A kind of even order Bernoulli-type operator with bivariate Shepard

1.   Introduction

The renowned integral Steffensen's inequality ^[1] is written as: Let $f$ and $g$ be integrable functions on $[a, b]$ such that $f$ is nonincreasing and $0\leq g(\mathfrak{J})\leq1$ on $[a, b]$. Then

where $\lambda = \int_{a}^{b}g(\mathfrak{J})d\mathfrak{J}.$

It is simple to notice that inequalities (1.1) are reversed if $f$ is nondecreasing.

The discrete version of the Steffensen inequality ^[2] states:

Theorem A. Assume that $\{f(k)\}_{k = 1}^n$ is a nonincreasing nonnegative real sequence and $\{g(k)\}_{k = 1}^n$ is a real sequence such that $0\leq g(k) \leq1$ for every $k$. Furthermore, let that $\lambda_1$, $\lambda_2\in\{1, \dots, n\}$ be such that $\lambda_2\leq \sum_{k = 1}^{n}g(k)\leq \lambda_1$. Then

Jakšetić et al. ^[3] established the following interesting results among many other similar results for a positive finite measure $\mu$. States:

Theorem B. Let $\hat{\mu}$ be a positive finite measure on $\mathfrak{B}([a, b])$, and let $f$, $g:[a, b\rightarrow \mathbb{R}$ be measurable functions on $[a, b]$ such that $f$ is nonincreasing and $0\leq g(\mathfrak{J})\leq 1$ for all $t\in[a, b]$. Further, let $\hat{\mu}([c, d]) = \int_{[a, b]}g(\mathfrak{J})d \hat{\mu}(\mathfrak{J})$, where $[c, d]\subseteq[a, b]$. Then

Also, the authors proved that:

Theorem C. Let $f$, $g:[a, b]\rightarrow \mathbb{R}$ be measurable functions on $[a, b]$ such that $f$ is nonincreasing and $0\leq g(\mathfrak{J})\leq 1$ for all $t\in[a, b]$. Further, let $\hat{\mu}([c, d]) = \int_{[a, b]}g(\mathfrak{J})d \hat{\mu}(\mathfrak{J})$, where $[c, d]\subseteq[a, b]$. If $\hat{\mu}$ is a positive finite measure on $\mathfrak{B}([a, b])$, then

In 1982, Pečarić ^[4] gave speculation of the Steffensen inequality as the following two hypotheses.

Theorem 1.1. Let $\hat{f}$, $\hat{g}$, $\hat{h}:[a, b]\rightarrow \mathbb{R}$ be integrable functions on $[a, b]$ such that $\hat{f}/\hat{h}$ is nonincreasing and $\hat{h}$ is nonnegative. Further, let $0\leq \hat{g}(\mathfrak{J})\leq 1$ $\forall \mathfrak{J}\in[a, b]$. Then

where $\hat{℘}$ is the solution of the equation

We get the reverse of (1.3), if $\hat{f(\mathfrak{J})}/\hat{h(\mathfrak{J})}$ is nondecreasing.

Theorem 1.2. Let $\hat{f}$, $\hat{g}$, $\hat{h}:[a, b]\rightarrow \mathbb{R}$ be integrable functions on $[a, b]$ such that $\hat{f}/\hat{h}$ is nonincreasing and $\hat{h}$ is nonnegative. Further, let $0\leq \hat{g}(\mathfrak{J})\leq 1$ $\forall \mathfrak{J}\in[a, b]$. Then

where $\hat{℘}$ gives us the solution of

We get the reverse of (1.4), if $\hat{f(\mathfrak{J})}/\hat{h(\mathfrak{J})}$ is nondecreasing.

Wu and Srivastava in ^[5] acquired the accompanying result.

Theorem 1.3. Let $\hat{f}$, $\hat{g}$, $\hat{h}:[a, b]\rightarrow \mathbb{R}$ be integrable functions on $[a, b]$ such that $\hat{f}$ is nonincreasing. Further, let $0\leq \hat{g}(\mathfrak{J})\leq \hat{h}(\mathfrak{J})$ $\forall \mathfrak{J}\in[a, b]$. Then the following integral inequalities hold true:

where $\hat{℘}$ gives us the solution of

The following interesting findings were published in ^[6].

Theorem 1.4. Suppose the integrability of $\hat{g}$, $\hat{h}$, $\hat{f}$, $\psi:[a, b]\rightarrow \mathbb{R}$ such that $\hat{f}$ is nonincreasing. Also suppose $0\leq\hat{\psi}(\mathfrak{J})\leq \hat{g}(\mathfrak{J})\leq \hat{h}(\mathfrak{J})-\hat{\psi}(\mathfrak{J})$ for all $\mathfrak{J}\in[a, b]$. Then

where $\hat{℘}$ is given by

Theorem 1.5. Under the hypotheses of Theorem 1.4. Then

where $\hat{℘}$ is given by

The calculus of time scales with the intention to unify discrete and continuous analysis (see ^[7]) was proposed by Hilger ^[8]. For more details on the time scales calculus we refer to the book by Bohner and Peterson ^[9].

Lately, several dynamic inequalities on time scales has been investigated by using exclusive authors who have been inspired with the aid of a few applications (see^{[10,11,12,13,14,15,16,17,18]}). Some authors created different results regarding fractional calculus on time scales to provide associated dynamic inequalities (see ^{[19,20,21,22,23,24,25,26,27]}).

In this article, we explore new generalizations of the integral Steffensen inequality given in ^[4,5,6] via general time scale measure space with a positive $\sigma$-finite measure. We also retrieve some of the integral inequalities known in the literature as special cases of our tests.

2.   Main results

In what follows $\mathfrak{B}([a, b]_{\mathbb{T}})$ is Borel $\sigma$-algebra $[a, b]$. Next, we enroll the accompanying suppositions for the verifications of our primary outcomes:

($S_1$) $([a, b]_{\mathbb{T}}, \mathfrak{B}([a, b]_{\mathbb{T}}), \hat{\mu})$ is time scale measure space with a positive $\sigma$-finite measure on $\mathfrak{B}([a, b]_{\mathbb{T}})$.

($S_2$) $\eta$, $\Upsilon$, $\Xi:[a, b]_\mathbb{T}\rightarrow \mathbb{R}$ are $\Delta \hat{\mu}$-integrable functions on $[a, b]_\mathbb{T}$.

($S_3$) $\eta/\Xi$ is nonincreasing and $\Xi$ is nonnegative.

($S_4$) $0\leq \Upsilon(\mathfrak{J})\leq 1$ for all $\mathfrak{J}\in[a, b]_\mathbb{T}$.

($S_5$) $\hat{℘}\in[0, \infty)$.

($S_6$) $\eta$ is nonincreasing.

($S_7$) $1\leq \Upsilon(\mathfrak{J})\leq \Xi(\mathfrak{J})$ for all $\mathfrak{J}\in [a, b]_{\mathbb{T}}$.

($S_8$) $0\leq \psi(\mathfrak{J})\leq \Upsilon(\mathfrak{J})\leq \Xi(\mathfrak{J})-\psi(\mathfrak{J})$ for all $\mathfrak{J}\in[a, b]_\mathbb{T}$.

($S_9$) $0\leq M\leq \Upsilon(\mathfrak{J})\leq 1-M$ for all $\mathfrak{J}\in[a, b]_\mathbb{T}$.

($S_{10}$) $0\leq \psi(\mathfrak{J})\leq \Upsilon(\mathfrak{J})\leq 1-\psi(\mathfrak{J})$ for all $\mathfrak{J}\in[a, b]_\mathbb{T}$.

$\hat{℘}$ is the solution of the equations listed below:

($S_{11}$) $\int_{[a, a+\hat{℘}]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu} = \int_{[a, b]_\mathbb{T}}\Xi(\mathfrak{J})\Upsilon(\mathfrak{J})\Delta \hat{\mu}$.

($S_{12}$) $\int_{[b-\hat{℘}, b]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu} = \int_{[a, b]_\mathbb{T}}\Xi(\mathfrak{J})\Upsilon(\mathfrak{J})\Delta \hat{\mu}$.

($S_{13}$) $\int_{[a, a+\hat{℘}]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu} = \int_{[a, b]_\mathbb{T}}\Upsilon(\mathfrak{J})\Delta \hat{\mu} = \int_{[b-\hat{℘}, b]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu}$.

($S_{14}$) $\int_{[a, a+\hat{℘}]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu} = \int_{[a, b]_\mathbb{T}}\Upsilon(\mathfrak{J})\Delta \hat{\mu}$.

($S_{15}$) $\int_{[b-\hat{℘}, b]_\mathbb{T}}\Xi(\mathfrak{J})\Delta \hat{\mu} = \int_{[a, b]_\mathbb{T}}\Upsilon(\mathfrak{J})\Delta \hat{\mu}$.

Presently, we are prepared to state and explain the principle results that make bigger numerous effects inside the literature.

Theorem 2.1. Let $S_1$, $S_2$, $S_3$, $S_4$, $S_5$ and $S_{11}$ be satisfied. Then

We get the reverse of (2.1), if $\eta/\Xi$ is nondecreasing.

Proof. From our hypotheses, we observe that,

The proof is complete.

Remark 2.1. In case of $\mathbb{T} = \mathbb{R}$ and related to Lebesgue measure in Theorem 2.1, we recollect [4,Theorem 1].

Theorem 2.2. Assumptions $S_1$, $S_2$, $S_3$, $S_4$, $S_5$ and $S_{12}$ imply

We get the reverse of (2.2), if $\eta/\Xi$ is nondecreasing.

Proof. From our hypotheses, we observe that,

Remark 2.2. By observing Lebesgue measure in Theorem 2.2, and $\mathbb{T} = \mathbb{R}$, we recapture [4,Theorem 2].

We will need the following lemma to prove the subsequent results.

Lemma 2.1. Let $S_1$, $S_2$, $S_5$ hold, such that

Then

and

Proof. The suppositions of the Lemma imply that

Firstly, we prove the validity of the integral identity (2.3). Indeed, by direct computation, and from our hypotheses, we find that

Since

we have

Combination of (2.5) and (2.6) led to the required integral identity (2.3) asserted by the Lemma. The integral identity (2.4) can be proved similarly. The proof is done.

Theorem 2.3. Suppose $S_1$, $S_2$, $S_5$, $S_6$, $S_7$ and $S_{13}$ give

Proof. From our hypotheses. In perspective of the considerations that the function $\eta$ is nonincreasing on $[a, b]$ and $0\leq \Upsilon(\mathfrak{J})\leq \Xi(\mathfrak{J})$ for all $\mathfrak{J}\in[a, b]$, we infer that

and

Using (2.3), (2.7) and (2.8), we find that

In the same way as above, we can prove that

The confirmation is finished by joining the integral inequalities (2.9) and (2.10).

Remark 2.3. We can reclaim [5,Theorem 1] with the use of Lebesgue measure in Theorem 2.3, and $\mathbb{T} = \mathbb{R}$.

Theorem 2.4. Assume $S_1$, $S_2$, $S_5$, $S_6$, $S_8$ and $S_{13}$ be fulfilled. Then

Proof. From our hypotheses. Clearly function $\eta$ is nonincreasing on $[a, b]$ and $0\leq \psi(\mathfrak{J})\leq \Upsilon(\mathfrak{J})\leq \Xi(\mathfrak{J})-\psi(\mathfrak{J})$ for all $\mathfrak{J}\in[a, b]$, we obtain

Also

Similarly, we find that

By combining (2.3), (2.4), (2.12) and (2.13), we arrive at the inequality (2.11) asserted by Theorem 2.

Remark 2.4. If we take $\mathbb{T} = \mathbb{R}$, and consider the Lebesgue measure in Theorem 2.4, we recapture [5,Theorem 2].

In the following theorem, we use the additional parameters $\hat{℘}_1$, $\hat{℘}_2\in[0, \infty)$.

Theorem 2.5. Let $S_1$, $S_2$, $S_5$, $S_6$, $S_9$ be satisfied, and

Then

Proof. By using straightforward calculations, we have

where we used the theorem's hypotheses

and

The function $\eta(\mathfrak{J})-\eta(b)$ is nonincreasing and integrable on $[a, b]$ and by applying Theorem 2 with $\Xi(\mathfrak{J}) = 1$, $\psi(\mathfrak{J}) = M$ and $\eta(\mathfrak{J})$ replaced by $\eta(\mathfrak{J})-\eta(b)$, hence

From (2.15) and (2.16) we obtain

which is the right-hand side inequality in (2.14).

Similarly, one can show that

which is the left-hand side inequality in (2.14).

Remark 2.5. [5,Theorem 3] can be obtained if $\mathbb{T} = \mathbb{R}$ and Lebesgue measure in Theorem 2.5.

Theorem 2.6. If $S_1$, $S_2$, $S_5$, $S_6$, $S_7$ and $S_{14}$ hold. Then

Proof. Follows similar to the proof of the right-hand side inequality in Theorem 2.

Remark 2.6. If we take $\mathbb{T} = \mathbb{R}$, and consider the Lebesgue measure in Theorem 2.6, we recapture [6,Theorem 2.12].

Corollary 2.1. Hypotheses $S_1$, $S_2$, $S_3$, $S_{10}$ and $S_{11}$ yield

Proof. Insert $\Upsilon(\mathfrak{J})\mapsto \Xi(\mathfrak{J})\Upsilon(\mathfrak{J})$, $\eta(\mathfrak{J})\mapsto \eta(\mathfrak{J})/\Xi(\mathfrak{J})$ and $\psi(\mathfrak{J})\mapsto \Xi(\mathfrak{J})\psi(\mathfrak{J})$ in Theorem 2.

Remark 2.7. [6,Corollary 2.3] can be recovered with the help of $\mathbb{T} = \mathbb{R}$, and Lebesgue measure in Corollary 2.1.

Theorem 2.7. If $S_1$, $S_2$, $S_5$, $S_6$, $S_7$ and $S_{15}$ hold. Then

Proof. Carry out the same proof of the left-hand side inequality in Theorem 2.

Remark 2.8. If we take $\mathbb{T} = \mathbb{R}$, and consider the Lebesgue measure in Theorem 2.7, we recapture [6,Theorem 2.13].

Corollary 2.2. Let $S_1$, $S_2$, $S_3$, $S_9$ and $S_{12}$, be fulfilled. Then

Proof. Proof can be completed by taking $\Upsilon(\mathfrak{J})\mapsto \Xi(\mathfrak{J})\Upsilon(\mathfrak{J})$, $\eta(\mathfrak{J})\mapsto \eta(\mathfrak{J})/\Xi(\mathfrak{J})$ and $\psi(\mathfrak{J})\mapsto \Xi(\mathfrak{J})\psi(\mathfrak{J})$ in Theorem 2.

Remark 2.9. By letting $\mathbb{T} = \mathbb{R}$, and consider the Lebesgue measure in Corollary 2.2, we recapture [6,Corollary 2.4].

3.   Conclusions

In this article, we explore new generalizations of the integral Steffensen inequality given in ^[4,5,6] via general time scale measure space with a positive $\sigma$-finite measure, we generalize a number of those inequalities to a general time scale measure space. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete and constant calculus.

Acknowledgments

This work was supported by Inho Hwang Incheon National University Research Grant 2021–2022.

Conflict of interest

The authors declare that there is no competing interest.