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 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 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 Mathematical Properties of Variable Topological Indices

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 43
Author(s):  
José M. Sigarreta
Keyword(s):  
Real Number ◽  
Large Class ◽  
Symmetric Functions ◽  
Topological Indices ◽  
Current Interest ◽  
Zagreb Indices ◽  
Mathematical Properties ◽  
Connectivity Indices ◽  
Optimal Bounds ◽  
New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

scholarly journals Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

2012 ◽  
Vol 58 (9) ◽  
pp. 6101-6109 ◽  
Author(s):  
Jiantao Jiao ◽  
Lin Zhang ◽  
Robert D. Nowak
Keyword(s):  
Prior Probabilities ◽  
Optimal Bounds

scholarly journals Optimal bounds on finding fixed points of contraction mappings

2010 ◽  
Vol 411 (16-18) ◽  
pp. 1742-1749 ◽  
Author(s):  
Ching-Lueh Chang ◽  
Yuh-Dauh Lyuu
Keyword(s):  
Fixed Points ◽  
Contraction Mappings ◽  
Optimal Bounds

scholarly journals Proof of a conjecture of Granath on optimal bounds of the Landau constants

2016 ◽  
Vol 204 ◽  
pp. 17-33 ◽  
Author(s):  
Chun-Ru Zhao ◽  
Wen-Gao Long ◽  
Yu-Qiu Zhao
Keyword(s):  
Optimal Bounds
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