scholarly journals The optimal convex combination bounds of arithmetic and harmonic means in terms of power mean

2012 ◽  
pp. 241-248 ◽  
Author(s):  
Wei-Feng Xia ◽  
Walther Janous ◽  
Yuming Chu
Keyword(s):  
Convex Combination ◽  
Power Mean ◽  
Harmonic Means

scholarly journals Global Bounds for the Generalized Jensen Functional with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2105
Author(s):  
Slavko Simić ◽  
Bandar Bin-Mohsin
Keyword(s):  
Power Mean ◽  
Jensen Functional ◽  
Exact Bounds ◽  
Harmonic Means ◽  
Stolarsky Means ◽  
Generalized Arithmetic

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

Sharp inequalities for bounding Seiffert mean in terms of the arithmetic, centroidal, and contra-harmonic means

2016 ◽  
Vol 66 (5) ◽  
Author(s):  
Wei-Dong Jiang ◽  
Jian Cao ◽  
Feng Qi
Keyword(s):  
Convex Combination ◽  
Parameter Family ◽  
Seiffert Mean ◽  
Second Seiffert Mean ◽  
Harmonic Means ◽  
The Second Seiffert Mean

AbstractIn the paper, the authors find two sharp and double inequalities for bounding the second Seiffert mean either by a one-parameter family of means derived from the centroidal mean or by a convex combination of the arithmetic and contra-harmonic means.

scholarly journals The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

2010 ◽  
Vol 2010 (1) ◽  
pp. 436457 ◽  
Author(s):  
Yu-Ming Chu ◽  
Ye-Fang Qiu ◽  
Miao-Kun Wang ◽  
Gen-Di Wang
Keyword(s):  
Convex Combination ◽  
Harmonic Means

scholarly journals Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yu-Ming Chu ◽  
Li-Min Wu ◽  
Ying-Qing Song
Keyword(s):  
Harmonic Mean ◽  
Power Mean ◽  
Double Inequality ◽  
The One ◽  
Harmonic Means

We present the best possible parametersα=α(r)andβ=β(r)such that the double inequalityMα(a,b)<Hr(a,b)<Mβ(a,b)holds for allr∈(0, 1/2)anda, b>0witha≠b, whereMp(a, b)=[(ap+bp)/2]1/p  (p≠0)andM0(a, b)=abandHr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b)are the power and one-parameter harmonic means ofaandb, respectively.

scholarly journals The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Feng Xia ◽  
Yu-Ming Chu ◽  
Gen-Di Wang
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Positive Real ◽  
Lower Power ◽  
Double Inequality ◽  
Logarithmic Means

Forp∈ℝ, the power meanMp(a,b)of orderp, logarithmic meanL(a,b), and arithmetic meanA(a,b)of two positive real valuesaandbare defined byMp(a,b)=((ap+bp)/2)1/p, forp≠0andMp(a,b)=ab, forp=0,L(a,b)=(b-a)/(log⁡b-log⁡a), fora≠bandL(a,b)=a, fora=bandA(a,b)=(a+b)/2, respectively. In this paper, we answer the question: forα∈(0,1), what are the greatest valuepand the least valueq, such that the double inequalityMp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b)holds for alla,b>0?

scholarly journals Schurm-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Wen Wang ◽  
Shiguo Yang
Keyword(s):  
Convex Functions ◽  
Symmetric Functions ◽  
Power Mean ◽  
Harmonic Means

We investigate the conditions under which the symmetric functionsFn,k(x,r)=∏1≤i1<i2<⋯<ik≤n ‍f(∑j=1k‍xijr)1/r,  k=1,2,…,n,are Schurm-power convex forx∈R++nandr>0. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving thepth power mean and the arithmetic, the geometric, or the harmonic means are presented.

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