scholarly journals Optimal Inequalities for Power Means

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yong-Min Li ◽  
Bo-Yong Long ◽  
Yu-Ming Chu ◽  
Wei-Ming Gong
Keyword(s):  
Power Mean ◽  
Power Means ◽  
Optimal Inequalities

We present the best possible power mean bounds for the productMpα(a,b)M-p1-α(a,b)for anyp>0,α∈(0,1), and alla,b>0witha≠b. Here,Mp(a,b)is thepth power mean of two positive numbersaandb.

scholarly journals VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

2020 ◽  
pp. 1-18 ◽  
Author(s):  
MOHSEN KIAN ◽  
MOHAMMAD SAL MOSLEHIAN ◽  
YUKI SEO
Keyword(s):  
Hilbert Space ◽  
Norm Inequality ◽  
Power Mean ◽  
Variable Operator ◽  
Linear Map ◽  
Power Means ◽  
Operator Mean ◽  
Positive Linear Map ◽  
Euclidean Mean

Abstract For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

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 Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Hongya Gao ◽  
Jianling Guo ◽  
Wanguo Yu
Keyword(s):  
Power Mean ◽  
Sharp Bounds ◽  
Power Means ◽  
Heronian Mean

For1<r<+∞, we find the least valueαand the greatest valueβsuch that the inequalityHα(a,b)<Ar(a,b)<Hβ(a,b)holds for alla,b>0witha≠b. Here,Hω(a,b)andAr(a,b)are the generalized Heronian and the power means of two positive numbersaandb, respectively.

scholarly journals A lower bound for ratio of power means

2004 ◽  
Vol 2004 (1) ◽  
pp. 49-53
Author(s):  
Feng Qi ◽  
Bai-Ni Guo ◽  
Lokenath Debnath
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Positive Real Number ◽  
Positive Sequence ◽  
Power Mean ◽  
Natural Numbers ◽  
Positive Real ◽  
Power Means

Letnandmbe natural numbers. Suppose that{ai}i=1n+mis an increasing, logarithmically convex, and positive sequence. Denote the power meanPn(r)for any given positive real numberrbyPn(r)=((1/n)∑i=1nair)1/r. ThenPn(r)/Pn+m(r)≥an/an+m. The lower bound is the best possible.

scholarly journals Sharp Power Mean Bounds for Sándor Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Yu-Ming Chu ◽  
Zhen-Hang Yang ◽  
Li-Min Wu
Keyword(s):  
Power Mean ◽  
Double Inequality ◽  
Power Means

We prove that the double inequalityMp(a,b)<X(a,b)<Mq(a,b)holds for alla,b>0witha≠bif and only ifp≤1/3andq≥log 2/(1+log 2)=0.4093…, whereX(a,b)andMr(a,b)are the Sándor andrth power means ofaandb, respectively.

scholarly journals One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Xiaoxue Li ◽  
Li Chen
Keyword(s):  
Exponential Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Linear Recurrence ◽  
Power Mean ◽  
Hybrid Power ◽  
Recurrence Formulas ◽  
Analytic Methods ◽  
Power Means ◽  
Hybrid Power Mean

The main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.

scholarly journals Optimal Lower Power Mean Bound for the Convex Combination of Harmonic and Logarithmic Means

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yu-Ming Chu ◽  
Shan-Shan Wang ◽  
Cheng Zong
Keyword(s):  
Convex Combination ◽  
Power Mean ◽  
Lower Power ◽  
Power Means ◽  
Logarithmic Means

We find the least valueλ∈(0,1)and the greatest valuep=p(α)such thatαH(a,b)+(1−α)L(a,b)>Mp(a,b)forα∈[λ,1)and alla,b>0witha≠b, whereH(a,b),L(a,b), andMp(a,b)are the harmonic, logarithmic, andp-th power means of two positive numbersaandb, respectively.

Gaussian ϕ(ρz) curves: Comparison with other models

1990 ◽  
Vol 48 (2) ◽  
pp. 192-193
Author(s):  
Alberto Riveros ◽  
Gustavo Castellano
Keyword(s):  
Good Description ◽  
Bulk Sample ◽  
Incident Electron Energy ◽  
Power Mean ◽  
General Shape ◽  
Ionization Cross ◽  
X Ray ◽  
Mass Absorption ◽  
Absorption Correction ◽  
Image Position

X ray characteristic intensity Ii , emerging from element i in a bulk sample irradiated with an electron beam may be obtained throughwhere the function ϕi(ρz) is the distribution of ionizations for element i with the mass depth ρz, ψ is the take-off angle and μi the mass absorption coefficient to the radiation of element i.A number of models has been proposed for ϕ(ρz), involving several features concerning the interaction of electrons with matter, e.g. ionization cross section, stopping power, mean ionization potential, electron backscattering, mass absorption coefficients (MAC’s). Several expressions have been developed for these parameters, on which the accuracy of the correction procedures depends.A great number of experimental data and Monte Carlo simulations show that the general shape of ϕ(ρz) curves remains substantially the same when changing the incident electron energy or the sample material. These variables appear in the parameters involved in the expressions for ϕ(ρz). A good description of this function will produce an adequate combined atomic number and absorption correction.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

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