Some integral inequalities for operator monotonic functions on Hilbert spaces

Abstract Let f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I, 0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t)dt∫01f((1-t)A+tB)dt≤14[ p(1)-p(0) ][ f(B)-f(A) ] \matrix{0 \hfill & { \le \int\limits_0^1 {p\left( t \right)f\left( {\left( {1 - t} \right)A + tB} \right)dt - \int\limits_0^1 {p\left( t \right)dt\int\limits_0^1 {f\left( {\left( {1 - t} \right)A + tB} \right)dt} } } } \hfill \cr {} \hfill & { \le {1 \over 4}\left[ {p\left( 1 \right) - p\left( 0 \right)} \right]\left[ {f\left( B \right) - f\left( A \right)} \right]} \hfill \cr } in the operator order. Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.


Introduction
Consider a complex Hilbert space (H, ·, · ). An operator T is said to be positive (denoted by T ≥ ) if Tx, x ≥ for all x ∈ H and also an operator T is said to be strictly positive (denoted by T > ) if T is positive and invertible. A real valued continuous function f (t) on ( , ∞) is said to be operator monotone if f (A) ≥ f (B) holds for any A ≥ B > .
In 1934, K. Löwner [7] had given a de nitive characterization of operator monotone functions as follows: We recall the important fact proved by Löwner and Heinz that states that the power function f : ( , ∞) → R, f (t) = t α is an operator monotone function for any α ∈ [ , ] .
Consider the family of functions de ned on ( , ∞) and p ∈ [− , ] \ { , } by We also have the functions of interest In [2] the authors showed that fp is operator monotone for ≤ p ≤ .
In the same category, we observe that the function is an operator monotone function for p ∈ ( , ], [3]. It is well known that the logarithmic function ln is operator monotone and in [3] the author obtained that the functions are also operator monotone functions on ( , ∞) . Let f be an operator monotonic function on an interval of real numbers I and A, B ∈ SA I (H) , the class of all selfadjoint operators with spectra in I. Assume that p : [ , ] → R is non-decreasing on [ , ]. In this paper we obtain, among others, that for A ≤ B and f an operator monotonic function on I, in the operator order.
Several other similar inequalities for either p or f is di erentiable, are also provided. Applications for power function and logarithm are given as well.

Main Results
For two Lebesgue integrable functions h, g : [a, b] → R, consider the Čebyšev functional: It is well known that, if h and g have the same monotonicity on [a, b] , then which is known in the literature as Čebyšev's inequality.
In 1935, Grüss [4] showed that provided that there exists the real numbers m, M, n, N such that The constant is best possible in (2.1) in the sense that it cannot be replaced by a smaller quantity.
For x ∈ H we can also consider the auxiliary function ϕ (A,B);x : [ , ] → R de ned by Proof. Let ≤ t < t ≤ and A ≤ B. Then and by operator monotonicity of f we get which is equivalent to that shows that the scalar function ϕ (A,B);x : [ , ] → R is monotonic nondecreasing for A ≤ B and for any x ∈ H. If we write the inequality (2.2) for the functions p and ϕ (A,B);x we get which can be written as for x ∈ H, and the rst inequality in (2.7) is obtained.
We also have that By writing Grüss' inequality for the functions ϕ (A,B);x and p, we get for x ∈ H and the second inequality in (2.7) is obtained. Proof. Let t ∈ ( , ) and h ≠ small enough such that t + h ∈ ( , ). Then Since f ∈ G ([A, B]) , hence by taking the limit over h → in (2.13) we get which proves (2.10).
Also, we have since f is assumed to be Gâteaux di erentiable in A. This proves (2.11).
The equality (2.12) follows in a similar way.

Lemma 2. Let f be an operator monotonic function on I and
This shows that ∇f ( −t)A+tB (B − A) ≥ for all t ∈ ( , ) . The inequalities (2.15) follow by (2.11) and (2.12).
The following inequality obtained by Ostrowski in 1970, [9] also holds Another, however less known result, even though it was obtained by Čebyšev in 1882, [1], states that (2.21) provided that h , g exist and are continuous on [a, b] and h ∞ = sup t∈ [a,b] h (t) . The constant cannot be improved in the general case. The case of euclidean norms of the derivative was considered by A. Lupaş in [5] in which he proved that provided that h, g are absolutely continuous and h , g ∈ L [a, b] . The constant π is the best possible. Using the above inequalities (2.21) and (2.22) and a similar procedure to the one employed in the proof of Theorem 3, we can also state the following result: provided the integrals in the second term are nite.