Refinements of bounds for the arithmetic mean by new Seiffert-like means

Wei-Mao Qian; Tie-Hong Zhao; Yu-Pei Lv; Wei-Mao Qian; Tie-Hong Zhao; Yu-Pei Lv

doi:10.3934/math.2021524

AIMS Mathematics

2021, Volume 6, Issue 8: 9036-9047. doi: 10.3934/math.2021524

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Research article

Refinements of bounds for the arithmetic mean by new Seiffert-like means

1.
School of Continuing Education, Huzhou Vocational & Technical College, Huzhou 313000, China
2.
Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
3.
Department of Mathematics, Huzhou University, Huzhou 313000, China

Received: 24 March 2021 Accepted: 09 June 2021 Published: 17 June 2021
MSC : 26D15, 26E60

In the article, we present the sharp upper and lower bounds for the arithmetic mean in terms of new Seiffert-like means, which give some refinements of the results obtained in ^[1]. As applications, two new inequalities for the sine and hyperbolic sine functions will be established.
- Seiffert-like mean,
- tangent mean,
- hyperbolic sine mean,
- sine mean,
- hyperbolic tangent mean,
- arithmetic mean
Citation: Wei-Mao Qian, Tie-Hong Zhao, Yu-Pei Lv. Refinements of bounds for the arithmetic mean by new Seiffert-like means[J]. AIMS Mathematics, 2021, 6(8): 9036-9047. doi: 10.3934/math.2021524

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Abstract

In the article, we present the sharp upper and lower bounds for the arithmetic mean in terms of new Seiffert-like means, which give some refinements of the results obtained in ^[1]. As applications, two new inequalities for the sine and hyperbolic sine functions will be established.

References

[1]	M. Nowicka, A. Witkowski, Optimal bounds for the arithmetic mean in terms of new seiffert-like means, Math. Inequal. Appl., 23 (2020), 383–392.
[2]	A. Witkowski, On Seiffert-like means, J. Math. Inequal., 9 (2015), 1071–1092.
[3]	P. A. Hästö, Optimal inequalities between Seiffert mean and power means, Math. Inequal. Appl., 7 (2004), 47–53.
[4]	Y.-M. Li, M.-K. Wang, Y.-M. Chu, Sharp power mean bounds for Seiffert mean, Appl. Math. J. Chin. Univ., 29 (2014), 101–107. doi: 10.1007/s11766-014-3008-6
[5]	E. Neuman, J. Sándor, On the Schwab-Borchardt mean, Math. Pannon., 14 (2003), 253–266.
[6]	F. Burk, The Geometric, Logarithmic, and Arithmetic Mean Inequality, Amer. Math. Monthly, 94 (1987), 527–528. doi: 10.1080/00029890.1987.12000678
[7]	Z.-H. Yang, Three families of two-parameter means constructed by trigonometric functions, J. Inequal. Appl., 2013 (2013), 541. doi: 10.1186/1029-242X-2013-541
[8]	Z.-H. Yang, Y.-M. Chu, An optimal inequalities chain for bivariate means, J. Math. Inequal., 9 (2015), 331–343.
[9]	Z.-H. Yang, Y.-M. Chu, Inequalities for certain means in two arguments, J. Inequal. Appl., 2015 (2015), 299. doi: 10.1186/s13660-015-0828-8
[10]	Z.-H. Yang, Y.-L. Jiang, Y.-Q. Song, Y.-M. Chu, Sharp inequalities for trigonometric functions, Abstr. Appl. Anal., 2014 (2014), 1–18.
[11]	Y. Chu, B. Liu, M. Wang, Refinements of bounds for the first and second Seiffert means, J. Math. Inequal., 7 (2013), 659–668.
[12]	Y. Chu, T. Zhao, B. Liu, Optimal bounds for Neuman-Sándor mean in terms of the convex combination of logarithmic and quadratic or contra-harmonic means, J. Math. Inequal., 8 (2014), 201–217.
[13]	Y. Chu, T. Zhao, Y. Song, Sharp bounds for Neuman-Sándor mean in terms of the convex combination of quadratic and first Seiffert means, Acta Math. Sci., 34 (2014), 797–806. doi: 10.1016/S0252-9602(14)60050-3
[14]	T.-H. Zhao, M.-K. Wang, Y.-M. Chu, Monotonicity and convexity involving generalized elliptic integral of the first kind, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115 (2021), 1–13. doi: 10.1007/s13398-020-00944-x
[15]	T.-H. Zhao, Z.-Y. He, Y.-M. Chu, Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput. Meth. Funct. Th., (2020), 1–14.
[16]	J.-J. Lei, J.-J. Chen, B.-Y. Long, Optimal bounds for the first Seiffert mean in terms of the convex combination of the logarithmic and Neuman-Sándor mean, J. Math. Inequal., 12 (2018), 365–377.
[17]	Y.-X. Li, M. A. Ali, H. Budak, M. Abbas, Y.-M. Chu, A new generalization of some quantum integral inequalities for quantum differentiable convex functions, Adv. Differ. Equ., 2021 (2021), 1–15. doi: 10.1186/s13662-020-03162-2
[18]	Y.-X. Li, M. H. Alshbool, Y.-P. Lv, I. Khan, M. Riza Khan, A. Issakhov, Heat and mass transfer in MHD Williamson nanofluid flow over an exponentially porous stretching surface, Case Stud. Therm. Eng., 26 (2021), 100975. doi: 10.1016/j.csite.2021.100975
[19]	Y.-X. Li, T. Muhammad, M. Bilal, M. Altaf Khan, A. Ahmadian, B. A. Pansera, Fractional simulation for Darcy-Forchheimer hybrid nanoliquid flow with partial slip over a spinning disk, Alex. Eng. J., 60 (2021), 4787–4796. doi: 10.1016/j.aej.2021.03.062
[20]	Y.-X. Li, A. Rauf, M. Naeem, M. A. Binyamin, A. Aslam, Valency-based topological properties of linear hexagonal chain and hammer-like benzenoid, Complexity, 2021 (2021), 1–16.
[21]	Y.-X. Li, F. Shan, M. Ijaz Khan, R. Chinram, Y. Elmasry, T.-C. Sun, Dynamics of Cattaneo-Christov double diffusion (CCDD) and arrhenius activation law on mixed convective flow towards a stretched Riga device, Chaos Solitons Fractals, 148 (2021), 111010. doi: 10.1016/j.chaos.2021.111010
[22]	F. Qi, W.-H. Li, A unified proof of several inequalities and some new inequalities involving Neuman-Sándor mean, Miskolc Math. Notes, 15 (2014), 665–675. doi: 10.18514/MMN.2014.1176
[23]	H. Sun, T. Zhao, Y. Chu, B. Liu, A note on the Neuman-Sándor mean, J. Math. Inequal., 8 (2014), 287–297.
[24]	M. Nowicka, A. Witkowski, Optimal bounds for the tangent and hyperbolic sine means, Aequat. Math., 94 (2020), 817–827.
[25]	M. Nowicka, A. Witkowski, Optimal bounds for the tangent and hyperbolic sine means II, J. Math. Inequal., 14 (2020), 23–33.
[26]	H.-Z. Xu, W.-M. Qian, Optimal Bounds of the Arithmetic Mean by Harmonic, Contra-harmonic and New Seiffert-like Means, Asian Res. J. Math., 16 (2020), 30–36.
[27]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
[28]	M. Biernacki, J. Krzyz, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sklodowska, 9 (1955), 135–147.
[29]	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. doi: 10.1016/j.jmaa.2015.03.043
[30]	Z.-H. Yang, J.-F. Tian, Sharp inequalities for the generalized elliptic integrals of the first kind, Ramanujan J., 48 (2019), 91–116. doi: 10.1007/s11139-018-0061-4
[31]	M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970.
[32]	F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math, 351 (2019), 1–5.

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