Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means

Wei Mao Qian; Wen Zhang; Yu Ming Chu

doi:10.18514/MMN.2019.2334

Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means

Wei Mao Qian
, Wen Zhang
, Yu Ming Chu

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

Abstract

In the article, we find the best possible parameters 1, ₂₂ on the interval [0,1/2 such that the double inequalities H(a,b₁) A(a,b)+(1-)T(a,b) H(a,b₁), G(a,b₂) A(a,b)+(1-)T(a,b) G(a,b₂), hold for all [small element of] [0,1] and a;b 0 with a [not asymptotically equivalent] b, where A(a,b) = (a+b)/2, T (a,b)= 2int;₀^/2a^cos²bsin2d/,HH(a,b)=2[a+(1-)b][b+(1-)a]/(a+b),G(a,b;)=[a+(1-b)][b+(1-a)] are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of a and b, respectively.

Original language	English
Pages (from-to)	1157-1166
Number of pages	10
Journal	Miskolc Mathematical Notes
Volume	20
Issue number	2
DOIs	https://doi.org/10.18514/MMN.2019.2334
State	Published - 2019

Keywords

Arithmetic mean
Geometric mean
Harmonic mean
Integral mean
Modified bessel function

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10.18514/MMN.2019.2334

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title = "Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means",

abstract = "In the article, we find the best possible parameters 1, 22 on the interval [0,1/2 such that the double inequalities H(a,b1) A(a,b)+(1-)T(a,b) H(a,b1), G(a,b2) A(a,b)+(1-)T(a,b) G(a,b2), hold for all [small element of] [0,1] and a;b 0 with a [not asymptotically equivalent] b, where A(a,b) = (a+b)/2, T (a,b)= 2int;0/2acos2bsin2d/,HH(a,b)=2[a+(1-)b][b+(1-)a]/(a+b),G(a,b;)=[a+(1-b)][b+(1-a)] are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of a and b, respectively.",

keywords = "Arithmetic mean, Geometric mean, Harmonic mean, Integral mean, Modified bessel function",

author = "Qian, \{Wei Mao\} and Wen Zhang and Chu, \{Yu Ming\}",

note = "Publisher Copyright: {\textcopyright}2019 Miskolc University Press.",

year = "2019",

doi = "10.18514/MMN.2019.2334",

language = "English",

volume = "20",

pages = "1157--1166",

journal = "Miskolc Mathematical Notes",

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publisher = "University of Miskolc",

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TY - JOUR

T1 - Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means

AU - Qian, Wei Mao

AU - Zhang, Wen

AU - Chu, Yu Ming

PY - 2019

Y1 - 2019

N2 - In the article, we find the best possible parameters 1, 22 on the interval [0,1/2 such that the double inequalities H(a,b1) A(a,b)+(1-)T(a,b) H(a,b1), G(a,b2) A(a,b)+(1-)T(a,b) G(a,b2), hold for all [small element of] [0,1] and a;b 0 with a [not asymptotically equivalent] b, where A(a,b) = (a+b)/2, T (a,b)= 2int;0/2acos2bsin2d/,HH(a,b)=2[a+(1-)b][b+(1-)a]/(a+b),G(a,b;)=[a+(1-b)][b+(1-a)] are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of a and b, respectively.

AB - In the article, we find the best possible parameters 1, 22 on the interval [0,1/2 such that the double inequalities H(a,b1) A(a,b)+(1-)T(a,b) H(a,b1), G(a,b2) A(a,b)+(1-)T(a,b) G(a,b2), hold for all [small element of] [0,1] and a;b 0 with a [not asymptotically equivalent] b, where A(a,b) = (a+b)/2, T (a,b)= 2int;0/2acos2bsin2d/,HH(a,b)=2[a+(1-)b][b+(1-)a]/(a+b),G(a,b;)=[a+(1-b)][b+(1-a)] are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of a and b, respectively.

KW - Arithmetic mean

KW - Geometric mean

KW - Harmonic mean

KW - Integral mean

KW - Modified bessel function

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DO - 10.18514/MMN.2019.2334

M3 - Article

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SN - 1787-2405

VL - 20

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EP - 1166

JO - Miskolc Mathematical Notes

JF - Miskolc Mathematical Notes

IS - 2

ER -

Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means

Abstract

Keywords

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