scholarly journals Sharp Bounds for the Weighted Geometric Mean of the First Seiffert and Logarithmic Means in terms of Weighted Generalized Heronian Mean

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Geometric Mean ◽  
Sharp Bounds ◽  
Double Inequality ◽  
Logarithmic Means ◽  
Optimal Bounds ◽  
Heronian Mean

Optimal bounds for the weighted geometric mean of the first Seiffert and logarithmic means by weighted generalized Heronian mean are proved. We answer the question: for what the greatest value and the least value such that the double inequality, , holds for all with are. Here, and denote the first Seiffert, logarithmic, and weighted generalized Heronian means of two positive numbers and respectively.

scholarly journals Optimal Bounds for Seiffert Mean in terms of One-Parameter Means

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Hua-Nan Hu ◽  
Guo-Yan Tu ◽  
Yu-Ming Chu
Keyword(s):  
Double Inequality ◽  
Seiffert Mean ◽  
Optimal Bounds

The authors present the greatest valuer1and the least valuer2such that the double inequalityJr1(a, b)<T(a, b)<Jr2(a, b)holds for alla, b>0witha≠b, whereT(a, b)andJp(a, b)denote the Seiffert andpth one-parameter means of two positive numbersaandb, respectively.

scholarly journals Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Yu-Ming Chu ◽  
Shou-Wei Hou
Keyword(s):  
Sharp Bounds ◽  
Double Inequality ◽  
Seiffert Mean ◽  
Contraharmonic Mean

We find the greatest valueαand the least valueβin(1/2,1)such that the double inequalityC(αa+(1-α)b,αb+(1-α)a)<T(a,b)<Cβa+1-βb,βb+(1-βa)holds for alla,b>0witha≠b. Here,T(a,b)=(a-b)/[2 arctan((a-b)/(a+b))]andCa,b=(a2+b2)/(a+b)are the Seiffert and contraharmonic means ofaandb, respectively.

scholarly journals Optimal Bounds for the Neuman-Sándor Mean in terms of the Convex Combination of the First and Second Seiffert Means

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Hao-Chuan Cui ◽  
Nan Wang ◽  
Bo-Yong Long
Keyword(s):  
Convex Combination ◽  
Double Inequality ◽  
Optimal Bounds

We find the least valueαand the greatest valueβsuch that the double inequalityαP(a,b)+(1-α)T(a,b)<M(a,b)<βP(a,b)+(1-β)T(a,b)holds for alla,b>0witha≠b, whereM(a,b), P(a,b), andT(a,b)are the Neuman-Sándor mean and the first and second Seiffert means of two positive numbersaandb, respectively.

scholarly journals Sharp bounds for the arithmetic-geometric mean

2014 ◽  
Vol 2014 (1) ◽  
pp. 192 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Ying-Qing Song ◽  
Yu-Ming Chu
Keyword(s):  
Geometric Mean ◽  
Sharp Bounds

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 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 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 Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Yu-Ming Chu ◽  
Bo-Yong Long ◽  
Wei-Ming Gong ◽  
Ying-Qing Song
Keyword(s):  
Sharp Bounds ◽  
Logarithmic Means

A double inequality for an integral mean in terms of the exponential and logarithmic means

2016 ◽  
Vol 75 (2) ◽  
pp. 180-189 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu ◽  
Zhen-Hang Yang
Keyword(s):  
Double Inequality ◽  
Logarithmic Means
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