Abstract
This research work considers the following inequalities: λA(a,b) + (1-λ)C(a,b) ≤ C(a,b) ≤ μA(a,b) + (1-μ)C(a,b) and C[λa + (1-λ)b, λb + (1-λ)a] ≤ C(a,b) ≤ C[μa + (1-μ)b, μb + (1-μ)a] with . The researchers attempt to find an answer as to what are the best possible parameters λ, μ that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert f(t) = λA(a,b) + (1-λ)C(a,b) - C(a,b) without the loss of generality. We assume that a>b and let to determine the condition for λ and μ to become f (t) ≤ 0. Secondly, we insert g(t) = μA(a,b) + (1-μ)C(a,b) - C(a,b) without the loss of generality. We assume that a>b and let to determine the condition for λ and μ to become g(t) ≥ 0.
Highlights
For a,b > 0 with a ≠ b, the Centroidal mean C (a,b), Harmonic mean A(a,b) and Contraharmonic mean C (a,b) are defined by: ( ) = C (a,b)3(a + b= ), A(a,b) a= + b ; C (a,b) a2 + b2 a+b respectively
This work finds out such inequality that arises in the search for determination of a point of reference about which some function of variants would be minimum or maximum
The theory of means has its roots in the work of the Pythagorean who introduced the harmonic, geometric, and arithmetic means
Summary
In [7], researchers studied what are the best possible parameters α1,α2 , β1 and β2 by two theorems: Theorem (1) the double inequality: - Long et al, proved that the following results: M0 (a,b) and Mtl3 (a,b) are the best possible lower and upper power bounds for the generalized logarithmic mean Lt (a,b) for any fixed t > 0 the double inequalities
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