A generalization of two refined Young inequalities

Abstract

We prove that if a, b > 0 and 0 ≤ ν ≤ 1, then for m = 1, 2, 3, . . ., we haveThis is a considerable generalization of two refinements of the classical Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications of this inequality, we give refined Young-type inequalities for the traces, determinants, and norms of positive definite matrices.

Convexity and matrix means

“…We refer the reader to [7][8][9]13,14,16] for some fresh refinements and discussion of Young's and related inequalities.…”

Some upper bounds for the Berezin number of Hilbert space operators

“…. In fact, 1 2 A ≤ w(A) ≤ A , for any A ∈ B(H). A functional Hilbert space is the Hilbert space of complex-valued functions on some set Ω such that the evaluation functional ϕ λ ( f ) = f (λ), λ ∈ Ω, are continuous on H. Then by the Riesz representation theorem for each λ ∈ Ω there exists a unique function k λ ∈ H such that f (λ) = f, k λ for all f ∈ H. The family {k λ : λ ∈ Ω} is called the reproducing kernel of the space H. For A a bounded linear operator on H, the Berezin symbol of A is the functionà on Ω defined bỹ…”

“…Let A, B, X ∈ B(H). Then (i) ber r (A) ≤1 2 ber(|A| r + |A * | r ) ∀r ≥ 1, (ii) ber(A * B) ≤ 1 2 ber(A * A + B * B), (iii) ber(A * XB) ≤1 2 ber(A * |X * |A + B * |X|B).…”

Further generalization, refinements of the Young inequality

“…For a generalized refinement of the weighted arithmetic-geometric mean inequality see [9]. Manasrah and Kittaneh [1] gave generalized refinements of the inequalities (1.3) and (1.4). as follows…”