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.

We prove that for all x > 0, we havewith the best possible constants α = 0 and .