Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means
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Research Article Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means ´ o´ Horvath, ´ 1 Khuram Ali Khan,2, 3 and J. Pecari ˇ c´ 2, 4 Laszl 1
Department of Mathematics, University of Pannonia, University Street 10, 8200 Veszpr´em, Hungary 2 Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan 3 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 4 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Khuram Ali Khan, [email protected] Received 26 November 2010; Accepted 23 February 2011 Academic Editor: Michel Chipot Copyright q 2011 L´aszlo´ Horv´ath et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A recent refinement of the classical discrete Jensen inequality is given by Horv´ath and Peˇcari´c. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.
1. Introduction and Preliminary Results A new refinement of the discrete Jensen inequality is given in 1. The following notations are also introduced in 1. Let X be a set, P X its power set, and |X| denotes the number of elements in X. Let u ≥ 1 and v ≥ 2 be fixed integers. Define the functions Sv,w : {1, . . . , u}v −→ {1, . . . , u}v−1 ,
1 ≤ w ≤ v,
Sv : {1, . . . , u}v −→ P {1, . . . , u}v−1 , Tv : P {1, . . . , u}v −→ P {1, . . . , u}v−1
1.1
2
Journal of Inequalities and Applications
by Sv,w i1 , . . . , iv : i1 , . . . , iw−1 , iw1 , . . . , iv , Sv i1 , . . . , iv
v
1 ≤ w ≤ v,
{Sv,w i1 , . . . , iv }, 1.2
w1
Tv I
⎧ ⎪ ⎨
Sv i1 , . . . , iv ,
I/ φ,
i1 ,...,iv ∈I
⎪ ⎩φ,
I φ.
Further, introduce the function αv,i : {1, . . . , u}v −→ N,
1 ≤ i ≤ u,
1.3
via αv,i i1 , . . . , iv : Number of occurrences of i in the sequence i1 , . . . , iv .
1.4
For each I ∈ P {1, . . . , u}v , let αI,i :
αv,i i1 , . . . , iv ,
1 ≤ i ≤ u.
i1 ,...,iv ∈I
1.5
It is easy to observe from the construction of the functions Sv , Sv,w , Tv and αv,i that they do not depend essentially on u, so we can write for short Sv for Suv , and so on. H1 The following considerations concern a subset Ik of {1, . . . , n}k satisfying αIk ,i ≥ 1,
1 ≤ i ≤ n,
1.6
where n ≥ 1 and k ≥ 2 are fixed integers. Next, we proceed inductively to define the sets Il ⊂ {1, . . . , n}l k − 1 ≥ l ≥ 1 by Il−1 : Tl Il ,
k ≥ l ≥ 2.
1.7
By 1.6, I1 {1, . . . , n} and this implies that αI1 ,i 1 for 1 ≤ i ≤ n. From 1.6, again, we have αIl ,i ≥ 1 k − 1 ≥ l ≥ 1, 1 ≤ i ≤ n. For every k ≥ l ≥ 2 and for any j1 , . . . , jl−1 ∈ Il−1 , let HIl j1 , . . . , jl−1 : i1 , . . . , il , m
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