scholarly journals Optimal Inequalities among Various Means of Two Arguments

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Ming-yu Shi ◽  
Yu-ming Chu ◽  
Yue-ping Jiang
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Optimal Inequalities

We establish two optimal inequalities among power meanMp(a,b)=(ap/2+bp/2)1/p, arithmetic meanA(a,b)=(a+b)/2, logarithmic meanL(a,b)=(a−b)/(log⁡a−log⁡b), and geometric meanG(a,b)=ab.

scholarly journals The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 380 ◽  
Author(s):  
Yongtao Li ◽  
Xian-Ming Gu ◽  
Jianxing Zhao
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Hölder Inequality ◽  
Power Mean ◽  
Weighted Arithmetic ◽  
Mathematical Equivalence ◽  
Power Mean Inequality ◽  
Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

scholarly journals Fractional Integral Inequalities for Some Convex Functions

2021 ◽  
Vol 104 (4) ◽  
pp. 14-27
Author(s):  
B.R. Bayraktar ◽  
◽  
A.Kh. Attaev ◽  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Arithmetic Mean ◽  
Integral Inequalities ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Real Numbers ◽  
Hadamard Inequality ◽  
Special Means

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

A relation of weak majorization and its applications to certain inequalities for means

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
Shan-He Wu ◽  
Huan-Nan Shi
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Weak Majorization

AbstractA relation of weak majorization for n-dimensional real vectors is established, the result is then used to derive some inequalities involving the power mean, the arithmetic mean and the geometric mean in n variables.

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 A new chain of inequalities involving the Toader-Qi, logarithmic and exponential means

2021 ◽  
pp. 28-28
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Logarithmic Mean

In this paper, we establish an interesting chain of sharp inequalities involving Toader-Qi mean, exponential mean, logarithmic mean, arithmetic mean and geometric mean. This greatly improves some existing results.

On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

2012 ◽  
Vol E95-B (2) ◽  
pp. 647-650
Author(s):  
Ning WANG ◽  
Julian CHENG ◽  
Chintha TELLAMBURA
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Log Ratio

scholarly journals Faecal contamination of drinking water during collection and household storage: the need to extend protection to the point of use

2003 ◽  
Vol 1 (3) ◽  
pp. 109-115 ◽  
Author(s):  
Thomas F. Clasen ◽  
Andrew Bastable
Keyword(s):  
Drinking Water ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Faecal Contamination ◽  
Household Level ◽  
Safe Water ◽  
Female Head ◽  
Point Of Use ◽  
Thermotolerant Coliforms ◽  
And Storage

Paired water samples were collected and analysed for thermotolerant coliforms (TTC) from 20 sources (17 developed or rehabilitated by Oxfam and 3 others) and from the stored household water supplies of 100 households (5 from each source) in 13 towns and villages in the Kailahun District of Sierra Leone. In addition, the female head of the 85 households drawing water from Oxfam improved sources was interviewed and information recorded on demographics, hygiene instruction and practices, sanitation facilities and water collection and storage practices. At the non-improved sources, the arithmetic mean TTC load was 407/100 ml at the point of distribution, rising to a mean count of 882/100 ml at the household level. Water from the improved sources met WHO guidelines, with no faecal contamination. At the household level, however, even this safe water was subject to frequent and extensive faecal contamination; 92.9% of stored household samples contained some level of TTC, 76.5% contained more than the 10 TTC per 100 ml threshold set by the Sphere Project for emergency conditions. The arithmetic mean TTC count for all samples from the sampled households was 244 TTC per 100 ml (geometric mean was 77). These results are consistent with other studies that demonstrate substantial levels of faecal contamination of even safe water during collection, storage and access in the home. They point to the need to extend drinking water quality beyond the point of distribution to the point of consumption. The options for such extended protection, including improved collection and storage methods and household-based water treatment, are discussed.

scholarly journals DUAL HESITANT FUZZY AGGREGATION OPERATORS

2015 ◽  
Vol 22 (2) ◽  
pp. 194-209 ◽  
Author(s):  
Dejian YU ◽  
Wenyu ZHANG ◽  
George HUANG
Keyword(s):  
Human Resources ◽  
Fuzzy Sets ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Aggregation Operators ◽  
Fuzzy Arithmetic ◽  
Hesitant Fuzzy Sets ◽  
Numerical Example ◽  
Fuzzy Environment ◽  
Fuzzy Aggregation

Dual hesitant fuzzy sets (DHFSs) is a generalization of fuzzy sets (FSs) and it is typical of membership and non-membership degrees described by some discrete numerical. In this article we chiefly concerned with introducing the aggregation operators for aggregating dual hesitant fuzzy elements (DHFEs), including the dual hesitant fuzzy arithmetic mean and geometric mean. We laid emphasis on discussion of properties of newly introduced operators, and give a numerical example to describe the function of them. Finally, we used the proposed operators to select human resources outsourcing suppliers in a dual hesitant fuzzy environment.

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