New H ebyšev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

Several new error bounds for the Cebysev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well.


Introduction
increasingly became essential.In 1882, Čebyšev [1] and the constant 1/12 is the best possible.
In 1973, Lupas ¸ [5] has improved Beesack et al. inequality (7), as follows: provided that ,  are two absolutely continuous functions on . The constant 1/ 2 is the best possible.
Another result when both functions are of bounded variation was considered in the same paper [6], as follows.
On other hand and in order to study the difference between two Riemann integral means, Barnett et al. [22] have proved the following estimates.( The constant 1/4 in the first inequality and 1/2 in the second inequality are the best possible.
After that, Cerone and Dragomir [23] have obtained the following three results as well.
In this paper by utilising amongst others the inequalities from Theorems 3-6, several new bounds for the Čebyšev functional T(, ) are provided.
The inequalities ( 15)-( 18) are used in an essential way to obtain new error bounds for the T(, ), which gives a significant application for these inequalities.Applications for functions of self-adjoint operators on complex Hilbert Spaces are provided as well.

The Case When 𝑓 Is of Bounded Variation
We may start with the following result.

Theorem 7. Let 𝑓, 𝑔 : [𝑎, 𝑏] → R be such that 𝑓 is of bounded variation on [𝑎, 𝑏] and 𝑔 is absolutely continuous on
where ‖ ⋅ ‖  are the usual Lebesgue norms; that is, Proof.Using integration by parts, we have As  is of bounded variation on [, ], by (22) we have In the inequality (15), setting  =  and  = , we get Substituting ( 24) into (23), we get , which proves the first inequality in (19).
Another result when  is of --Hölder type is as follows.
In the third part of inequality (18), setting  =  and  = , we get Substituting (32) into (31), we get and thus the proof is finished.

The Case When 𝑓 Is Lipschitzian
In this section, we give some new bounds when  is -Lipschitzian.
Theorem 10.Let ,  : [, ] → R be such that  is -Lipschitzian on [, ] and  is an absolutely continuous on where for the last inequality we used the inequality (15), with  =  and  = , (see (24)).
In the inequality ( 16), setting  =  and  = , we get Substituting (37) into (36), we get where sup which proves the second and the third inequalities in (34).(45) In the third part of inequality (18), setting  =  and  = , we get Substituting ( 46) into (45), we get which completes the proof.
When the integrator is of bounded variation we have the following.
Proof.As  is -Lipschitzian on [, ] and  is of bounded variation on [, ], by (22) and using (23), we have In the second inequality of ( 18 and the proof is completed.
Chinese Journal of Mathematics 7 When both functions are Lipschitzian we have the following.
Proof.As  and  are  1 -and  2 -Lipschitzian on [, ], respectively, by (35) and using (36), we have In the second inequality of ( 18), setting  =  and  = , we get We leave the details to the interested reader.
Then for every  ∈ R the operator is a projection which reduces .The properties of these projections are collected in the following fundamental result concerning the spectral representation of bounded self-adjoint operators in Hilbert spaces; see for instance [24, page 256].
We recall the following result (see [25]) that provides an upper bound for the total variation of the function R ∋   → ⟨  , ⟩ ∈ C on an interval [, ].The inequality (72) was also proved in the recent monographs [26,27] and will be utilized in the following.
After these preparations we can state and prove the following trapezoidal type inequality for functions of selfadjoint operators on Hilbert spaces.
for any ,  ∈ .Substituting these values into (74) we deduce after simple calculations the desired result (73).
The above inequality (73) can be utilized for different particular functions of interest in Operator Theory, such as the power, logarithmic, and exponential functions.
∞            ∞ , p 302] Beesack et al. have proved the following Čebyšev inequality for absolutely continuous functions whose first derivatives belong to   spaces: 1/                         ,