A new generalization of two refined Young inequalities and applications

In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . ,

r0m(am2-bm2)2≤r0m(bm+1-am+1b-a-(m+1)(ab)m2)≤(αa+(1-α)b)m-(aαb1-α)m,\matrix{ {r_0^m{{\left( {{a^{{m \over 2}}} - {b^{{m \over 2}}}} \right)}^2}} & { \le r_0^m\left( {{{{b^{m + 1}} - {a^{m + 1}}} \over {b - a}} - \left( {m + 1} \right){{\left( {ab} \right)}^{{m \over 2}}}} \right)} \cr {} & { \le {{\left( {\alpha a + \left( {1 - \alpha } \right)b} \right)}^m} - {{\left( {{a^\alpha }{b^{1 - \alpha }}} \right)}^m},} \cr }

where r₀ = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.