Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean

In the article, we prove that λ ₁=1/2+√ [ (√2+ log (1+√2) )/2 ]1/ν-1/2 , μ ₁=1/2+√6 ν /(12ν ) , λ ₂=1/2+√ [(π +2)/4 ] 1/ν-1/2 and μ ₂=1/2+√3ν /(6ν ) are the best possible parameters on the interval $[1/2, 1]$ such that the double inequalities $${aligned}& Cν[λ ₁x+(1-λ ₁)y, λ ₁y+(1-λ ₁)x]A 1-ν(x, y) \& < RQA(x, y)< Cν[μ ₁x+(1-μ ₁)y, μ ₁y+(1-μ ₁)x]A1-ν(x, y), \& Cν[λ ₂x+(1-λ ₂)y, λ ₂y+(1-λ ₂)x]A 1-ν(x, y) \& < RAQ(x, y)< Cν[μ ₂x+(1-μ ₂)y, μ ₂y+(1-μ ₂)x]A1-ν(x, y) {aligned}$$ hold for all $x, y>0$ with $x≠ y$ and $ν ∈ [1/2, ∞ )$ , where $A(x, y)$ is the arithmetic mean, $C(x, y)$ is the contraharmonic mean, and $RQA(x, y)$ and $RAQ(x, y)$ are two Neuman means.