## Abstract

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

$$\begin{array}{}{\displaystyle K(\frac{b}{a},2{)}^{r}\le \frac{a{\mathrm{\nabla}}_{\nu}b}{a{\mathrm{\u266f}}_{\nu}b}\le K(\frac{b}{a},2{)}^{R},}\end{array}$$

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

$$\begin{array}{}{\displaystyle e({h}^{-1},\nu )\le \frac{a{\mathrm{\nabla}}_{\nu}b}{a{\mathrm{\u266f}}_{\nu}b}\le e(h,\nu ),}\end{array}$$

where *h* = max
$\begin{array}{}{\displaystyle \{\frac{a}{b},\frac{b}{a}\}}\end{array}$
and
*e*(*t*,*ν*) = exp (4*ν*(1 – *ν*)(*K*(*t*,2)–1)
$\begin{array}{}{\displaystyle (1-\frac{1}{2t})).}\end{array}$
Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved.

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