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

Abstract

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

1. Introduction

In recent years the approximation problem of the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑢(𝑡) via the famous Čebyšev functional 𝒯=1𝑓,𝑔𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑔(𝑡)−1𝑑𝑡𝑏−𝑎^𝑏_𝑎𝑓(𝑡)1𝑑𝑡⋅𝑏−𝑎^𝑏_𝑎𝑔(𝑡)𝑑𝑡 (1) increasingly became essential. In 1882, Čebyšev [1] derived an interesting result involving two absolutely continuous functions whose first derivatives are continuous and bounded and is given by 𝒯≤1𝑓,𝑔()12𝑏−𝑎²𝑓^′_∞𝑔^′_∞, (2) and the constant 1/12 is the best possible.

In 1935, Grüss [2] proved another result for two integrable mappings 𝑓, 𝑔 such that 𝜙≤𝑓(𝑥)≤Φ and 𝛾≤𝑓(𝑥)≤Γ; the inequality 𝒯≤1𝑓,𝑔4Φ−𝜙Γ−𝛾 (3) holds, and the constant 1/4 is the best possible.

In [3, p 302] Beesack et al. have proved the following Čebyšev inequality for absolutely continuous functions whose first derivatives belong to 𝐿_𝑝 spaces: 𝒯≤𝑓,𝑔𝑏−𝑎42^𝑝−1𝑝𝑝+1^1/𝑝2^𝑞−1𝑞𝑞+1^1/𝑞𝑓^′_𝑝𝑔^′_𝑞, (4) where ‖ℎ‖_𝑝∶=(∫^𝑏_𝑎|ℎ(𝑡)|^𝑝𝑑𝑡)^1/𝑝, ∀𝑝>1, and (1/𝑝)+(1/𝑞)=1.

For the constant 𝜔1𝑝,𝑞∶=42^𝑝−1𝑝𝑝+1^1/𝑝2^𝑞−1𝑞𝑞+1^1/𝑞 (5) we have 18≤1≤𝜔𝑝,𝑞4 (6) for all 𝑞=𝑝/(𝑝−1), 𝑝>1. Furthermore, we have the following particular cases in (4).(1)If 𝑝=𝑞=2, we have 𝒯𝑓,𝑔≤𝑏−𝑎8𝑓^′₂𝑔^′₂. (7)(2)If 𝑞→∞, we have 𝒯𝑓,𝑔≤𝑏−𝑎4𝑓^′₁𝑔^′_∞. (8)

In 1970, Ostrowski [4] has proved the following combination of the Čebyšev and Grüss results: 𝒯≤1𝑓,𝑔8(𝑏−𝑎)(𝑀−𝑚)𝑔^′_∞, (9) where 𝑔 is absolutely continuous with 𝑔^′∈𝐿_∞[𝑎,𝑏] and 𝑓 is Lebesgue integrable on [𝑎,𝑏] and satisfying 𝑚≤𝑓(𝑡)≤𝑀, for all 𝑡∈[𝑎,𝑏]. The constant 1/8 is the best possible.

In 1973, Lupaş [5] has improved Beesack et al. inequality (7), as follows: 𝒯≤()𝑓,𝑔𝑏−𝑎𝜋²𝑓^′₂𝑔^′₂, (10) provided that 𝑓, 𝑔 are two absolutely continuous functions on [𝑎,𝑏] with 𝑓^′,𝑔^′∈𝐿₂[𝑎,𝑏], where ‖ℎ^′‖₂∶=(∫^𝑏_𝑎|ℎ^′(𝑡)|²𝑑𝑡)^1/2. The constant 1/𝜋² is the best possible.

More recently, and using the identity ([3], page 246), 𝒯=1𝑓,𝑔𝑏−𝑎^𝑏_𝑎𝑓(𝑡)−𝑓(𝑎)+𝑓(𝑏)2×𝑔(𝑡)−1𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠𝑑𝑡. (11) Dragomir [6] has proved the following inequality.

Theorem 1. Let 𝑓∶[𝑎,𝑏]→ℂ be of bounded variation on [𝑎,𝑏] and 𝑔∶[𝑎,𝑏]→ℂ a Lebesgue integrable function on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔2^𝑏_𝑎𝑓⋅1𝑏−𝑎^𝑏_𝑎𝑔(𝑡)−1𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠𝑑𝑡, (12) where ⋁^𝑏_𝑎(𝑓) denotes the total variation of 𝑓 on the interval [𝑎,𝑏]. The constant 1/2 is best possible in (12).

Another result when both functions are of bounded variation was considered in the same paper [6], as follows.

Theorem 2. If 𝑓,𝑔∶[𝑎,𝑏]→ℂ are of bounded variation on [𝑎,𝑏], then 𝒯≤1𝑓,𝑔4^𝑏_𝑎𝑓⋅^𝑏_𝑎𝑔. (13) The constant 1/4 is best possible in (13).

Many authors have studied the functional (1) and, therefore, several bounds under various assumptions have been obtained; for more new results and generalizations the reader may refer to [6–21].

On other hand and in order to study the difference between two Riemann integral means, Barnett et al. [22] have proved the following estimates.

Theorem 3. Let 𝑓∶[𝑎,𝑏]→ℝ be an absolutely continuous function with the property that 𝑓^′∈𝐿_∞[𝑎,𝑏]; that is, 𝑓^′_∞∶=esssup_{[]𝑡∈𝑎,𝑏}𝑓^′(𝑡). (14) Then for 𝑎≤𝑐<𝑑≤𝑏, we have the inequality 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)1𝑑𝑡−𝑑−𝑐^𝑑_𝑐𝑓(𝑠)≤1𝑑𝑠4+(𝑎+𝑏)/2−(𝑐+𝑑)/2(𝑏−𝑎)−(𝑑−𝑐)²×[(𝑏−𝑎)−(𝑑−𝑐)]𝑓^′_∞≤12[(𝑏−𝑎)−(𝑑−𝑐)]𝑓^′_∞. (15)
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.

Theorem 4. Let 𝑓∶[𝑎,𝑏]→ℝ be an absolutely continuous mapping. Then for 𝑎≤𝑐<𝑑≤𝑏, we have the inequalities 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)1𝑑𝑡−𝑑−𝑐^𝑑_𝑐𝑓(𝑠)≤()𝑑𝑠𝑏−𝑎𝑞+1^1/𝑞𝜌1+1+𝜌^𝑞1/𝑞×𝑣^𝑞+1+𝜆^𝑞+11/𝑞𝑓^′_𝑝,𝑓^′∈𝐿_𝑝[],1𝑎,𝑏1≤𝑝<∞,𝑝+1𝑞1=1;21−𝜌+|𝑣−𝜆|𝑓^′₁,𝑓^′∈𝐿₁[𝑎,𝑏], (16) where (𝑏−𝑎)𝑣=𝑐−𝑎, (𝑏−𝑎)𝜌=𝑑−𝑐 and (𝑏−𝑎)𝜆=𝑏−𝑑. Both inequalities in (16) are sharp.

Theorem 5. Assume that the mapping 𝑓∶[𝑎,𝑏]→ℝ is of 𝑟-𝐻-Hölder type on [𝑎,𝑏]. Then for 𝑎≤𝑐<𝑑≤𝑏, we have the inequality 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)1𝑑𝑡−𝑑−𝑐^𝑑_𝑐𝑓(𝑠)()𝑑𝑠≤𝐻𝑐−𝑎^𝑟+1+(𝑏−𝑑)^𝑟+1()−(.𝑟+1)[(𝑏−𝑎𝑑−𝑐)] (17) Inequality (17) is best possible in the sense that we cannot put in the right-hand side a constant less than 1.

Theorem 6. Let 𝑓∶[𝑎,𝑏]→ℝ be a mapping of bounded variation on [𝑎,𝑏]. The following bounds hold: 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)1𝑑𝑡−𝑑−𝑐^𝑑_𝑐𝑓(𝑠)≤()𝑑𝑠𝑏−𝑎−𝑑−𝑐2+𝑐+𝑑2−𝑎+𝑏2∨^𝑏_𝑎𝑓;𝐿()𝑏−𝑎𝑐−𝑎²+(𝑏−𝑑)²2)−(;[(𝑏−𝑎𝑑−𝑐)]𝑖𝑓𝑓𝑖𝑠𝐿-𝐿𝑖𝑝𝑠𝑐ℎ𝑖𝑡𝑧𝑖𝑎𝑛𝑏−𝑑𝑓(𝑏)−𝑏−𝑎𝑐−𝑎𝑓(𝑎)+()𝑏−𝑎𝑐+𝑑−𝑎+𝑏𝑓𝑠𝑏−𝑎⁰;𝑖𝑓𝑓𝑖𝑠𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔, (18) where 𝑠₀=(𝑐𝑏−𝑎𝑑)/((𝑏−𝑎)−(𝑑−𝑐))∈[𝑐,𝑑].

In this paper by utilising amongst others the inequalities from Theorems 3–6, several new bounds for the Čebyšev functional 𝒯(𝑓,𝑔) are provided.

The inequalities (15)–(18) are used in an essential way to obtain new error bounds for the 𝒯(𝑓,𝑔), which gives a significant application for these inequalities. Applications for functions of self-adjoint operators on complex Hilbert Spaces are provided as well.

2. 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 [𝑎,𝑏]; then 𝒯≤𝑓,𝑔_𝑏𝑎𝑓⋅18(𝑏−𝑎)𝑔^′_∞,𝑔^′∈𝐿_∞[];1𝑎,𝑏21+(1/2^𝑞)2𝑞+1^1/𝑞𝑔^′_𝑝,𝑔^′∈𝐿_𝑝[];1𝑎,𝑏8𝑔^′₁,𝑔^′∈𝐿₁[𝑎,𝑏], (19) where ‖⋅‖_𝑝 are the usual Lebesgue norms; that is, ‖ℎ‖_𝑝∶=^𝑏_𝑎|ℎ(𝑡)|^𝑝𝑑𝑡^1/𝑝‖ℎ‖,𝑓𝑜𝑟𝑝≥1,_∞∶=esssup_{[]𝑡∈𝑎,𝑏}|ℎ(𝑡)|. (20)

Proof. Using integration by parts, we have 𝒯1𝑓,𝑔=−𝑏−𝑎^𝑏_𝑎^𝑡_𝑎𝑔(𝑠)𝑑𝑠−𝑡−𝑎𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠𝑑𝑓(𝑡). (21) It is known that for a continuous function 𝑝∶[𝑎,𝑏]→ℝ and a function 𝜈∶[𝑎,𝑏]→ℝ of bounded variation, the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑝(𝑡)𝑑𝜈(𝑡) exists and one has the inequality ^𝑏_𝑎𝑝(𝑡)𝑑𝜈(𝑡)≤sup_{[]𝑡∈𝑎,𝑏}𝑝(𝑡)^𝑏_𝑎(𝜈). (22) As 𝑓 is of bounded variation on [𝑎,𝑏], by (22) we have −1𝑏−𝑎^𝑏_𝑎^𝑡_𝑎𝑔(𝑠)𝑑𝑠−𝑡−𝑎𝑏−𝑎^𝑏_𝑎𝑔(𝑠)(𝑡≤1𝑑𝑠𝑑𝑓)𝑏−𝑎sup^𝑡_𝑎𝑔(𝑠)𝑑𝑠−𝑡−𝑎𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠⋅^𝑏_𝑎𝑓=1𝑏−𝑎sup_{[]𝑡∈𝑎,𝑏}1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑠)−1𝑑𝑠𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠^𝑏_𝑎𝑓. (23) In the inequality (15), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)≤1𝑑𝑠2(𝑏−𝑡)𝑔^′_∞. (24) Substituting (24) into (23), we get 𝒯≤1𝑓,𝑔⋅1𝑏−𝑎2𝑔^′_∞sup_{[]𝑡∈𝑎,𝑏}{(𝑡−𝑎)(𝑏−𝑡)}^𝑏_𝑎𝑓=18(𝑏−𝑎)𝑔^′_∞^𝑏_𝑎𝑓, (25) since sup_{𝑡∈[𝑎,𝑏]}{(𝑡−𝑎)(𝑏−𝑡)}, occurs at 𝑡=(𝑎+𝑏)/2, therefore, sup_{𝑡∈[𝑎,𝑏]}{(𝑡−𝑎)(𝑏−𝑡)}=(1/4)(𝑏−𝑎)², which proves the first inequality in (19).
In the inequality (16), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)≤()𝑑𝑠𝑏−𝑡^1+(1/𝑞)𝑞+1^1/𝑞(𝑏−𝑎)^1/𝑞×1+𝑡−𝑎𝑏+𝑡−2𝑎^𝑞1/𝑞𝑔^′_𝑝,𝑔^′∈𝐿_𝑝[],1𝑎,𝑏2𝑏−𝑡𝑔𝑏−𝑎^′1,𝑔^′∈𝐿₁[𝑎,𝑏]. (26) Substituting (26) into (23), we get 𝒯≤1𝑓,𝑔𝑏−𝑎^𝑏_𝑎𝑓×sup_{[]𝑡∈𝑎,𝑏}()⋅()𝑡−𝑎𝑏−𝑡^1+(1/𝑞)𝑞+1^1/𝑞(𝑏−𝑎)^1/𝑞×1+𝑡−𝑎𝑏+𝑡−2𝑎^𝑞1/𝑞𝑔^′_𝑝,𝑔^′∈𝐿_𝑝[],1𝑎,𝑏2𝑏−𝑡𝑔𝑏−𝑎^′1,𝑔^′∈𝐿₁[]≤𝑎,𝑏_𝑏𝑎𝑓×1+(1/2)^𝑞1/𝑞2^1+1/𝑞1+𝑞^1/𝑞𝑔^′_𝑝,𝑔^′∈𝐿_𝑝[];1𝑎,𝑏8𝑔^′₁,𝑔^′∈𝐿₁[𝑎,𝑏], (27) where 𝑝>1 and (1/𝑝)+(1/𝑞)=1, which proves the second and the third inequalities in (19).

Another result when 𝑔 is of 𝑟-𝐻-Hölder type is as follows.

Theorem 8. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is of bounded variation on [𝑎,𝑏] and 𝑔 is of 𝑟-𝐻-Hölder type on [𝑎,𝑏]; then 𝒯𝑓,𝑔≤𝐻𝑟^𝑟(𝑟+1)^𝑟+1(𝑏−𝑎)^𝑟𝑏_𝑎𝑓. (28)

Proof. As 𝑓 is of bounded variation and 𝑔 is of 𝑟-𝐻-Hölder type on [𝑎,𝑏], by (23) and using (17) we have 𝒯≤1𝑓,𝑔𝑏−𝑎sup_{[]𝑡∈𝑎,𝑏}1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢^𝑏_𝑎𝑓≤1𝐻𝑏−𝑎𝑟+1sup_{[]𝑡∈𝑎,𝑏}(𝑡−𝑎)(𝑏−𝑡)^𝑟𝑏_𝑎𝑓𝑟=𝐻^𝑟(𝑟+1)^𝑟+1(𝑏−𝑎)^𝑟𝑏_𝑎𝑓, (29) since sup_{𝑡∈[𝑎,𝑏]}(𝑡−𝑎)(𝑏−𝑡)^𝑟=(𝑟^𝑟/(𝑟+1)^𝑟+1)(𝑏−𝑎)^𝑟+1, which completes the proof.

Theorem 9. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is of bounded variation on [𝑎,𝑏] and 𝑔 is monotonic nondecreasing on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔6(𝑏−𝑎)𝑔(𝑏)−𝑔(𝑎)^𝑏_𝑎𝑓. (30)

Proof. As 𝑓 and 𝑔 is of bounded variation on [𝑎,𝑏] and 𝑔 is monotonic nondecreasing on [𝑎,𝑏], by (22) and using (23) we have 𝒯≤1𝑓,𝑔𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢𝑑𝑡^𝑏_𝑎𝑓. (31) In the third part of inequality (18), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠≤𝑏−𝑡𝑔(𝑏)(𝑎.𝑏−𝑎−𝑔) (32) Substituting (32) into (31), we get 𝒯≤1𝑓,𝑔⋅𝑔(𝑏)(𝑎)𝑏−𝑎−𝑔𝑏−𝑎^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)𝑑𝑡^𝑏_𝑎𝑓=16(𝑏−𝑎)𝑔(𝑏)−𝑔(𝑎)^𝑏_𝑎𝑓, (33) and thus the proof is finished.

3. The Case When 𝑓 Is Lipschitzian

In this section, we give some new bounds when 𝑓 is 𝐿-Lipschitzian.

Theorem 10. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is 𝐿-Lipschitzian on [𝑎,𝑏] and 𝑔 is an absolutely continuous on [𝑎,𝑏]; then 𝒯1𝑓,𝑔≤𝐿()12𝑏−𝑎²𝑔^′_∞,𝑔^′∈𝐿_∞[];𝑞𝑎,𝑏₂1+(1/2^𝑞)^1/𝑞2𝑞+13𝑞+1𝑞+1^1/𝑞×(𝑏−𝑎)²𝑔^′_𝑝,𝑔^′∈𝐿_𝑝[];1𝑎,𝑏(𝑔12𝑏−𝑎)^′₁,𝑔^′∈𝐿₁[𝑎,𝑏], (34) where 𝑝>1 and (1/𝑝)+(1/𝑞)=1.

Proof. Using the fact that for a Riemann integrable function 𝑝∶[𝑐,𝑑]→ℝ and 𝐿-Lipschitzian function 𝜈∶[𝑐,𝑑]→ℝ, one has the inequality ^𝑑_𝑐𝑝(𝑡)𝑑𝜈(𝑡)≤𝐿^𝑑_𝑐𝑝(𝑡)𝑑𝑡. (35) As 𝑓^′ is 𝐿-Lipschitzian on [𝑎,𝑏], by (35) we have 𝒯≤𝐿𝑓,𝑔𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)≤1𝑑𝑢𝑑𝑡2()𝐿𝑔𝑏−𝑎^′∞𝑏_𝑎()=1𝑡−𝑎)(𝑏−𝑡𝑑𝑡𝐿()12𝑏−𝑎²𝑔^′_∞, (36) where for the last inequality we used the inequality (15), with 𝑑=𝑡 and 𝑎=𝑐, (see (24)).
In the inequality (16), setting 𝑑=𝑡 and 𝑎=𝑐, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)≤()𝑑𝑠𝑏−𝑡^1+(1/𝑞)𝑞+1^1/𝑞(𝑏−𝑎)^1/𝑞×1+𝑡−𝑎𝑏+𝑡−2𝑎^𝑞1/𝑞𝑔^′_𝑝𝑔^′∈𝐿_𝑝[],1𝑎,𝑏1≤𝑝<∞,𝑝+1𝑞1=1;2𝑏−𝑡𝑔𝑏−𝑎^′1,𝑔^′∈𝐿₁[𝑎,𝑏]. (37) Substituting (37) into (36), we get 𝒯≤𝐿𝑓,𝑔𝑔𝑏−𝑎^′𝑝𝑞+1^1/𝑞(𝑏−𝑎)^1/𝑞×^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)^1+(1/𝑞)×1+𝑡−𝑎𝑏+𝑡−2𝑎^𝑞1/𝑞𝑔𝑑𝑡,_′∈𝐿_𝑝[],1𝑎,𝑏2𝑔^′₁𝑏−𝑎^𝑏_𝑎()𝑔𝑡−𝑎)(𝑏−𝑡𝑑𝑡,_′∈𝐿₁[],≤𝐿𝑎,𝑏𝑔𝑏−𝑎^′𝑝𝑞+1^1/𝑞(𝑏−𝑎)^1/𝑞×sup_{[]𝑡∈𝑎,𝑏}1+𝑡−𝑎𝑏+𝑡−2𝑎^𝑞1/𝑞×^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)^1+(1/𝑞)𝑔𝑑𝑡,_′∈𝐿_𝑝[],1𝑎,𝑏()12𝑏−𝑎²𝑔^′₁,𝑔^′∈𝐿₁[],𝑔𝑎,𝑏=𝐿^′𝑝𝑞+1^1/𝑞11+2^𝑞1/𝑞×(𝑏−𝑎)²𝑞²2𝑞+13𝑞+1,𝑔^′∈𝐿_𝑝[],1𝑎,𝑏(𝑔12𝑏−𝑎)^′₁,𝑔^′∈𝐿₁[𝑎,𝑏], (38) where sup_{𝑡∈[𝑎,𝑏]}[1+((𝑡−𝑎)/(𝑏+𝑡−2𝑎))^𝑞]^1/𝑞=(1+(1/2^𝑞))^1/𝑞, and ^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)^1+(1/𝑞)=()𝑑𝑡𝑏−𝑎^3+(1/𝑞)1₀(1−𝑡)𝑡^1+(1/𝑞)=()𝑑𝑡𝑏−𝑎^3+(1/𝑞)𝑞²,2𝑞+13𝑞+1 (39) which proves the second and the third inequalities in (34).

Theorem 11. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is 𝐿-Lipschitzian on [𝑎,𝑏] and 𝑔 is of 𝑟-𝐻-Hölder type on [𝑎,𝑏]; then 𝒯()𝑓,𝑔≤𝐿𝐻𝑏−𝑎^𝑟+1(𝑟+1)²().𝑟+2 (40)

Proof. As 𝑓 is 𝐿-Lipschitzian and 𝑔 is of 𝑟-𝐻-Hölder type on [𝑎,𝑏], by (36) and using (17) we have 𝒯≤𝐿𝑓,𝑔𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)≤𝑑𝑢𝑑𝑡𝐿𝐻(𝑟+1)(𝑏−𝑎)^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)^𝑟()𝑑𝑡=𝐿𝐻𝑏−𝑎^𝑟+1(𝑟+1)²(),𝑟+2 (41) where, for the last inequality, a simple calculation yields that ^𝑏_𝑎(𝑡−𝑎)(𝑏−𝑡)^𝑟𝑑𝑡=(𝑏−𝑎)^𝑟+21₀(1−𝑡)𝑡^𝑟=()𝑑𝑡𝑏−𝑎^𝑟+2(),𝑟+1)(𝑟+2 (42) which completes the proof.

Corollary 12. In Theorem 11, if 𝑔 is 𝑀-Lipschitzian on [𝑎,𝑏], then we have 𝒯≤1𝑓,𝑔()12𝐿𝑀𝑏−𝑎². (43)

Theorem 13. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is of bounded variation on [𝑎,𝑏] and 𝑔 is monotonic nondecreasing on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔6𝐿(𝑏−𝑎)²𝑔(𝑏)−𝑔(𝑎). (44)

Proof. As 𝑓 is 𝐿-Lipschitzian on [𝑎,𝑏] and 𝑔 is monotonic nondecreasing on [𝑎,𝑏], by (35) and using (36) we have 𝒯≤𝐿𝑓,𝑔𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢𝑑𝑡. (45) In the third part of inequality (18), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠≤𝑏−𝑡𝑔(𝑏)(𝑎.𝑏−𝑎−𝑔) (46) Substituting (46) into (45), we get 𝒯≤𝐿𝑓,𝑔𝑔(𝑏)(𝑎)𝑏−𝑎−𝑔𝑏−𝑎^𝑏_𝑎()=1𝑡−𝑎)(𝑏−𝑡𝑑𝑡6𝐿(𝑏−𝑎)𝑔(𝑏)−𝑔(𝑎), (47) which completes the proof.

4. More Inequalities

In this section we give other related results.

Theorem 14. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 and 𝑔 are of bounded variation on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔4^𝑏_𝑎𝑔^𝑏_𝑎𝑓. (48) The constant 1/4 is the best possible.

Proof. As 𝑓 and 𝑔 is of bounded variation on [𝑎,𝑏], by (22) and using (23), we have 𝒯≤1𝑓,𝑔𝑏−𝑎sup_{[]𝑡∈𝑎,𝑏}1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢^𝑏_𝑎𝑓. (49) In the first inequality of (18), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠≤𝑏−𝑡𝑏−𝑎^𝑏_𝑎𝑔. (50) Substituting (50) into (49), we get 𝒯≤1𝑓,𝑔(𝑏−𝑎)²sup_{[]𝑡∈𝑎,𝑏}{(𝑡−𝑎)(𝑏−𝑡)}^𝑏_𝑎𝑔^𝑏_𝑎𝑓=14^𝑏_𝑎𝑔^𝑏_𝑎𝑓, (51) which proves the inequality. The sharpness case trivially holds by taking 𝑓(𝑡)=𝑔(𝑡)=sgn(𝑡−((𝑎+𝑏)/2)), which completes the proof.

When the integrator is of bounded variation we have the following.

Theorem 15. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be such that 𝑓 is 𝐿-Lipschitzian on [𝑎,𝑏] and 𝑔 is of bounded variation on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔6𝐿(𝑏−𝑎)^𝑏_𝑎𝑔. (52)

Proof. As 𝑓 is 𝐿-Lipschitzian on [𝑎,𝑏] and 𝑔 is of bounded variation on [𝑎,𝑏], by (22) and using (23), we have 𝒯≤𝐿𝑓,𝑔𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢𝑑𝑡. (53) In the second inequality of (18), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)𝑑𝑠≤𝑏−𝑡𝑏−𝑎^𝑏_𝑎𝑔. (54) Substituting (54) into (53), we get 𝒯≤𝐿𝑓,𝑔(𝑏−𝑎)^2𝑏𝑎(𝑡−𝑎)(𝑏−𝑡)𝑑𝑡^𝑏_𝑎𝑔=16𝐿(𝑏−𝑎)^𝑏_𝑎𝑔, (55) and the proof is completed.

When both functions are Lipschitzian we have the following.

Theorem 16. Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be, respectively, such that 𝑓 and 𝑔 are 𝐿₁- and 𝐿₂-Lipschitzian on [𝑎,𝑏]; then 𝒯≤1𝑓,𝑔𝐿12¹𝐿₂(𝑏−𝑎)². (56) The constant 1/12 is the best possible.

Proof. As 𝑓 and 𝑔 are 𝐿₁- and 𝐿₂-Lipschitzian on [𝑎,𝑏], respectively, by (35) and using (36), we have 𝒯≤𝐿𝑓,𝑔¹𝑏−𝑎^𝑏_𝑎1(𝑡−𝑎)𝑡−𝑎^𝑡_𝑎𝑔(𝑢)−1𝑑𝑢𝑏−𝑎^𝑏_𝑎𝑔(𝑢)𝑑𝑢𝑑𝑡. (57) In the second inequality of (18), setting 𝑑=𝑡 and 𝑐=𝑎, we get 1𝑡−𝑎^𝑡_𝑎𝑔(𝑠)1𝑑𝑠−𝑏−𝑎^𝑏_𝑎𝑔(𝑠)≤1𝑑𝑠2𝐿₂(𝑏−𝑡). (58) Substituting (58) into (57), we get 𝒯≤1𝑓,𝑔2()𝐿𝑏−𝑎¹𝐿₂^𝑏_𝑎()=1𝑡−𝑎)(𝑏−𝑡𝑑𝑡𝐿12¹𝐿₂(𝑏−𝑎)², (59) which proves the inequality. The sharpness case trivially holds by taking 𝑓(𝑡)=𝑔(𝑡)=𝑡, which completes the proof.

Remark 17. Let 𝑔 be as in Theorems 7–16. By applying the same techniques used in the corresponding proofs of each theorem, we may obtain several inequalities for monotonic nondecreasing integrator 𝑓 using the fact that for a monotonic nondecreasing function 𝜈∶[𝑎,𝑏]→ℝ and continuous function 𝑝∶[𝑎,𝑏]→ℝ, one has the inequality ^𝑏_𝑎𝑝(𝑡)𝑑𝜈(𝑡)≤^𝑏_𝑎𝑝(𝑡)𝑑𝜈(𝑡). (60) We leave the details to the interested reader.

5. Applications for Self-Adjoint Operators

We denote by ℬ(𝐻) the Banach algebra of all bounded linear operators on a complex Hilbert space (𝐻;⟨⋅,⋅⟩). Let 𝐴∈ℬ(𝐻) be self-adjoint and let 𝜑_𝜆 be defined for all 𝜆∈ℝ as follows: 𝜑_𝜆(𝑠)∶=1,for−∞<𝑠≤𝜆,0,for𝜆<𝑠<+∞. (61) Then for every 𝜆∈ℝ the operator 𝐸_𝜆∶=𝜑_𝜆(𝐴) (62) is a projection which reduces 𝐴.

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].

Theorem 18 (Spectral Representation Theorem). Let 𝐴 be a bonded self-adjoint operator on the Hilbert space 𝐻 and let 𝑚=min{𝜆∣𝜆∈𝑆𝑝(𝐴)}=∶min𝑆𝑝(𝐴) and 𝑀=max{𝜆∣𝜆∈𝑆𝑝(𝐴)}=∶max𝑆𝑝(𝐴). Then there exists a family of projections {𝐸_𝜆}_𝜆∈ℝ, called the spectral family of 𝐴, with the following properties:(a)𝐸_𝜆≤𝐸_𝜆′ for 𝜆≤𝜆^′;(b)𝐸_𝑚−0=0, 𝐸_𝑀=𝐼 and 𝐸_𝜆+0=𝐸_𝜆 for all 𝜆∈ℝ;(c)we have the representation 𝐴=^𝑀_𝑚−0𝜆𝑑𝐸_𝜆. (63)
More generally, for every continuous complex-valued function 𝜑 defined on ℝ and for every 𝜀>0 there exists a 𝛿>0 such that 𝜑(𝐴)−^𝑛_𝑘=1𝜑𝜆^′_𝑘𝐸_𝜆𝑘−𝐸_𝜆𝑘−1≤𝜀 (64) whenever 𝜆₀<𝑚=𝜆₁<⋯<𝜆_𝑛−1<𝜆_𝑛𝜆=𝑀,_𝑘−𝜆_𝑘−1𝜆≤𝛿𝑓𝑜𝑟1≤𝑘≤𝑛,_′𝑘∈𝜆_𝑘−1,𝜆_𝑘𝑓𝑜𝑟1≤𝑘≤𝑛 (65) this means that 𝜑(𝐴)=^𝑀_𝑚−0𝜑(𝜆)𝑑𝐸_𝜆, (66) where the integral is of Riemann-Stieltjes type.

Corollary 19. With the assumptions of Theorem 18 for 𝐴, 𝐸_𝜆, and 𝜑 we have the representations 𝜑(𝐴)𝑥=^𝑀_𝑚−0𝜑(𝜆)𝑑𝐸_𝜆𝜑(𝐴)=𝑥∀𝑥∈𝐻,𝑥,𝑦^𝑀_𝑚−0𝜑(𝜆)𝑑𝐸_𝜆𝑥,𝑦∀𝑥,𝑦∈𝐻. (67) In particular, 𝜑(𝐴)𝑥,𝑥=^𝑀_𝑚−0𝜑(𝜆)𝑑𝐸_𝜆𝑥,𝑥∀𝑥∈𝐻. (68) Moreover, we have the equality 𝜑(𝐴)𝑥²=^𝑀_𝑚−0𝜑(𝜆)²𝑑𝐸_𝜆𝑥²∀𝑥∈𝐻. (69)

We recall the following result (see [25]) that provides an upper bound for the total variation of the function ℝ∋𝜆↦⟨𝐸_𝜆𝑥,𝑦⟩∈ℂ on an interval [𝛼,𝛽].

Theorem 20. Let {𝐸_𝜆}_𝜆∈ℝ be the spectral family of the bounded self-adjoint operator 𝐴 and let 𝑚=min𝑆𝑝(𝐴) and 𝑀=max𝑆𝑝(𝐴). Then for any 𝑥,𝑦∈𝐻 and 𝛼<𝛽 we have the inequality ^𝛽_𝛼𝐸_(⋅)𝑥,𝑦²≤𝐸_𝛽−𝐸_𝛼𝑥,𝑥𝐸_𝛽−𝐸_𝛼𝑦,𝑦, (70) where ⋁^𝛽_𝛼(⟨𝐸_(⋅)𝑥,𝑦⟩) denotes the total variation of the function ⟨𝐸_(⋅)𝑥,𝑦⟩ on [𝛼,𝛽].

Remark 21. For 𝛼=𝑚−𝜀 with 𝜀>0 and 𝛽=𝑀 we get from (70) the inequality ^𝑀_𝑚−𝜀𝐸_(⋅)≤1𝑥,𝑦_𝐻−𝐸_𝑚−𝜀𝑥,𝑥^1/21_𝐻−𝐸_𝑚−𝜀𝑦,𝑦^1/2 (71) for any 𝑥,𝑦∈𝐻.
This implies, for any 𝑥,𝑦∈𝐻, that ^𝑀_𝑚−0𝐸_(⋅)𝑥,𝑦≤‖𝑥‖𝑦, (72) where ⋁^𝑀_𝑚−0(⟨𝐸_(⋅)𝑥,𝑦⟩) denotes the limit lim_𝜀→0+[⋁^𝑀_𝑚−𝜀(⟨𝐸_(⋅)𝑥,𝑦⟩)].

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 self-adjoint operators on Hilbert spaces.

Theorem 22. Let 𝐴 be a bonded self-adjoint operator on the Hilbert space 𝐻 and let 𝑚=min{𝜆∣𝜆∈𝑆𝑝(𝐴)}=∶min𝑆𝑝(𝐴) and 𝑀=max{𝜆∣𝜆∈𝑆𝑝(𝐴)}=∶max𝑆𝑝(𝐴). If 𝑓∶[𝑚,𝑀] →ℂ is such that its derivative 𝑓 is of bounded variation on [𝑚,𝑀], then we have the inequality 𝑓(𝐴)−𝑓(𝑚)𝑀1_𝐻−𝐴+𝑓(𝑀)𝐴−𝑚1_𝐻≤1𝑀−𝑚𝑥,𝑦4(𝑀−𝑚)^𝑀_𝑚𝑓^𝑀_𝑚−0𝐸_(⋅)≤1𝑥,𝑦4(𝑀−𝑚)^𝑀_𝑚𝑓‖𝑥‖𝑦 (73) for any 𝑥,𝑦∈𝐻.

Proof. Utilising the inequality (48) for the function of bounded variation 𝑔(𝜆)=⟨𝐸_𝜆𝑥,𝑦⟩ and the continuous function 𝑓 we have ^𝑀_𝑚−0𝑓(𝜆)𝑑𝐸_𝜆(𝑚)∫𝑥,𝑦−𝑓^𝑀𝑚−0𝐸_𝜆𝑥,𝑦𝑑𝜆(𝑀−∫𝑀−𝑚−𝑓)𝑥,𝑦^𝑀𝑚−0𝐸_𝜆𝑥,𝑦𝑑𝜆≤1𝑀−𝑚4(𝑀−𝑚)^𝑀_𝑚𝑓^𝑀_𝑚−0𝐸_(⋅)≤1𝑥,𝑦4(𝑀−𝑚)^𝑀_𝑚𝑓‖𝑥‖𝑦 (74) for any 𝑥,𝑦∈𝐻.
By the Spectral Representation Theorem we have ^𝑀_𝑚−0𝑓(𝜆)𝑑𝐸_𝜆𝑥,𝑦=𝑓(𝐴)𝑥,𝑦_𝑀𝑚−0,𝐸_𝜆𝑥,𝑦𝑑𝜆=𝐸_𝜆𝑥,𝑦𝜆^𝑀_𝑚−0−^𝑀_𝑚−0𝜆𝑑𝐸_𝜆−=𝑥,𝑦=𝑀𝑥,𝑦𝐴𝑥,𝑦𝑀1_𝐻−𝐴𝑥,𝑦 (75) for any 𝑥,𝑦∈𝐻.
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.

If we take 𝑓(𝑡)=𝑡^𝑝 with 𝑝≥1, then for any positive operator 𝐴 with 𝑆𝑝(𝐴)⊂[𝑚,𝑀]⊂[0,∞) we have the inequality 𝐴^𝑝−𝑚^𝑝𝑀1_𝐻−𝐴+𝑀^𝑝𝐴−𝑚1_𝐻≤1𝑀−𝑚𝑥,𝑦4(𝑀−𝑚)𝑀^𝑝−𝑚^𝑝𝑀_𝑚−0𝐸_(⋅)≤1𝑥,𝑦4(𝑀−𝑚)𝑀^𝑝−𝑚^𝑝‖𝑥‖𝑦 (76) for any 𝑥,𝑦∈𝐻.

If we take the function 𝑓(𝑡)=ln𝑡, then for any positive definite operator 𝐴 with 𝑆𝑝(𝐴)⊂[𝑚,𝑀]⊂(0,∞) we have the inequality ln𝐴−ln𝑚𝑀1_𝐻−𝐴+ln𝑀𝐴−𝑚1_𝐻𝑀𝑀−𝑚𝑥,𝑦≤ln𝑚^{𝑀(𝑀−𝑚)/4}_𝑚−0𝐸_(⋅)𝑀𝑥,𝑦≤ln𝑚^{(𝑀−𝑚)/4}‖𝑥‖𝑦 (77) for any 𝑥,𝑦∈𝐻.

Finally, if we take 𝑓(𝑡)=exp(𝑡), then we have for any self-adjoint operator 𝐴 with 𝑆𝑝(𝐴)⊂[𝑚,𝑀]⊂ℝ the inequality (𝐴)−(𝑚)expexp𝑀1_𝐻−𝐴+exp(𝑀)𝐴−𝑚1_𝐻≤1𝑀−𝑚𝑥,𝑦4(𝑀−𝑚)exp𝑀−exp𝑚^𝑀_𝑚−0𝐸_(⋅)≤1𝑥,𝑦4(𝑀−𝑚)exp𝑀−exp𝑚‖𝑥‖𝑦 (78) for any 𝑥, 𝑦∈𝐻.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.