\beta\)-function are also obtained.
Reviewer: Chrysoula G. Kokologiannaki (Patras)
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11B83 | Special sequences and polynomials |
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
| 26D07 | Inequalities involving other types of functions |
| 26D15 | Inequalities for sums, series and integrals |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 41A80 | Remainders in approximation formulas |
Keywords:
Turan inequality; Bernoulli numbers; Euler numbers; generating functions; remainders in asymptotic expansions; completely monotonic functions of positive order; ratio of series of functions; \(\beta\)-functionReferences:
| [1] | Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z. 256 pp 895– (2007) · Zbl 1125.26022 · doi:10.1007/s00209-007-0111-x |
| [2] | Baricz, Turán type inequalities for modified Bessel functions, Bull. Aust. Math. Soc. 82 pp 254– (2010) · Zbl 1206.33007 · doi:10.1017/S000497271000002X |
| [3] | Berg, Bounds on Turán determinants, J. Approx. Theory 161 (1) pp 127– (2009) · Zbl 1190.33012 · doi:10.1016/j.jat.2008.08.010 |
| [4] | Brändén, Iterated sequences and the geometry of zeros, J. Reine Angew. Math. 658 pp 115– (2011) |
| [5] | Csordas, Conjectures and theorems in the theory of entire functions, Numer. Algorithms. 25 pp 109– (2000) · Zbl 0968.30012 · doi:10.1023/A:1016604906346 |
| [6] | Csordas, The Riemann hypothesis and the Turán inequalities, Trans. Am. Math. Soc. 296 pp 521– (1986) |
| [7] | Csordas, Moment inequalities and the Riemann hypothesis, Constr. Approx. 4 pp 175– (1988) · Zbl 0696.30007 · doi:10.1007/BF02075457 |
| [8] | Dimitrov, Higher order Turán inequalities, Proc. Am. Math. Soc. 126 (7) pp 2033– (1998) · Zbl 0891.30016 · doi:10.1090/S0002-9939-98-04438-4 |
| [9] | Dimitrov, Sharp Turán inequalities via very hyperbolic polynomials, J. Math. Anal. Appl. 376 pp 385– (2011) · Zbl 1210.30003 · doi:10.1016/j.jmaa.2010.12.014 |
| [10] | Dimitrov, Higher order Turán inequalities for the Riemann {\(\xi\)}-function, Proc. Am. Math. Soc. 139 pp 1013– (2011) · Zbl 1214.11092 · doi:10.1090/S0002-9939-2010-10515-4 |
| [11] | Frenzen, Error bounds for asymptotic expansions of the ratio of two gamma functions, SIAM J. Math. Anal. 18 (3) pp 890– (1987) · Zbl 0625.41022 · doi:10.1137/0518067 |
| [12] | Ikeda, Some inequalities for Bernoulli’s polynomials and related functions, Monatsh. Math. 68 pp 224– (1964) · Zbl 0129.28402 · doi:10.1007/BF01298510 |
| [13] | Ismail, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal. 10 (5) pp 884– (1979) · Zbl 0427.60021 · doi:10.1137/0510083 |
| [14] | S. Karlin G. Szego On certain determinants whose elements are orthogonal polynomials, J. Anal. Math. 8 1 157 · Zbl 0116.27901 |
| [15] | Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal. pp 11– (2008) |
| [16] | Knopp, Theory and Application of Infinite Series (1947) · JFM 54.0222.09 |
| [17] | Koumandos, Remarks on some completely monotonic functions, J. Math. Anal. Appl. 324 pp 1458– (2006) · Zbl 1108.26008 · doi:10.1016/j.jmaa.2005.12.017 |
| [18] | Koumandos, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function, J. Math. Anal. Appl. 355 pp 33– (2009) · Zbl 1169.33001 · doi:10.1016/j.jmaa.2009.01.042 |
| [19] | Koumandos, On the asymptotic expansion of the logarithm of Barnes triple gamma function, Math. Scand. 105 pp 287– (2009) · Zbl 1184.33002 · doi:10.7146/math.scand.a-15119 |
| [20] | Krasikov, Turán inequalities for three-term recurrences with monotonic coefficients, J. Approx. Theory. 163 pp 1269– (2011) · Zbl 1228.42029 · doi:10.1016/j.jat.2011.04.004 |
| [21] | Laforgia, On some Turán-type inequalities, J. Inequal. Appl. 2006 pp 1– · Zbl 1095.33002 · doi:10.1155/JIA/2006/29828 |
| [22] | Lorch, On the composition of completely monotonic functions and completely monotonic sequences and related questions, J. Lond. Math. Soc. 28 pp 31– (1983) · Zbl 0547.26010 · doi:10.1112/jlms/s2-28.1.31 |
| [23] | Pedersen, Completely monotonic functions related to logarithmic derivatives of entire functions, Anal. Appl. Singap. 9 (4) pp 409– (2011) · Zbl 1231.26011 · doi:10.1142/S0219530511001911 |
| [24] | Prudnikov, Integrals and Series, Elementary Functions Vol. 1 (1986) |
| [25] | Simic, Turán’s inequality for Appel polynomials, J. Inequal. Appl. Vol. 2006 pp 7– |
| [26] | Skovgaard, On inequalities of Turán-type, Math. Scand. 2 pp 65– (1954) · Zbl 0055.29904 · doi:10.7146/math.scand.a-10396 |
| [27] | Szego, On an inequality of P. Turán concerning Legendre polynomials, Bull. Am. Math. Soc. 54 pp 401– (1948) · Zbl 0032.27502 · doi:10.1090/S0002-9904-1948-09017-6 |
| [28] | Szwarc, Harmonic Analysis and Hypergroups, Delhi 1995 pp 165– (1997) · Zbl 0827.33008 |
| [29] | Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics (1996) · Zbl 0856.33001 · doi:10.1002/9781118032572 |
| [30] | Turán, On the zeros of the polynomials of Legendre, Časopis Pro Pěstování Matematiky Vol. (1950) |
| [31] | Widder, A classification of generating functions, Trans. Am. Math. Soc. 39 pp 244– (1936) · JFM 62.0471.03 · doi:10.1090/S0002-9947-1936-1501847-6 |
| [32] | Widder, The Laplace Transform (1946) · Zbl 0060.24801 |
| [33] | Whittaker, A Course of Modern Analysis (2000) |
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