Abstract
We present several new Young-type inequalities for positive real numbers and we apply our results to obtain the matrix analogues. Among others, for real numbers ,
and
, with
and
, we prove the inequalities
where
and
are, respectively, the (weighted) arithmetic and geometric means of the positive real numbers
and
with
. In addition, we show that both bounds are sharp. An example of a matrix analogue for the case
is the double-inequality
for positive definite matrices
. Our results extend some fresh inequalities established by Kittaneh, Manasrah, Hirzallah and Feng. Estimates for the quotient
and its matrix analogues given by Furuichi and Minculete are also improved.