Some improvements of the young mean inequality and its reverse

The main objective of the present paper, is to obtain some new versions of Young-type inequalities with respect to two weighted arithmetic and geometric means and their reverses, using two inequalities $$\begin{array}{} \displaystyle K\Big(\frac{b}{a},2\Big)^r\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq K\Big(\frac{b}{a},2\Big)^R, \end{array}$$ where r = min{ν, 1 – ν}, R = max{ν,1 – ν} and K(t,2) = $\begin{array}{} \displaystyle \frac{(t+1)^2}{4t} \end{array}$ is the Kantorovich constant, and $$\begin{array}{} \displaystyle e(h^{-1},\nu)\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq e(h,\nu), \end{array}$$ where h = max $\begin{array}{} \displaystyle \{\frac{a}{b},\frac{b}{a}\} \end{array}$ and e(t,ν) = exp (4ν(1 – ν)(K(t,2)–1) $\begin{array}{} \displaystyle (1-\frac{1}{2t})\big). \end{array}$ Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved.