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 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.

Sharp bounds for Seiffert mean in terms of arithmetic and geometric means

2013 ◽  
Vol 7 ◽  
pp. 1765-1773
Author(s):  
Baoyu Liu ◽  
Weiming Gong ◽  
Yingqing Song ◽  
Yuming Chu
Keyword(s):  
Geometric Means ◽  
Sharp Bounds ◽  
Seiffert Mean

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 Two Parameter Bounds for the Seiffert Mean

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Hui Sun ◽  
Ying-Qing Song ◽  
Yu-Ming Chu
Keyword(s):  
Parameter Family ◽  
Sharp Bounds ◽  
Seiffert Mean ◽  
Two Parameter ◽  
Parameter Bounds

We obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).

scholarly journals Sharp Generalized Seiffert Mean Bounds for Toader Mean

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Yu-Ming Chu ◽  
Miao-Kun Wang ◽  
Song-Liang Qiu ◽  
Ye-Fang Qiu
Keyword(s):  
Elliptic Integrals ◽  
Double Inequality ◽  
Seiffert Mean ◽  
Toader Mean ◽  
Complete Elliptic Integrals

Forp∈[0,1], the generalized Seiffert mean of two positive numbersaandbis defined bySp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)],  0<p≤1,  a≠b;  (a+b)/2,  p=0,  a≠b;  a,  a=b. In this paper, we find the greatest valueαand least valueβsuch that the double inequalitySα(a,b)<T(a,b)<Sβ(a,b)holds for alla,b>0witha≠b, and give new bounds for the complete elliptic integrals of the second kind. Here,T(a,b)=(2/π)∫0π/2a2cos⁡2θ+b2sin⁡2θdθdenotes the Toader mean of two positive numbersaandb.

scholarly journals Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhi-Jun Guo ◽  
Yu-Ming Chu ◽  
Ying-Qing Song ◽  
Xiao-Jing Tao
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Sharp Bounds ◽  
Contraharmonic Mean

We give several sharp bounds for the Neuman meansNAHandNHA(NCAandNAC) in terms of harmonic meanH(contraharmonic meanC) or the geometric convex combination of arithmetic meanAand harmonic meanH(contraharmonic meanCand arithmetic meanA) and present a new chain of inequalities for certain bivariate means.

scholarly journals A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

2016 ◽  
Vol 99 (113) ◽  
pp. 237-242 ◽  
Author(s):  
Wei-Dong Jiang ◽  
Feng Qi
Keyword(s):  
Arithmetic Mean ◽  
Double Inequality ◽  
Toader Mean ◽  
Contraharmonic Mean

We find the greatest value ? and the least value ? such that the double inequality C(?a +(1-?)b, ?b + (1-?)a) < ?A(a,b) + (1-?)T(a, b)< C(?a + (1-?)b, ?b + (1-?)a) holds for all ? ? (0,1) and a, b > 0 with a ? b, where C(a,b), A(a,b), and T(a,b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.

scholarly journals New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 901
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian ◽  
Ya-Ru Zhu
Keyword(s):  
Bessel Function ◽  
Modified Bessel Function ◽  
Sharp Bounds ◽  
Double Inequality

Let I v x be he modified Bessel function of the first kind of order v. We prove the double inequality sinh t t cosh 1 / q q t < I 0 t < sinh t t cosh 1 / p p t holds for t > 0 if and only if p ≥ 2 / 3 and q ≤ ln 2 / ln π . The corresponding inequalities for means improve already known results.

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