New inequalities of Wilker’s type for circular functions. (English) Zbl 1484.33003
Summary: In the article, we establish three new Wilker type inequalities involving tangent and sine functions by use of a double inequality for the ratio of two consecutive non-zero Bernoulli numbers.
MSC:
| 33B10 | Exponential and trigonometric functions |
| 26D05 | Inequalities for trigonometric functions and polynomials |
Keywords:
Wilker type inequality; circular function; tangent function; sine function; Bernoulli numberReferences:
| [1] | S. Rashid; M. A. Noor; K. I. Noor, t al. <i>Hermite-Hadamrad type inequalities for the class of</i> <i>convex functions on time scale</i, Mathematics, 7 (2019) |
| [2] | M. A. Latif; S. Rashid; S. S. Dragomir, t al. <i>Hermite-Hadamard type inequalities for co-ordinated</i> <i>convex and qausi-convex functions and their applications</i, J. Inequal. Appl., 2019 (2019) · Zbl 1499.26141 |
| [3] | M. Adil Khan; N. Mohammad; E. R. Nwaeze, t al. <i>Quantum Hermite-Hadamard inequality by</i> <i>means of a Green function</i, Adv. Difference Equ., 2020 (2020) · Zbl 1482.26026 |
| [4] | A. Iqbal; M. Adil Khan; S. Ullah, t al. <i>Some new Hermite-Hadamard-type inequalities associated</i> <i>with conformable fractional integrals and their applications</i, J. Funct. Spaces, 2020 (2020) · Zbl 1436.26020 |
| [5] | M. U. Awan; S. Talib; Y. M. Chu, t al. <i>Some new refinements of Hermite-Hadamard-type</i> <i>inequalities involving</i> Ψ<sub><i>k</i></sub><i>-Riemann-Liouville fractional integrals and applications</i, Math. Probl. Eng., 2020 (2020) |
| [6] | M. U. Awan; N. Akhtar; S. Iftikhar, t al. <i>Hermite-Hadamard type inequalities for n-polynomial</i> <i>harmonically convex functions</i, J. Inequal. Appl., 2020 (2020) · Zbl 1503.26024 |
| [7] | H. Z. Xu; Y. M. Chu; W. M. Qian, i>Sharp bounds for the Sándor-Yang means in terms of arithmetic</i> <i>and contra-harmonic means</i, J. Inequal. Appl., 2018 (2018) · Zbl 1497.26042 |
| [8] | W. M. Qian; Y. Y. Yang; H. W. Zhang, t al. <i>Optimal two-parameter geometric and arithmetic</i> <i>mean bounds for the Sándor-Yang mean</i, J. Inequal. Appl., 2019 (2019) · Zbl 1499.26226 |
| [9] | W. M. Qian; W. Zhang; Y. M. Chu, i>Bounding the convex combination of arithmetic and integral</i> <i>means in terms of one-parameter harmonic and geometric means</i, Miskolc Math. Notes, 20, 1157-1166 (2019) · Zbl 1449.26053 |
| [10] | M. K. Wang; Z. Y. He; Y. M. Chu, i>Sharp power mean inequalities for the generalized elliptic</i> <i>integral of the first kind</i, Comput. Methods Funct. Theory, 20, 111-124 (2020) · Zbl 1437.33018 · doi:10.1007/s40315-020-00298-w |
| [11] | B. Wang; C. L. Luo; S. H. Li, t al. <i>Sharp one-parameter geometric and quadratic means bounds</i> <i>for the Sándor-Yang means</i, RACSAM, 114 (2020) · Zbl 1434.26075 |
| [12] | T. H. Zhao; Y. M. Chu; H. Wang, i>Logarithmically complete monotonicity properties relating to the</i> <i>gamma function</i, Abstr. Appl. Anal., 2011 (2011) · Zbl 1221.33008 |
| [13] | W. M. Qian; Z. Y. He; Y. M. Chu, i>Approximation for the complete elliptic integral of the first kind</i, RACSAM, 114 (2020) · Zbl 1434.33023 |
| [14] | Z. H. Yang; W. M. Qian; W. Zhang, t al. <i>Notes on the complete elliptic integral of the first kind</i, Math. Inequal. Appl., 23, 77-93 (2020) · Zbl 1440.33020 |
| [15] | M. K. Wang; H. H. Chu; Y. M. Li, t al. <i>Answers to three conjectures on convexity of three functions</i> <i>involving complete elliptic integrals of the first kind</i, Appl. Anal. Discrete Math., 14, 255-271 (2020) · Zbl 1474.33081 |
| [16] | T. H. Zhao; L. Shi; Y. M. Chu, i>Convexity and concavity of the modified Bessel functions of the first</i> <i>kind with respect to Hölder means</i, RACSAM, 114 (2020) · Zbl 1436.26028 |
| [17] | M. Adil Khan; M. Hanif; Z. A. Khan, t al. <i>Association of Jensen’s inequality for s-convex function</i> <i>with Csiszár divergence</i, J. Inequal. Appl., 2019 (2019) · Zbl 1499.26087 |
| [18] | S. Khan; M. Adil Khan; Y. M. Chu, i>Converses of the Jensen inequality derived from the Green</i> <i>functions with applications in information theory</i, Math. Methods Appl. Sci., 43, 2577-2587 (2020) · Zbl 1447.26027 · doi:10.1002/mma.6066 |
| [19] | S. Khan; M. Adil Khan; Y. M. Chu, i>New converses of Jensen inequality via Green functions with</i> <i>applications</i, RACSAM, 114 (2020) · Zbl 1437.26030 |
| [20] | S. Rashid; M. A. Noor; K. I. Noor, t al. <i>Ostrowski type inequalities in the sense of generalized</i> K<i>-fractional integral operator for exponentially convex functions</i, AIMS Math., 5, 2629-2645 (2020) · Zbl 1484.26038 · doi:10.3934/math.2020171 |
| [21] | X. M. Hu; J. F. Tian; Y. M. Chu, t al. <i>On Cauchy-Schwarz inequality for N-tuple diamond-alpha</i> <i>integral</i, J. Inequal. Appl., 2020 (2020) · Zbl 1503.26057 |
| [22] | S. Rashid; F. Jarad; Y. M. Chu, i>A note on reverse Minkowski inequality via generalized proportional</i> <i>fractional integral operator with respect to another function</i, Math. Probl. Eng., 2020 (2020) |
| [23] | I. A. Baloch; Y. M. Chu, i>Petrović-type inequalities for harmonic h-convex functions</i, J. Funct. Space., 2020 (2020) · Zbl 1436.26018 |
| [24] | S. Rashid; F. Jarad; H. Kalsoom, t al. <i>On Pólya-Szegö and Ćebyšev type inequalities via</i> <i>generalized k-fractional integrals</i, Adv. Difference Equ., 2020 (2020) · Zbl 1482.26039 |
| [25] | S. Rafeeq; H. Kalsoom; S. Hussain, t al. <i>Delay dynamic double integral inequalities on time scales</i> <i>with applications</i, Adv. Difference Equ., 2020 (2020) · Zbl 1487.26051 |
| [26] | S. Z. Ullah; M. Adil Khan; Z. A. Khan, t al. <i>Integral majorization type inequalities for the</i> <i>functions in the sense of strong convexity</i, J. Funct. Spaces, 2019 (2019) · Zbl 1429.26029 |
| [27] | S. Z. Ullah; M. Adil Khan; Y. M. Chu, i>A note on generalized convex functions</i, J. Inequal. Appl., 2019 (2019) · Zbl 1499.26039 |
| [28] | S. Rashid; F. Jarad; M. A. Noor, t al. <i>Inequalities by means of generalized proportional fractional</i> <i>integral operators with respect another function</i, Math., 7 (2019) |
| [29] | M. K. Wang; M. Y. Hong; Y. F. Xu, t al. <i>Inequalities for generalized trigonometric and hyperbolic</i> <i>functions with one parameter</i, J. Math. Inequal., 14, 1-21 (2020) · Zbl 1437.26016 |
| [30] | S. Rashid; R. Ashraf; M. A. Noor, t al. <i>New weighted generalizations for di fferentiable</i> <i>exponentially convex mappings with application</i, AIMS Math., 5, 3525-3546 (2020) · Zbl 1484.26037 · doi:10.3934/math.2020229 |
| [31] | H. L. Montgomery, J. D. Vaaler, J. Delany, et al. Elementary |
| [32] | J. B. Wilker, J. S. Sumner, A. A. Jagers, et al. Solutions of Elementary |
| [33] | L. Zhu, A new simple proof of Wilker’s inequality, Math. Inequal. Appl., 8 (2005), 749-750. · Zbl 1084.26008 |
| [34] | L. Zhu, A new elementary proof of Wilker’s inequalities, Math. Inequal. Appl., 11 (2008), 149-151. · Zbl 1138.26307 |
| [35] | L. Zhu, On Wilker-type inequalities, Math. Inequal. Appl., 10 (2007), 727-731. · Zbl 1144.26020 |
| [36] | Z. J. Sun, L. Zhu, On new Wilker-type inequalities, ISRN Math. Anal., 2011 (2011), 681702. · Zbl 1226.26011 |
| [37] | L. Zhu, New inequalities of Wilker’s type for hyperbolic functions, AIMS Math., 5 (2019), 376-384. · Zbl 1484.26021 |
| [38] | Z. H. Yang, Y. M. Chu, M. K. Wang, Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl., 428 (2015), 587-604. · Zbl 1321.26019 |
| [39] | A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, 2004. · Zbl 1078.00011 |
| [40] | J. L. Li, An identity related to Jordan’s inequality, Int. J. Math. Math. Sci., 2006 (2006), 76782. · Zbl 1143.26010 |
| [41] | Z. H. Yang, J. F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math., 364 (2020), 112359. · Zbl 1472.11233 |
| [42] | L. Zhu, A source of inequalities for circular functions, Comput. Math. Appl., 58 (2009), 1998-2004. · Zbl 1189.26029 |
| [43] | M. Masjed-Jamei, S. S. Dragomir, H. M. Srivastava, Some generalizations of the Cauchy-Schwarz and the Cauchy-Bunyakovsky inequalities involving four free parameters and their applications, Math. Comput. Modelling, 49 (2009), 1960-1968. · Zbl 1171.26331 |
| [44] | M. Masjed-Jamei, S. S. Dragomir, A new generalization of the Ostrowski inequality and applications, Filomat, 25 (2011), 115-123. · Zbl 1265.26098 |
| [45] | M. Masjed-Jamei, A main inequality for several special functions, Comput. Math. Appl., 60 (2010), 1280-1289. · Zbl 1201.33001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
