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SHARP BOUNDS FOR THE CONVEX COMBINATIONS OF ARITHMETIC, LOGARITHMIC AND GEOMETRIC MEANS IN TERMS OF HARMONIC MEAN

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Poincare Journal of Analysis & Applications
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unpublished

In this paper, we find the greatest values r 1 and r 2 , and the least values s 1 and s 2 in (0, 1/2) such that the double inequalities H[r 1 a + (1 − r 1)b, r 1 b + (1 − r 1)a] < αA(a, b)+(1 − α)L(a, b) < H[s 1 a+(1−s 1)b, s 1 b+(1−s 1)a] and H[r 2 a+(1−r 2)b, r 2 b+(1−r 2)a] < αA(a, b) + (1 − α)G(a, b) < H[s 2 a + (1 − s 2)b, s 2 b + (1 − s 2)a] hold for all a, b > 0 with a b and any α ∈ (0, 1), where H(a, b), G(a, b), L(a, b) and A(a, b) are the harmonic, geometric, logarithmic and

fatcat:iqwns4pvfbhnfeoqvogzmicyz4