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The logarithmic mean and the power mean of positive numbers - Octogon Mathematical Magazine (Brasov)
... In this part, we recall some basic definitions. The following definitions are from papers [1,2,7,[9][10][11]. ...
... In [4,5], there is a generalization of the logarithmic mean and in [6,7], the following relation between L(a, b) and M α (a, b) was proven in various ways for arbitrary positive numbers a, b: ...
The aim of this article is to investigate the relations between the exponent of the convergence of sequences and other characteristics defined for monotone sequences of positive numbers. Another main goal is to characterize such monotone sequences (an) of positive numbers that, for each n≥2, satisfy the equality an=K(an−1,an+1), where the function K:R+×R+→R+ is the mean, i.e., each value of K(x,y) lies between min{x,y} and max{x,y}. Well-known examples of such sequences are, for example, arithmetic (geometric) progression, because starting from the second term, each of its terms is equal to the arithmetic (geometric) mean of its neighboring terms. Furthermore, this accomplishment generalized and extended previous results, where the properties of the logarithmic sequence (an) are referred to, i.e., in such a sequence that every n≥2 satisfies an=L(an−1,an+1), where L(x,y) is the logarithmic mean of positive numbers x,y defined as follows: L(x,y):=y−xlny−lnxifx≠y,xifx=y.
... It is well known that M 0 (a, b) = √ a.b and M α (a, b) is increasing with respect to α (see [6]). In paper [3] the following relation between L(a, b) and M α (a, b) is proved for arbitrary positive numbers a, b: ...
The number L(a, b) = a b ln a ln b for a 6= b and L(a, a) = a, is said to be the logarithmic mean of the positive numbers a, b. We shall say that a sequence (an) 1 n=1 with positive terms is a logarithmic sequence if an = L(an 1, an+1). In the present paper some basic estimations of the terms of logarithmic sequences are investigated.
For p∈ℝ, the p-th power mean M p (a,b), arithmetic mean A(a,b), geometric mean G(a,b), and harmonic mean H(a,b) of two positive numbers a and b are defined by M p (a,b)=a p +b p w,p≠0,ab,p=0, A(a,b)=(a+b)/2, G(a,b)=ab, and H(a,b)=2ab/(a+b), respectively. In this paper, we answer the questions: For α∈(0,1), what are the greatest values p, r and m, and the least values q, s and n, such that the inequalities M p (a,b)≤A α (a,b)G 1-α (a,b)≤M q (a,b), M r (a,b)≤G α (a,b)H 1-α (a,b)≤M s (a,b) and M m (a,b)≤A α (a,b)H 1-α (a,b)≤M n (a,b) hold for all a,b>0?
For p ∈ ℝ , the power mean of order p of two positive numbers a and b is defined by M p (a, b) = ((ap + bp) / 2)1/p, for p ≠ 0, and Mp (a, b) = √a b, for p = 0. In this paper, we answer the question: what are the greatest value p and the least value q such that the double inequality Mp (a, b)≤ Aα (a, b) Gβ (a, b) H1 -α-β (a, b) ≤ Mq (a, b) holds for all a, b > 0 and α,β> 0 with α+β < 1 ? Here A (a, b) = (a + b) / 2, G (a, b) = √a b, and H (a, b) = 2 a b / (a + b) denote the classical arithmetic, geometric, and harmonic means, respectively.
For p. R, the power mean of order p of two positive numbers a and b is defined by M-p (a, b) = ((a(p) + b(p))/2)(1/p), p not equal 0, and M-p (a, b) = root ab, p = 0. In this paper, we establish two sharp inequalities as follows: (2/3)G(a, b) + (1/3)H(a, b) >= M-1/3(a, b) and (1/3)G(a, b) + (2/3)H(a, b) >= M-2/3(a, b) for all a, b > 0. Here G(a, b) = root ab and H(a, b) = 2ab/(a + b) denote the geometric mean and harmonic mean of a and b, respectively. Copyright (C) 2009 Y.-M. Chu and W.-F. Xia.
We establish two optimal inequalities among power mean Mp(a,b)=(ap/2+bp/2)1/p, arithmetic mean A(a,b)=(a+b)/2, logarithmic mean L(a,b)=(a−b)/(loga−logb), and geometric mean G(a,b)=ab.
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