An optimal double inequality among the one-parameter, arithmetic and harmonic means

Authors

  • Wang Miao-Kun Huzhou Teachers College
  • Qiu Ye-Fang Huzhou Teachers College
  • Chu Yu-Ming Huzhou Teachers College

Keywords:

one-parameter mean, arithmetic mean, harmonic mean

Abstract

For \(p\in\mathbb{R}\), the one-parameter mean \(J_{p}(a,b)\), arithmetic mean \(A(a,b)\), and harmonic mean \(H(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}\(A(a,b)=\tfrac{a+b}{2}\), and \(H(a,b)=\tfrac{2ab}{a+b}\),respectively. In this paper, we answer the question: For \(\alpha\in(0,1)\), what are the greatest value \(r_{1}\) and the least value \(r_{2}\) such that the double inequality \(J_{r_{1}}(a,b)<\alpha A(a,b)+(1-\alpha)H(a,b)<J_{r_{2}}(a,b)\) holds for all \(a,b>0\) with \(a\neq b\)?

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References

H. Alzer, On Stolarsky's mean value family, Internat. J. Math. Ed. Sci. Tech., 20(1), pp. 186-189, 1987.

H. Alzer, Über eine einparametrige Familie Von Mittelwerten, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., 1987, pp. 1-9, 1988.

H. Alzer, Über eine einparametrige Familie Von Mittelwerten II, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., 1988, pp. 23-29, 1989.

F. Qi, The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Math. Educ., 5(3), pp. 63-90, 2003.

W.-S. Cheung and F. Qi, Logarithmic convexity of the one-parameter mean values, Taiwanese J. Math., 11(1), pp. 231-237, 2007, https://doi.org/10.11650/twjm/1500404648

F. Qi, P. Cerone, S.S. Dragomir and H.M. Srivastava, Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values, Appl. Math. Comput., 208(1), pp. 129-133, 2009, https://doi.org/10.1016/j.amc.2008.11.023

N.-G. Zheng, Z.-H. Zhang and X.-M. Zhang, Schur-convexity of two types of one-parameter mean values in n variables, J. Inequal. Appl., Art. ID 78175, 10 pages, 2007, https://doi.org/10.1155/2007/78175.

P.S. Bullen, D.S. Mitrinović and P.M. Vasić, Means and Their Inequalities, D. Reidel Pubishing Co., Dordrecht, 1988.

H. Alzer and W. Janous, Solution of problem 8^{∗}, Crux Math., 13, pp. 173-178, 1987, https://doi.org/10.1007/bf01436200

Q.-J. Mao, Power mean, logarithmic mean and Heronian dual mean of two positive number, J. Suzhou Coll. Edu., 16(1-2), pp. 82-85, 1999.

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Published

2010-08-01

How to Cite

Miao-Kun, W., Ye-Fang, Q., & Yu-Ming, C. (2010). An optimal double inequality among the one-parameter, arithmetic and harmonic means. Rev. Anal. Numér. Théor. Approx., 39(2), 169–175. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art9

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