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

In this paper, we start by presenting multiple-term refinement of Young?s and Young?s type inequalities and its reverse using different weights, which extends and unifies two recent and important results due to M. Khosravi (Math. Slovaca 68 (2018), 803-810) and L. Nasiri et al. (Asian-European Journal Math. 15, (2022) No. 07). Further, we mainly present some new real power inequalities of Young?s inequality, extending a key results of Alzer et al. (Linear Multilinear Algebra 63(3) (2015), 622-635), D. Q. Huy et al. (Linear Algebra Appl. 656 (2023), 368-384) and J. Zhao (Bull. Malys. Math. Sci. Soc. 46, No 52 (2023)). As applications of these results, we provide some inequalities for matrices, unitarily invariant norms and traces.