On Two Double Inequalities (Optimal Bounds and Sharps Bounds) for Centroidal Mean in Terms of Contraharmonic and Arithmetic Means

摘要

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.