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  • Author or Editor: Silvestru Sever Dragomir x
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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transform

D w , μ t : = 0 w λ λ + t 1 d μ λ ,

where the integral is assumed to exist for t > 0.

We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) -f(t)] t -1 is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function f 0 + f + 0 t f t t 2 is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s difference

D w , μ A + D w , μ B 2 D w , μ A + B 2

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

Open access

Assume that Aj , j ∈ {1, … , m} are positive definite matrices of order n. In this paper we prove among others that, if 0 < l In Aj , j ∈ {1, … , m} in the operator order, for some positive constant l, and In is the unity matrix of order n, then

o 1 2 k = 1 m P k 1 P k det 2 A j l I n 1 / 2 2 1 j < k m P j P k det A j + A k l I n 1 / 2 j = 1 m P j det A j 1 / 2 det k = 1 m P k A k 1 / 2 ,

where Pk ≥ 0 for k ϵ {1, …, m} and j = 1 m P j = 1 .

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In this paper we obtain some new power and hölder type trace inequalities for positive operators in Hilbert spaces. As tools, we use some recent reverses and refinements of Young inequality obtained by several authors.

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In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transform

M ( w , μ ) ( T )  := 0 w ( λ ) T ( λ + T ) 1 d μ ( λ ) ,

where the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, then

0 1 24 δ M ( w , μ ) ( β ) M ( w , μ ) A + B 2 0 1 M ( w , μ ) ( ( 1 t ) A + t B ) d t 1 24 Δ M ( w , μ ) ( α )

and

0 1 12 δ M ( w , μ ) ( β ) 0 1 M ( w , μ ) ( ( 1 t ) A + t B ) d t M ( w , μ ) ( A ) + M ( w , μ ) ( B ) 2 1 12 Δ M ( w , μ ) ( α ) ,

where M ( w , μ ) is the second derivative of M ( w , μ ) as a real function.

Applications for power function and logarithm are also provided.

Open access