Sharps Bounds for Power Mean in Terms of Contraharmonic Mean

In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters α, β that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that a > b and let for 1) and a < b, (t small) for 2) to determine the condition for α and β to become f(t) ≤ 0 and g(t) ≥ 0.

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

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.