Concise high precision approximation for the complete elliptic integral of the first kind

1.   Introduction

Riemann-Liouville fractional integral given by

Many different concepts of fractional derivative maybe found in ^[9,10,11]. In ^[12] studied a conformable derivative:

The time scale conformable derivatives was introduced by Benkhettou et al. ^[17].

Further, in recent years, numerous mathematicians claimed that non-integer order derivatives and integrals are well suited to describing the properties of many actual materials, such as polymers. Fractional derivatives are a wonderful tool for describing memory and learning. a variety of materials and procedures inherited properties is one of the most significant benefits of fractional ownership. For more concepts and definition on time scales see ^{[13,14,15,16,17,18,19,33,34,35]}.

Continuous version of Steffensen's inequality ^[7] is written as: For $0\leq g(\wp)\leq1$ on $\in[a, b]$. Then

where $\lambda = \int_{a}^{b}g(\wp)dt.$

Supposing $f$ is nondecreasing gets the reverse of (1.1).

Also, the discrete inequality of Steffensen ^[6] is: For $\lambda_2\leq \sum_{\ell = 1}^{n}g(\ell)\leq \lambda_1$. Then

Recently, a large number of dynamic inequalities on time scales have been studied by a small number of writers who were inspired by a few applications (see ^{[1,2,3,4,8,28,29,30,31,32,36,37,40,41,42,44,48,49,50,51,52,53]}).

In ^[5] Jakšetić et al. proved that, if $\hat{\mu}([c, d]) = \int_{[a, b]}g(\wp)d \hat{\mu}(\wp)$, where $[c, d]\subseteq[a, b]$. Then

and

Anderson, in ^[3], studied the inequality:

In ^[47] the authors have proved, for

and

If there exists a constant $A$ such that $r(\wp)/\zeta(\wp)-At$ is monotonic on the intervals $[m, k]$, $[k, n]$, and

then

In particularly, Anderson ^[3] proved

where $m, n\in\mathbb{T}_\kappa$ with $m < n$, $r$, $g:[m, n]_{\mathbb{T}}\rightarrow \mathbb{R}$ are $\nabla$-integrable functions such that $r$ is of one sign and nonincreasing and $0\leq g(\wp)\leq1$ on $[m, n]_{\mathbb{T}}$ and $\lambda = \int_{m}^{n}g(\wp)\nabla \wp$, $n-\lambda, m+\lambda \in \mathbb{T}$.

We prove the next two needed results:

Theorem 1.1. Assume $q > 0$ with $0\leq g(\wp)\leq \zeta(\wp)$ $\forall\wp\in[m, n]_{\mathbb{T}}$ and $\lambda$ is given from $\int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp$, then

Also, provided with $0\leq g(\wp)\leq \zeta(\wp)$ and $\int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp$, we have

We get the reverse inequalities of (1.4) and (1.5) when assuming $r/\zeta$ is nondecreasing.

Theorem 1.2. Assume $\psi$ is integrable on time scales interval $[m, n]$, with $\zeta(\wp)-\psi(\wp)\geq g(\wp)\geq\psi(\wp)\geq 0 \; \forall\wp\in[m, n]_{\mathbb{T}}$ and $\int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp$ and $g$, $r$ and $\zeta$ are $\Delta_{\alpha}$-integrable functions, $\zeta(\wp)\geq g(\wp)\geq0$, we have

and

Proof. The proof techniques of Theorems 1.6 and 1.7 are like to that in ^[4] and is removed.

Several authors proved conformable Hardy's inequality ^[20,21], conformable Hermite-Hadamard's inequality ^[22,23,24], conformable inequality of Opial's ^[26,27] and conformable inequality of Steffensen's ^[25]. In ^[45] Anderson proved the followong results:

Theorem 1.3. ^[45] Suppose $\alpha\in(0, 1]$ and $r_1$, $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$. Suppose $\prod: [r_1, r_2]\rightarrow [0, \infty)$ and $\Gamma: [r_1, r_2]\rightarrow [0, 1]$ are $\alpha$-fractional integrable functions on $[r_1, r_2]$ with $\Pi$ is decreasing, we get

where $\aleph = \frac{\alpha(r_2-r_1)}{r_2^{\alpha}-r_1^{\alpha}}\int_{r_1}^{r_2}\Gamma(\zeta)d_{\alpha}\zeta\in[0, r_2-r_1]$.

In ^[46] the authors gave an extension for Theorem 1.8:

Theorem 1.4. Assume $\alpha\in(0, 1]$ and $r_1$, $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$. Suppose $\prod, \Gamma, \Sigma: [r_1, r_2]\rightarrow [0, \infty)$ are integrable on $[r_1, r_2]$ with the decreasing function $\Pi$ and $0\leq \Gamma\leq \Sigma$, we get

where $\aleph = \frac{(r_2-r_1)}{\int_{r_1}^{r_2}\Sigma(\zeta)d_{\alpha}\zeta}\int_{r_1}^{r_2}\Gamma(\zeta) d_{\alpha}\zeta\in[0, r_2-r_1]$.

In this paper, we prove and explore several novel speculations of the Steffensen inequality obtained in ^[47] through the conformable integral containing time scale concept. We furthermore recover certain known results as special cases of our results.

2.   Main results

Lemma 2.1. Assume $\zeta > 0$ is $rd$-continuous function on $[m, n]\cap\mathbb{T}$, $g$, $r$ be $rd$-continuous on $[m, n]\cap\mathbb{T}$ such that $r/\zeta$ nonincreasing function and $0\leq g(\wp)\leq 1$ $\forall\wp\in[m, n]\cap\mathbb{T}$. Then

$(\Lambda_{1})$

where $\lambda$ is given by

$(\Lambda_{2})$

such that

(2.1) and (2.2) are reversed when $r/\zeta$ is nondecreasing.

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.4), (1.5) to get $(\Lambda_{1})$ and $(\Lambda_{2})$ simultaneously.

Lemma 2.2. Under the same hypotheses of Lemma 2.1. with $\psi$ be integrable functions on $[m, n]\cap\mathbb{T}$ and $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$ for all $\wp\in[m, n]_{\mathbb{T}}$. Then

where $\lambda$ is obtained from

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$, $r(\wp)\mapsto r(\wp)/h(\wp)$ and $\psi(\wp)\mapsto \zeta(\wp)\psi(\wp)$ in (1.6).

Lemma 2.3. Under the same conditions of Lemma 2.1. Then

where $\lambda$ is obtained from

Proof. Taking $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.7).

Theorem 2.1. Under the same conditions of Lemma 2.3 such that $k\in(m, n)$ and $\lambda_1$, $\lambda_2$ are given from

$(\Lambda_{3})$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

(2.4) is reversed if $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.3).

$(\Lambda_{4})$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.5) satisfied, then we reverse (2.6).

$(\Lambda_{5})$ If $\lambda_1$, $\lambda_2$ be the same as in $(\Lambda_{3})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ so that

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.7) satisfied, the inequality in (2.8) is reversed.

$(\Lambda_{6})$ If $\lambda_1$, $\lambda_2$ be defined as in $(\Lambda_{4})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ and

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.9) satisfied, we reverse (2.10).

Proof. $(\Lambda_{3})$ Consider $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$, and $R_{1}(\ell) = r^{\sigma}(\ell)-A\phi(\ell)\zeta(\ell)$, since $A$ is given in Definition 2.1. Since $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$, using Lemma 2.1$(\Lambda_{1})$, we deduce

As $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nondecreasing, using Lemma 2.1$(\Lambda_{2})$, we obtain

(2.11) and (2.12) imply that

Hence, if (2.3) is hold, then (2.4) holds. For $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$, we get the some steps.

$(\Lambda_{4})$ Let $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$, also $R_{1}(x) = r^{\sigma}(x)-A\phi(x)\zeta(x)$, where $A$ as in Definition 2.1. $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nonincreasing, so from Lemma 2.1$(\Lambda_{1})$ we obtain

Using Lemma 2.1$(\Lambda_{1})$ we have

Thus, from (2.13), (2.14), we get

Therefore, if $\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp$ is satisfied, then (2.8) holds. Follow the same steps for $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$.

Using Lemma 2.3 and repeat the steps of Theorem 2.1$(\Lambda_{3})$ and Theorem 2.1$(\Lambda_{4})$ in the proof of $(\Lambda_{5})$ and $(\Lambda_{6})$ respectively.

Corollary 2.1. The inequalities (2.4), (2.6), (2.8) and (2.10) of Theorem 2.1 letting $\mathbb{T} = \mathbb{R}$ takes

Corollary 2.2. We get [47,Theorems 8,10,21 and 22], if we put $\alpha = 1$ and $\phi(\wp) = \wp$ in Corollary 2.1 $[(i), (ii), (iii), (iv)]$ simultaneously.

Corollary 2.3. In Corollary 2.1 taking $\mathbb{T} = \mathbb{Z}$, the results (2.15)–(2.18) will be equivalent to

Theorem 2.2. Under the assumptions in Lemma 2.1 with $0\leq g(\wp)\leq \zeta(\wp)$ and $\lambda_1$, $\lambda_2$ be defined as

$(\Lambda_{7})$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

$(\Lambda_{8})$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.19), (2.21) satisfied, we get the reverse of (2.20) and (2.22).

Proof. By using Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ and by putting $g\mapsto g/h$ and $f\mapsto fh$, we get the proof of $(\Lambda_{7})$ and $(\Lambda_{8})$.

Corollary 2.4. In Theorem 2.2 $[(\Lambda_{7}), (\Lambda_{8})]$, assuming $\mathbb{T} = \mathbb{R}$, the following results obtains:

Corollary 2.5. In Corollary 2.4 $[(i), (ii)]$, when we put $\alpha = 1$ and $\phi(\wp) = \wp$ then [47,Theorems 16 and 17] gotten.

Corollary 2.6. In (2.23) and (2.24) letting $\mathbb{T} = \mathbb{Z}$, gets

Theorem 2.3. Using the same conditions in Lemma 2.3. Letting $w:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be integrable with $0\leq g(\wp)\leq w(\wp)$ $\forall \wp\in[m, n]\cap{\mathbb{T}}$ and

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

The inequalities in (2.26) and (2.28) are reversible if $r^{\sigma}/\zeta\in\mathbb{AH}_2^c [a, b]$ and (2.25), (2.27) hold.

Proof. In Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$, $\zeta$ changes $wq$, $g$ changes $g/w$ and $r$ changes $rw$.

Corollary 2.7. In (2.26) and (2.28). Letting $\mathbb{T} = \mathbb{R}$, we have

Corollary 2.8. In Corollary 2.7 $[(i), (ii)]$, letting $\alpha = 1$ and $\phi(\wp) = \wp$ we get [47,Theorems 18 and 19].

Corollary 2.9. In (2.29) and (2.30), crossing $\mathbb{T} = \mathbb{Z}$, gets

Theorem 2.4. Using the same conditions in Lemma 2.1, and Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with $\psi:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be a integrable: $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$.

$(\Lambda_{11})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

$(\Lambda_{12})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

then

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.31) and (2.33) satisfied, we get the reverse of (2.32) and (2.34).

Proof. The same steps of Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with Lemma 2.1, $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nonincreasing, $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nondecreasing.

Corollary 2.10. In Theorem 2.4 $[(\Lambda_{11}), (\Lambda_{12})]$, letting $\mathbb{T} = \mathbb{R}$ we get:

Corollary 2.11. In (2.35) and (2.36), we put $\alpha = 1$, with $\phi(\wp) = \wp$ we get [47,Theorems 23 and 24].

Corollary 2.12. Our results (2.35) and (2.36), by using $\mathbb{T} = \mathbb{Z}$ gets

3.   Conclusions

In this work, we explore new generalizations of the integral Steffensen inequality given in ^[38,39,43] by the utilization of the $\alpha$-conformable derivatives and integrals, A few of these results are generalised to time scales. We also obtained the discrete and continuous case of our main results, in order to gain some fresh inequalities as specific cases.

Acknowledgments

The authors extend their appreciation to the Research Supporting Project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.