Abstract
In the paper, in view of two monotonicity rules for the ratios of two functions and of two Maclaurin power series expansions, the authors establish several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals.
Acknowledgement
The authors appreciate the anonymous referees for their careful reading, valuable comments, and helpful suggestions to the original version of this paper.
References
[1] Anderson, G. D.—Vamanamurthy, M. K.—Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.Suche in Google Scholar
[2] Bagul, Y. J.—Dhaigude, R. M.—Chesneau, C.—Kostić, M.: Tight exponential bounds for hyperbolic tangent, Jordan J. Math. Stat. 15(4A) (2022), 807–821.Suche in Google Scholar
[3] Banjac, B.—Malešević, B.—Mićović, M.—Mihailović, B.—Savatović, M.: The best possible constants approach for Wilker–Cusa–Huygens inequalities via stratification, Appl. Anal. Discrete Math. 18(1) (2024), 244–288;10.2298/AADM240308012BSuche in Google Scholar
[4] Biernacki, M.—Krzyż, J.: On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 9 (1955), 135–145.Suche in Google Scholar
[5] Callan, D.: Solution to E 3306, Amer. Math. Monthly 98(10) (1991), 962–964.10.2307/2324162Suche in Google Scholar
[6] Chen, S.—Liu, Z.: Automated proof of mixed trigonometric-polynomial inequalities, J. Symbolic Comput. 101 (2020), 318–329.10.1016/j.jsc.2019.10.002Suche in Google Scholar
[7] Gearhart, W. B.—Shultz, H. S.: The function
[8] Guo, B.-N.—Li, W.—Qi, F.: Proofs of Wilker’s inequalities involving trigonometric functions. In: Inequality Theory and Applications, Vol. 3 (Chinju/Masan, 2001), Nova Science Publishers, Hauppauge, NY, 2003, 109–112.Suche in Google Scholar
[9] Guo, B.-N.—Qiao, B.-M.—Qi, F.—Li, W.: On new proofs of Wilker’s inequalities involving trigonometric functions, Math. Inequal. Appl. 6(1) (2003), 19–22.10.7153/mia-06-02Suche in Google Scholar
[10] Jiang, W.-D.—Luo, Q.-M.—And QI, F.: Refinements and sharpening of some Huygens and Wilker type inequalities, Turkish J. Anal. Number Theory 2 (2014), 134–139.10.12691/tjant-2-4-6Suche in Google Scholar
[11] Li, W.-H.—Guo, B.-N.: Several inequalities for bounding sums of two (hyperbolic) sine cardinal functions, Filomat 38(11) (2024), 3937–3943.Suche in Google Scholar
[12] Li, W.-H.—Miao, P.—Guo, B.-N.: Bounds for the NeumanSándor mean in terms of the arithmetic and contra-harmonic means, Axioms11(5) (2022), Art. No. 236.10.3390/axioms11050236Suche in Google Scholar
[13] Li, W.-H.—Qi, F.: Harmonic mean inequalities for generalized hyperbolic functions, Montes Taurus J. Pure Appl. Math. 6(3) (2024), 199–207.Suche in Google Scholar
[14] Li, W.-H.—Shen, Q.-X.—Guo, B.-N.: Several double inequalities for integer powers of the sinc and sinhc functions with applications to the Neuman–Sándor mean and the first Seiffert mean, Axioms 11(7) (2022), Art. No. 304.10.3390/axioms11070304Suche in Google Scholar
[15] Malešević, B.—Makragić, M.: A method for proving some inequalities on mixed trigonometric polynomial functions, J. Math. Inequal. 10(3) (2016), 849–876.10.7153/jmi-10-69Suche in Google Scholar
[16] Pinelis, I.: l’Hospital rules for monotonicity and the Wilker-Anglesio Inequality, Amer. Math. Monthly 111 (2004), 905–909.10.1080/00029890.2004.11920156Suche in Google Scholar
[17] Sánchez-Reyes, J.: The hyperbolic sine cardinal and the catenary, College Math. J. 43(4) (2012), 285–290.10.4169/college.math.j.43.4.285Suche in Google Scholar
[18] Sumner, J. S.—Jagers, A. A.—Vowe, M.—Anglesio, J.: Inequalities involving trigonometric functions, Amer. Math. Monthly 98(3) (1991), 264–267.10.2307/2325035Suche in Google Scholar
[19] Wilker, J. B.: Problem E3306, Amer. Math. Monthly 96(1) (1989), 55–55.10.2307/2323260Suche in Google Scholar
[20] Wu, S.—Debnath, L.: Wilker-type inequalities for hyperbolic functions, Appl. Math. Lett. 25(5) (2012), 837–842.10.1016/j.aml.2011.10.028Suche in Google Scholar
[21] Wu, S.-H.—Srivastava, H. M.: A weighted and exponential generalization of Wilkers inequality and its applications, Integral Transforms Spec. Funct. 18(8) (2007), 529–535.10.1080/10652460701284164Suche in Google Scholar
[22] Zhang, L.—Zhu, L.: A new elementary proof of Wilkers inequalities, Math. Inequal. Appl. 11(1) (2008), 149–151.10.7153/mia-11-09Suche in Google Scholar
[23] Zhu, L.: New inequalities of Wilkers type for circular functions, AIMS Math. 5(5) (2020), 4874–4888.10.3934/math.2020311Suche in Google Scholar
[24] Zhu, L.: New inequalities of Wilkers type for hyperbolic functions, AIMS Math. 5(1) (2020), 376–384.10.3934/math.2020025Suche in Google Scholar
[25] Zhu, L.: On Wilker-type inequalities, Math. Inequal. Appl. 10(4) (2007), 727–731.10.7153/mia-10-67Suche in Google Scholar
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Artikel in diesem Heft
- Algebraic structures formalizing the logic of effect algebras incorporating time dimension
- Walks on tiled boards
- Composition on FLew-algebras
- On high power sums of a hybrid arithmetic function
- On indices of quintic number fields defined by x5 + ax + b
- Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
- Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
- A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
- On a periodic problem for super-linear second-order ODEs
- Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
- Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
- Self referred equations with an integral boundary condition
- Approximation by matrix means of double Vilenkin-Fourier series
- Maps on self-adjoint operators preserving some relations related to commutativity
- Chains in the Rudin-Frolík order
- Relative versions of first and second countability in hyperspaces
- Proportion estimation in multistage pair ranked set sampling
- The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference