scholarly journals Note on weighted Carleman-type inequality

2005 ◽  
Vol 2005 (3) ◽  
pp. 475-481 ◽  
Author(s):  
Chao-Ping Chen ◽  
Wing-Sum Cheung ◽  
Feng Qi
Keyword(s):  
Type Inequality ◽  
Arithmetic Mean ◽  
Logarithmic Mean ◽  
Double Inequality ◽  
Carleman Type

A double inequality involving the constanteis proved by using an inequality between the logarithmic mean and arithmetic mean. As an application, we generalize the weighted Carleman-type inequality.

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 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.

scholarly journals Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wei-Mao Qian ◽  
Yu-Ming Chu
Keyword(s):  
Unique Solution ◽  
Trigonometric Functions ◽  
Logarithmic Mean ◽  
Double Inequality ◽  
Logarithmic Means ◽  
Inverse Trigonometric Functions

We prove that the double inequalityLp(a,b)<U(a,b)<Lq(a,b)holds for alla,b>0witha≠bif and only ifp≤p0andq≥2and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep0=0.5451⋯is the unique solution of the equation(p+1)1/p=2π/2on the interval(0,∞),U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], andLp(a,b)=[(ap+1-bp+1)/((p+1)(a-b))]1/p  (p≠-1,0),L-1(a,b)=(a-b)/(log⁡a-log⁡b)andL0(a,b)=(aa/bb)1/(a-b)/eare the Yang, andpth generalized logarithmic means ofaandb, respectively.

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 A reverse Hölder type inequality for the logarithmic mean and generalizations

2000 ◽  
Vol 41 (3) ◽  
pp. 401-409 ◽  
Author(s):  
John Maloney ◽  
Jack Heidel ◽  
Josip Pečarić
Keyword(s):  
Type Inequality ◽  
Logarithmic Mean ◽  
Image Position ◽  
Reverse Hölder

AbstractAn inequality involving the logarithmic mean is established. Specifically, we show thatwhere . Then several generalizations are given.

scholarly journals Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Yu-Ming Chu ◽  
Bo-Yong Long
Keyword(s):  
Logarithmic Mean ◽  
Double Inequality ◽  
Classical Means ◽  
Harmonic Means

We answer the question: forα,β,γ∈(0,1)withα+β+γ=1, what are the greatest valuepand the least valueq, such that the double inequalityLp(a,b)<Aα(a,b)Gβ(a,b)Hγ(a,b)<Lq(a,b)holds for alla,b>0witha≠b? HereLp(a,b),A(a,b),G(a,b), andH(a,b)denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbersaandb, respectively.

scholarly journals On HT-convexity and Hadamard-type inequalities

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Ping Bai ◽  
Shu-Hong Wang ◽  
Feng Qi
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Arithmetic Mean ◽  
Logarithmic Mean ◽  
New Class

AbstractIn the paper, the authors define a new notion of “HT-convex function”, present some Hadamard-type inequalities for the new class of HT-convex functions and for the product of any two HT-convex functions, and derive some inequalities for the arithmetic mean and the p-logarithmic mean. These results generalize corresponding ones for HA-convex functions and MT-convex functions.

Sign in / Sign up

Export Citation Format

Share Document