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 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 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 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?

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

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