Generalizations of AM-GM-HM means inequalities

: In this paper, we showed some generalized reﬁnements and reverses of arithmetic-geometric-harmonic means (AM-GM-HM) inequalities due to Sababheh [J. Math. Inequal. 12 (2018), 901–920]. Among other results, it was shown that if a , b > 0, 0 < p ≤ t < 1 and m ∈ N + , then


Introduction
Let M n (C) denote the algebra of all n×n complex matrices and M ++ n (C) be the set of positive definite matrices in M n (C).For A and B are two Hermitian matrices, A > B means that A − B ∈ M ++ n (C).Let B(H) denote the C * -Algebra of all bounded linear operators on a complex Hilbert space H.An operator A ∈ B(H) is called positive if Ax, x ≥ 0 for all x ∈ H, denoted by A ≥ 0. The set of all positive invertible operators is denoted by B ++ (H).For two self-adjoint operators A, B ∈ B(H), A > B means (A − B) ∈ B ++ (H).
As usual, we denote the arithmetic-geometric-harmonic means (AM-GM-HM) as for 0 < p ≤ t < 1 and m ≥ 1.In fact, Alzer-Fonseca-Kovačec's inequalities have become one of the most important extensions to Young's inequalities for the past few years.Liao and Wu [5] replicated (1.1) as follows for a, b > 0, 0 < p ≤ t < 1 and m ≥ 1. Sababheh [7] improved (1.2) under some conditions: For a, b > 0 and In the same paper [7], the author also showed that: We refer the readers to [3,4,8,9] and references therein for some other results about the AM-GM-HM means inequality.
Following the ideas of Yang and Wang [10], we will give some generalizations of inequalities (1.3)-(1.5)and a generalized reverse of inequality (1.5).As applications, we obtain some inequalities for operators and determinants.

Main results
Firstly, we give the generalization of inequalities (1.3) and (1.5).Without loss of generality, we may assume 0 < p ≤ t < 1 in the following theorem. and Taking x = b a , we complete the proof of (2.1).Similarly, letting and That is and
Next, we give some inequalities for operators and determinants by Theorem 1 and Corollary 2. We will list some necessary lemmas in front of each theorem.
Lemma 4 ([6, p.3]).Let X ∈ B(H) be self-adjoint and f and g be continuous real functions such that f (t) ≥ g(t) for all t ∈ S p(X) (the spectrum of X), then f (X) ≥ g(X). and (ii) The inequalities (2.5) and (2.6) are reversed if A ≤ I.
Proof.We only prove the first inequality.The other inequalities are shown similarly.Let a = 1 in inequality (2.1), then we get (2.7) The operator A has a positive spectrum, then by Lemma 4 and inequality (2.7) we get Finally, multiplying inequality (2.8) by A m 2 on both left and right sides, we can get (2.5) directly.
Before we give some inequalities for determinants as promised, we should recall some basic signs.The singular values of a matrix A are defined by s j (A), j = 1, 2, • • • , n, and we denote the values of {s j (A)} as a nonincreasing order.To obtain our results, we need a following lemma.

Corollary 2 .
by (1.5)) taking x = b a , as desired.Letting a = b, b = a, p = 1 − t and t = 1 − p in Theorem 1, we have the following results: Let a, b > 0, 0 < p ≤ t < 1 and m ∈ N + .If a ≥ b, then we have and only if a = b.Proof.Under the conditions, we have 1 ≤ s j (A − 1 2 BA − 1 2 ) for B ≥ A. Putting a = 1 and b = s j (T ), whereT = A − 1 2 BA − 1 2in the inequality (2.1), then we have1∇ p s j (T ) = 1, 2, • • • , n.It is a fact that the determinant of a positive definite matrix is a product of its singular values, and we have det(I∇ t T ) n (C), 0 < p ≤ t < 1 and m ∈ N + .If B ≥ A, then we have m − 1! p s j (T ) m 1∇ t s j (T ) m − 1! t s j (T ) m ≤ j (T ) m − 1! p s j (T ) m + 1! t s j (T ) m 1 nby(2.11)