On Logarithmic Convexity for Power Sums and Related Results
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Research Article On Logarithmic Convexity for Power Sums and Related Results ˇ c´ 1, 2 and Atiq Ur Rehman2 J. Pecari 1 2
Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Abdus Salam School of Mathematical Sciences, GC University, Lahore 54660, Pakistan
Correspondence should be addressed to Atiq Ur Rehman, [email protected] Received 28 March 2008; Revised 23 May 2008; Accepted 29 June 2008 Recommended by Martin j. Bohner We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results. Copyright q 2008 J. Peˇcari´c and A. U. Rehman. 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.
1. Introduction and preliminaries Let x x1 , . . . , xn , p p1 , . . . , pn denote two sequences of positive real numbers with n i1 pi 1. The well-known Jensen Inequality 1, page 43 gives the following, for t < 0 or t > 1: t n n t p i xi ≥ p i xi 1.1 i1
i1
and vice versa for 0 < t < 1. Simi´c 2 has considered the difference of both sides of 1.1. He considers the function defined as ⎧ n n t t ⎪ i1 pi xi − i1 pi xi ⎪ ⎪ , t/ 0, 1; ⎪ ⎪ ⎪ tt − 1 ⎪ ⎪ ⎪ ⎪ ⎨ n n λt log 1.2 pi xi − pi log xi , t 0; ⎪ ⎪ i1 i1 ⎪ ⎪ ⎪ ⎪ n n n ⎪ ⎪ ⎪ ⎪ pi xi log pi xi , t 1; ⎩ pi xi log xi − i1
and has proved the following theorem.
i1
i1
2
Journal of Inequalities and Applications
Theorem 1.1. For −∞ < r < s < t < ∞, then t−s s−r λt−r . s ≤ λr λt
1.3
Anwar and Peˇcari´c 3 have considerd further generalization of Theorem 1.1. Namely, they introduced new means of Cauchy type in 4 and further proved comparison theorem for these means. In this paper, we will give some results in the case where instead of means we have power sums. Let x be positive n-tuples. The well-known inequality for power sums of order s and r, for s > r > 0 see 1, page 164, states that 1/s 1/r n n s r xi < xi . 1.4 i1
i1
Moreover, if p p1 , . . . , pn is a positive n-tuples such that pi ≥ 1 i 1, . . . , n, then for s > r > 0 see 1, page 165, we have 1/s 1/r n n s r p i xi < p i xi . 1.5 i1
i1
Let us note that 1.5 can also be obtained from the following theorem 1, page 152. Theorem 1.2. Let x and p be two nonnegative n-tuples such that xi ∈ 0, a i 1, . . . , n and n p i x i ≥ xj ,
for j 1, . . . , n,
i1
n
pi xi ∈ 0, a.
1.6
i1
If fx/x is an increasing function, then n n f p i xi ≥ pi fxi . i1
1.7
i1
Remark 1.3. Let us note that if fx/x is a strictly increasing function, then equality in 1.7 is valid if we have equalities in 1.6 instead of inequalities, that is, x1 · · · xn and n1 pi 1. The following similar result is also valid 1, page 153. Theorem 1.4. Let fx/x
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