An Inequality for Complete Symmetric Functions

Ilori, Samuel A.

doi:10.4153/cmb-1978-086-x

Cited by 2 publications

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“…It has been investigated by many mathematicians and there are many interesting results in the literature. See, for example, , , , , , , , , , , , , , and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Some properties of a class of symmetric functions and its applications

Sun

Chen

2014

Mathematische Nachrichten

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For x=(x1,...,xn)∈[0,1)n∪(1,+∞)n, the symmetric functions Fn(x,r) are defined by Fn(x,r)=Fn(x1,x2,...,xn;r)=∑i1+i2+⋯+in=r()x11−x1i1()x21−x2i2⋯()xn1−xnin,where r=1,2,...,n,..., and i1,i2,...,in are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of Fn(x,r) are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.

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“…It has been investigated by many mathematicians and there are many interesting results in the literature. See, for example, , , , , , , , , , , , , , and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Some properties of a class of symmetric functions and its applications

Sun

Chen

2014

Mathematische Nachrichten

View full text Add to dashboard Cite

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“…It has been investigated by many mathematicians and many interesting results are in literature. See, for example, [5], [8], [17], [24], [27], [33], [34], [36], [37], [39], [44], [45], [50] and the references therein. Another important function ϕ n (x, r) is defined by ϕ n (x, r) = F n (x, r) F n (x, r − 1)…”

Section: Introductionmentioning

confidence: 99%

The Schur multiplicative and harmonic convexities of the complete symmetric function

Chu

Wang

Zhang

2011

Mathematische Nachrichten

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Key words Complete symmetric function, Schur multiplicative convexity, Schur harmonic convexity, theory of majorization MSC (2010) 26B25 This paper investigates the Schur multiplicative and harmonic convexities of the complete symmetric function Fn (x, r) = i 1 + i 2 + ···+ i n = r x i 1 1 x i 2 2 . . . x i n n and the function ϕn (x, r) = F n (x ,r ) F n (x ,r −1) , where i1 , i2 , . . . , in are nonnegative integers and r ≥ 1. As applications, some analytic inequalities are established by use of the theory of majorization.

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An Inequality for Complete Symmetric Functions

Abstract: Consider the identitywhere aj, …, am are positive real numbers. Then for r = 1, 2, 3, … Tr =Tr(a1, …, am) is called the rth complete symmetric function in a1, …, am (T0=l).

Cited by 2 publications

References 3 publications

Some properties of a class of symmetric functions and its applications

Some properties of a class of symmetric functions and its applications

The Schur multiplicative and harmonic convexities of the complete symmetric function

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