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Generalized Jensen’s functional on time scales via extended Montgomery identity

Abstract

In the paper, we use Jensen’s inequality for diamond integrals and generalize it for n-convex functions with the help of an extended Montgomery identity. Moreover, the bounds have been suggested for identities associated with the generalized Jensen-type functional.

Introduction and preliminaries

Montgomery identity is well known in the literature. It is utilized to obtain a number of revolutionary inequalities such as trapezoid inequality, Ostrowski-type inequality, Čebyšev inequality, Grüss inequality, and Mohajani inequality. Mitrinović et al. in [17] proved that if \(\chi :[b_{1}, b_{2}] \rightarrow \mathbb{R}\) is differentiable on \([b_{1},b_{2}]\) and \(\chi ':[b_{1},b_{2}] \rightarrow \mathbb{R}\) is integrable on \([b_{1},b_{2}]\), then the Montgomery identity is

$$ \chi (t) = \frac{1}{b_{2} - b_{1}} \int _{b_{1}}^{b_{2}} \chi (p) \,dp + \int _{b_{1}}^{b_{2}} \mathfrak{P}(t,p) \chi ^{\prime }(p) \,dp, $$
(1)

where

$$ \mathfrak{P}(t,p) = \textstyle\begin{cases} {\frac{{p - b_{1}}}{{b_{2} - b_{1}}},}&{b_{1} \le p \le t,} \\ {\frac{{p - b_{2}}}{{b_{2} - b_{1}}},}&{t < p \le b_{2}.} \end{cases} $$
(2)

Bohner et al. in [7] proved the Montgomery identity on time scales and discussed it for discrete, continuous, and quantum cases. In case of \(\mathbb{T} =\mathbb{R}\), it becomes (1).

Suppose that a probability density function \(\psi :[b_{1},b_{2}] \rightarrow [0, \infty )\), i.e., an integrable function which satisfies \(\int _{b_{1}}^{b_{2}}\psi (p)\,dp=1\). Also \(\xi (p)=\int _{b_{1}}^{p}\psi (t)\,dt\) when \(p \in [b_{1},b_{2}]\), \(\xi (p)=0\) when \(p< b_{1}\) and \(\xi (p)=1\) when \(p>b_{2}\). Then the weighted generalization [19] of the Montgomery identity is

$$ \chi (t)= \int _{b_{1}}^{b_{2}}\psi (p)\chi (p)\,dp+ \int _{b_{1}}^{b_{2}} \mathfrak{P}_{\psi }(t,p)\chi '(p)\,dp , $$
(3)

where the weighted Peano kernel is

$$ {\mathfrak{P}_{\psi }}(t,p) = \textstyle\begin{cases} {\xi (p),}&{b_{1} \le p \le t,} \\ {\xi (p) - 1,}&{t < p \le b_{2}.} \end{cases} $$
(4)

Sarikaya et al. [20] proved the weighted Montgomery identity on time scales.

To obtain our main results, we use the extended Montgomery identities by means of Taylor’s formula I and II given in [1, 2].

Theorem 1

Let E be an open interval in \(\mathbb{R}\) and \(b_{1},b_{2}\in E\), s.t., \(b_{1} < b_{2} \). Assume that a function \(\chi :E \rightarrow \mathbb{R}\), s.t., \({\chi ^{(n - 1)}}\) is absolutely continuous for \(n \in \mathbb{N}\), then

$$\begin{aligned} \chi (t) =& \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {\chi (p)} \,dp + \sum _{\mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b_{1}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b_{1}})}^{\mathfrak{z} + 2}}}}{{{b_{2}} - {b_{1}}}}} \\ & {}- \sum_{\mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b_{2}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b_{2}})}^{\mathfrak{z} + 2}}}}{{{b_{2}} - {b_{1}}}}} + \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {{R_{n}}(t,s){ \chi ^{(n)}}(s)\,ds} , \end{aligned}$$
(5)

where

$$ {R_{n}}(t,s) = \textstyle\begin{cases} {\frac{{ - {{(t - s)}^{n}}}}{{n({b_{2}} - {b_{1}})}} + \frac{{t - {b_{1}}}}{{{b_{2}} - {b_{1}}}}{{(t - s)}^{n - 1}},}&{{b_{1}} \le s \le t,} \\ {\frac{{ - {{(t - s)}^{n}}}}{{n({b_{2}} - {b_{1}})}} + \frac{{t - {b_{2}}}}{{{b_{2}} - {b_{1}}}}{{(t - s)}^{n - 1}},}&{t < s \le {b_{2}}.} \end{cases} $$
(6)

Theorem 2

Let E be an open interval in \(\mathbb{R}\) and \(b_{1},b_{2}\in E\), s.t., \(b_{1} < b_{2} \). Assume that a function \(\chi :E \rightarrow \mathbb{R}\), s.t., \({\chi ^{(n - 1)}}\) is absolutely continuous for \(n \in \mathbb{N}\), then

$$\begin{aligned} \chi (t ) =& \frac{1}{{{b _{2}} - {b _{1}}}} \int _{{b _{1}}}^{{b _{2}}} {\chi (t)\,dt + \sum _{\mathfrak{z} = 0}^{n - 2} {{\chi ^{( \mathfrak{z} + 1)}}(t ) \frac{{{{({b _{1}} - t )}^{\mathfrak{z} + 2}} + {{({b _{2}} - t )}^{\mathfrak{z} + 2}}}}{{(\mathfrak{z} + 2)!({b _{2}} - {b _{1}})}}} } \\ & {}+ \frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {{{\hat{R}}_{n}}(t ,s){ \chi ^{(n)}}(s)\,ds,} \end{aligned}$$
(7)

where

$$ {\hat{R}_{n}}(t ,s) = \textstyle\begin{cases} {\frac{{ - 1}}{{n({b _{2}} - {b _{1}})}}{{({b _{1}} - s)}^{n}},}&{{b _{1}} \le s \le t ,} \\ {\frac{{ - 1}}{{n({b _{2}} - {b _{1}})}}{{({b _{2}} - s)}^{n}},}&{t < s \le {b _{2}}.} \end{cases} $$
(8)

For \(n = 1\), \(\sum_{\mathfrak{z} = 0}^{n - 2}\cdot \cdot \cdot \) is empty, therefore (5) and (7) reduce to Montgomery identity (1).

On convex functions

Suppose that a function μ: \(E \rightarrow \mathbb{R}\), it is termed convex if \(\forall y_{1}, y_{2} \in E\) and \(\gamma \in [0, 1]\), the inequality

$$ \mu \bigl(\gamma {y_{1}} + (1 - \gamma ){y_{2}}\bigr) \le \gamma \mu ({y_{1}}) + (1 - \gamma )\mu ({y_{2}}) $$
(9)

holds. If \({y_{1}} \ne {y_{2}}\) and \(\gamma \in (0,1)\) in (9), then μ is strictly convex. Moreover, μ is called concave function if (9) is reversed and inequality (9) is strictly concave if \(\forall {y_{1}} \ne {y_{2}}\) and \(\gamma \in (0,1)\) [19].

Divided differences are truly assigned to Newton, and Augustus de Morgan in 1842 used the term divided difference. The divided differences are beneficial if the functions have different degrees of smoothness. Divided differences are discussed in [13].

Definition 1

Suppose μ: \([p, q] \rightarrow \mathbb{R}\), \(n\in N\) and mutually distinct points \(\eta _{0},\eta _{1},\ldots ,\eta _{n}\) of \([p, q]\). Then the nth order divided difference of the function μ is

$$\begin{aligned}& [ {{\eta _{i}};\mu } ] = \mu ({\eta _{i}}),\quad i = 0, \ldots,n, \\& [{\eta _{0}},\ldots,{\eta _{n}};\mu ] = \frac{{ [ {{\eta _{1}},\ldots,{\eta _{n}};\mu } ] - [ {{\eta _{0}},\ldots,{\eta _{n - 1}};\mu } ]}}{{{\eta _{n}} - {\eta _{0}}}}. \end{aligned}$$
(10)

n-convex function is defined on the basis of nth-order divided difference [19].

Suppose \(n\geq 0\), then a function \(\mu :[p, q] \rightarrow \mathbb{R}\) is termed n-convex iff \((n+1)\) distinct points \(y_{0},y_{1},\ldots ,y_{n} \in [p, q]\),

$$ [ {{y_{0}},{y_{1}},\ldots ,{y_{n}};\mu } ] \ge 0 $$

holds. The reverse effect of the above inequality implies that μ is n-concave. The strict effect of the above inequality implies that μ is a strictly n-convex (n-concave) function. The n convexity of a function μ is examined by the following theorem [19].

Theorem 3

If \(\mu ^{n}\) exists, then μ is n-convex if and only if \(\mu ^{n} \geq 0\).

On time scales

In 1988, Hilger presented the notion of time scales calculus and proposed the unification of discrete and continuous time dynamical systems [14]. Gradually, this perspective of unification has been affixed by the extension and generalization characteristics.

Δ and calculus is the initial approach to study time scales calculus. For detailed study of calculus on time scales, readers are reffered to [6, 8, 9].

An arbitrary and nonempty closed subset of real numbers is called time scales \(\mathbb{T}\). The real numbers \(\mathbb{R}\) and the integers \(\mathbb{Z}\) are most familiar examples of time scales.

Suppose \(p\in \mathbb{T}\), then the mappings \(\sigma (t)\), \(\rho (t)\): \(\mathbb{T} \to \mathbb{T}\) indicate the forward and backward jump operators respectively on time scale \(\mathbb{T}\) and are defined as follows:

$$\begin{aligned}& \sigma (t) = inf \{\mathfrak{e}\in \mathbb{T} : \mathfrak{e} > t\}, \\& \rho (t) = sup \{\mathfrak{e}\in \mathbb{T} :\mathfrak{e} < t\}. \end{aligned}$$

Take a point \(t \in \mathbb{T}\), then t is stated as follows:

right-scattered, provided \(\sigma (t) > t\);

left-scattered, provided \(\rho (t) < t\);

isolated, provided \(\rho (t) < t < \sigma (t)\);

right-dense, provided \(\sigma (t) = t\);

left-dense, provided \(\rho (t) = t\);

dense, provided \(\rho (t) = t = \sigma (t)\).

• The graininess functions \(\sigma , \nu :\mathbb{T} \rightarrow [0, +\infty )\) are stated as

$$\begin{aligned}& \sigma (t):= \sigma (t)-t, \\& \nu (t):= t-\rho (t). \end{aligned}$$

Assume that a function μ: \(\mathbb{T} \rightarrow \mathbb{R}\), then it is termed:

rd-continuous if it is continuous at all right-dense points in \(\mathbb{T}\) and its left-sided limits are finite at all left-dense points in \(\mathbb{T}\);

ld-continuous if it is continuous at all left-dense points in \(\mathbb{T}\) and its right-sided limits are finite at all right-dense points in \(\mathbb{T}\).

We present the sets \(\mathbb{T}^{k}\), \(\mathbb{T}_{k}\), and \(\mathbb{T}^{\ast }\) that originated from the time scale \(\mathbb{T}\).

\(\mathbb{T}^{k} = \mathbb{T} - t_{1}\), provided \(\mathbb{T}\) contains a left-scattered maximum \(t_{1}\); unless \(\mathbb{T}^{k} = \mathbb{T}\).

\(\mathbb{T}_{k} = \mathbb{T} - t_{2}\), provided \(\mathbb{T}\) contains a right-scattered minimum \(t_{2}\); unless \(\mathbb{T}_{k} = \mathbb{T}\).

Lastly \(\mathbb{T}^{\ast } = \mathbb{T}^{k} \cap \mathbb{T}_{k}\).

Suppose that a function μ: \(\mathbb{T} \rightarrow \mathbb{R}\) and \(t \in \mathbb{T}^{k}\), then we describe \(\mu ^{\triangle }(t)\) (in case it exists) if, for given \(\epsilon > 0, \exists \) a neighborhood W of t so that \(\forall y_{1} \in W\)

$$ \bigl\vert {\mu \bigl(\sigma (t)\bigr) - \mu (y_{1}) - {\mu ^{\Delta }}(t)\bigl[\sigma (t) - y_{1}\bigr]} \bigr\vert \le \epsilon \bigl\vert {\sigma (t) - y_{1}} \bigr\vert . $$

Then μ is delta differentiable on \(\mathbb{T}^{k}\) provided \(\mu ^{\Delta }(t)\) exists \(\forall t \in \mathbb{T}^{k}\).

In the same way, for \(t \in \mathbb{T}_{k}\), we say \(\mu ^{\nabla }(t)\) (in case it exists) if, for given \(\epsilon > 0, \exists \) a neighborhood W of t so that \(\forall y_{1} \in W\)

$$ \bigl\vert {\mu \bigl(\rho (t)\bigr) - \mu (y_{1}) - {\mu ^{\nabla }}(t)\bigl[\rho (t) - y_{1}\bigr]} \bigr\vert \le \epsilon \bigl\vert {\rho (t) - y_{1}} \bigr\vert . $$

Then μ is nabla differentiable on \(\mathbb{T}_{k}\) provided \(\mu ^{\nabla }(t)\) exists \(\forall t \in \mathbb{T}_{k}\).

Suppose that \(\mu (t)\) is differentiable in the Δ and sense on the time scales \(\mathbb{T}\). Let \(t \in \mathbb{T}\), \(0 \le \alpha \le 1\), then \(\diamond _{\alpha }\)-dynamic derivative \({\mu ^{{\diamond _{\alpha }}}}(t)\) is

$$ {\mu ^{{\diamond _{\alpha }}}}(t) = \alpha {\mu ^{\Delta }}(t) + (1 - \alpha ){\mu ^{\nabla }}(t). $$

Hence, μ is \(\diamond _{\alpha }\)-differentiable iff Δ and derivatives of μ exist.

Remark 1

The \(\diamond _{\alpha }\)-derivative changes to

• Δ derivative, when \(\alpha = 1\);

derivative, when \(\alpha = 0\);

• “weighted dynamic derivative”, when \(\alpha \in (0,1)\).

For a complete advancement of the calculus on the \(\diamond _{\alpha }\)-derivative and \(\diamond _{\alpha }\)-integrals, refer to [21]. The refinement of \(\diamond _{\alpha }\)-derivative on time scales \(\mathbb{T}\) is known as symmetric derivative given in [11], which is as follows:

Let \(t \in \mathbb{T}_{k}^{k}\) and μ: \(\mathbb{T} \rightarrow \mathbb{R}\). Then \(\mu ^{\diamond }(t) \in \mathbb{R}\) (in case it exists) provided \(\epsilon > 0\), there is a neighborhood \(V \subset \mathbb{T}\) of t, s.t.,

$$ \begin{aligned} &\bigl\vert {\bigl[{\mu ^{\sigma }}(t) - \mu (y) + \mu (2t - y) - {\mu ^{\rho }}(t)\bigr] - {\mu ^{\diamond }}(t)\bigl[\sigma (t) + 2t - 2y - \rho (t)\bigr]} \bigr\vert \\ &\quad \le \epsilon \bigl\vert {\sigma (t) + 2t - 2y - \rho (t)} \bigr\vert \end{aligned} $$

holds \(\forall y \in V\) for which \(2t - y \in V\).

Remark 2

The symmetric derivative is

• classic symmetric derivative, when \(\mathbb{T}=\mathbb{R}\).

• symmetric difference operator, when \(\mathbb{T}=h\mathbb{Z}\), \(h>0\).

• q-symmetric difference operator, when \(\mathbb{T}=q\mathbb{Z}\), \(0< q<1\).

A detailed discussion of symmetric derivatives on time scales is given in [11].

Suppose μ: \(\mathbb{T} \rightarrow \mathbb{R}\), then \(U:\mathbb{T} \rightarrow \mathbb{R}\) is interpreted as delta antiderivative of μ if \(U^{\Delta }(t)=\mu (t)\) holds \(\forall t\in \mathbb{T}^{k}\). The delta integral of μ is

$$ \int _{{\ell _{1}}}^{{\ell _{2}}} {\mu (t)\Delta t = U({\ell _{2}}) - U({\ell _{1}}).} $$

Suppose \(s:\mathbb{T} \rightarrow \mathbb{R}\), then \(S:\mathbb{T} \rightarrow \mathbb{R}\) is interpreted as nabla antiderivative of s if \(S^{\nabla }(t)=s(t)\) holds \(\forall t \in \mathbb{T}_{k}\). The nabla integral of s is

$$ \int _{{\ell _{1}}}^{{\ell _{2}}} {s(t)\nabla t = S({\ell _{2}}) - S({\ell _{1}}).} $$

Suppose \(\ell _{1}, t \in \mathbb{T}\) and ħ: \(\mathbb{T} \rightarrow \mathbb{R}\). Then \(\diamond _{\alpha }\)-integral is stated as

$$ \int _{{\ell _{1}}}^{{t}} {\hbar (\mathfrak{s}){\diamond _{\alpha }}} \mathfrak{s} = \alpha \int _{{\ell _{1}}}^{{t}} { \hbar (\mathfrak{s})\Delta \mathfrak{s} + (1 - \alpha )} \int _{{\ell _{1}}}^{{t}} {\hbar (\mathfrak{s})\nabla \mathfrak{s},} $$

where \(0 \le \alpha \le 1\). Since the diamond-α integral is a mixed form of Δ and integrals, generally there does not exist

$$ { \biggl( { \int _{{\ell _{1}}}^{{t}} {\hbar (\mathfrak{s}){ \diamond _{\alpha }}\mathfrak{s}} } \biggr)^{{\diamond _{\alpha }}}} = \hbar (t),\quad t \in \mathbb{T}. $$

In the special case, for each \(t \in \mathbb{T}_{k}^{k}\), if a function is Δ and differentiable at the same time, then μ is symmetric differentiable and

$$ {\mu ^{\diamond }}(t) = \lambda (t){\mu ^{\Delta }}(t) + \bigl(1 - \lambda (t)\bigr){ \mu ^{\nabla }}(t), $$

where

$$ \lambda (t) = \lim_{t \to y} \frac{{\sigma (t) - y}}{{\sigma (t) + 2t - 2y - \rho (t)}} $$
(11)

is a real-valued function.

Remark 3

Clearly \(0\leq \lambda (t) \leq 1\) and

$$ \lambda (t) = \textstyle\begin{cases} {\frac{1}{2}}&t \text{ is dense,} \\ {\frac{{\sigma (t) - t}}{{\sigma (t) - \rho (t)}}}&t \text{ is not dense.} \end{cases} $$

Suppose μ: \([\ell _{1}, \ell _{2}]_{\mathbb{T}} \rightarrow \mathbb{R}\), \(\ell _{1} < \ell _{2}\), and \(\lambda (\cdot )\) is defined in (11). If λμ is Δ integrable and \((1 - \lambda )\mu \) is integrable on \([\ell _{1}, \ell _{2}]_{\mathbb{T}}\), then -integral of μ on \([\ell _{1}, \ell _{2}]_{\mathbb{T}}\) is stated as

$$ \int _{{\ell _{1}}}^{{\ell _{1}}} {\mu (t)\diamond t = \int _{{\ell _{1}}}^{{\ell _{1}}} {\lambda (t)\mu (t)\Delta t + \int _{{\ell _{1}}}^{{\ell _{1}}} {\bigl(1 - \lambda (t)\bigr)\mu (t) \nabla t.} } } $$

More details on diamond integrals are given in [12].

Jensen’s inequality

Inequality proved by Jensen [15] in 1906 is popular in mathematical analysis. For continuous and discrete analysis, it is used to formulate many classical inequalities. Therefore the developments in many other inequalities are based on the developments in Jensen’s inequality. Jensen’s inequality in a discrete version [15] is as follows:

Suppose that an interval E in \(\mathbb{R}\) and a convex function \(\chi :E \rightarrow \mathbb{R}\), \(v=(v_{1},\ldots ,v_{n})\) is a real n-tuple and \(y=(y_{1},\ldots ,y_{n})\) is a positive n-tuple \((n \in \mathbb{N} )\). Then

$$ \chi \biggl( { \frac{{\sum_{\mathfrak{j} = 1}^{n} {{v_{\mathfrak{j}}}{y_{\mathfrak{j}}}} }}{{\sum_{\mathfrak{j} = 1}^{n} {{v_{\mathfrak{j}}}} }}} \biggr) \le \frac{{\sum_{\mathfrak{j} = 1}^{n} {{v_{\mathfrak{j}}}\chi ({y_{\mathfrak{j}}})} }}{{\sum_{\mathfrak{j} = 1}^{n} {{v_{\mathfrak{j}}}} }}. $$
(12)

The strict convexity of χ implies that (12) is strict unless \(y_{1}=\cdots = y_{n}\) and χ is concave if (12) holds in reverse direction.

Jensen’s inequality in an integral form [16] is as follows:

Suppose \(b_{1}, b_{2} \in \mathbb{R}\) with \(b_{1} < b_{2}\) and an interval E in \(\mathbb{R}\). If \(\zeta \in C([b_{1},b_{2}], E)\) and a convex function \(\chi \in C(E, \mathbb{R})\), then

$$ \chi \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} {\zeta ( p )\,dp} }}{{{b _{2}} - {b _{1}}}}} \biggr) \le \frac{{\int _{{b _{1}}}^{{b _{2}}} {\chi ( {\zeta ( p )} )\,dp} }}{{{b _{2}} - {b _{1}}}}. $$
(13)

Anwar et al. in [4] extended the Jensen’s inequality for Δ-integrals as follows:

Suppose \(\zeta \in {C_{rd}}({[b_{1},b_{2}]_{\mathbb{T}}, E})\) and a convex function \(\chi \in C(E, \mathbb{R})\) with \(E \subset \mathbb{R}\). Let \(b_{1}, b_{2} \in \mathbb{T}\) with \(b_{1} < b_{2}\) and \(\hbar \in {C_{rd}}({\chi :[b_{1},b_{2}]_{\mathbb{T}}, \mathbb{R}})\) satisfying \(\int _{{b _{1}}}^{{b _{2}}} {|\hbar (p)|\Delta p > 0}\), then

$$ \chi \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \zeta (p)\Delta p} }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \Delta p} }}} \biggr) \le \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \chi (\zeta (p))\Delta p} }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \Delta p} }}. $$
(14)

Özkan et al. in [18] showed that if we apply nabla integrals instead of delta integrals, then (14) also holds. In [21], Sheng et al. presented \(\diamond _{\alpha }\)-dynamic derivative and \(\diamond _{\alpha }\)-dynamic integral for providing more balanced approximations with respect to computations.

Jensen’s inequality for \(\diamond _{\alpha }\)-integral is given in [3].

Suppose \(b_{1}, b_{2} \in \mathbb{T}\) with \(b_{1} < b_{2}\) and \(\alpha \in [0, 1]\). Let \(\hbar \in {C}({[b_{1},b_{2}]_{\mathbb{T}}, \mathbb{R}})\) satisfying \(\int _{{b _{1}}}^{{b _{2}}} {|\hbar (p)|{\diamond _{\alpha }}p > 0}\). If \(\zeta \in {C}({\chi :[b_{1},b_{2}]_{\mathbb{T}}, E})\) and a convex function \(\chi \in C(E, \mathbb{R})\) with an interval \(E \subset \mathbb{R}\), then we have

$$ \chi \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} \vert {\hbar (p)} \vert \zeta (p){\diamond _{\alpha }}p}}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert {\diamond _{\alpha }}p} }}} \biggr) \le \frac{{\int _{{b _{1}}}^{{b _{2}}} \vert {\hbar (p)} \vert \chi (\zeta (p)){\diamond _{\alpha }}p}}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert {\diamond _{\alpha }}p} }}. $$
(15)

Da Cruz et al. in [11] described diamond integral (a generalization of diamond-α integral) in terms of “approximate symmetric integral” on time scales \(\mathbb{T}\). Bibi et al. proved Jensen’s inequality related to diamond integrals in [5].

Theorem 4

Suppose \(b_{1}, b_{2} \in \mathbb{T}\) with \(b_{1} < b_{2}\), and \(\hbar \in C([b_{1},b_{2}]_{\mathbb{T}}, \mathbb{R})\) which satisfies \(\int _{b_{1}}^{b_{2}} {\hbar (p)\diamond p}\neq 0\). Also assume \(\zeta \in C([b_{1},b_{2}]_{\mathbb{T}}, E)\) and a convex function \(\chi \in C( E, \mathbb{R})\), where \(E=[w, W]\) with \(w = {\min_{p \in {{\chi :[b_{1},b_{2}]}_{\mathbb{T}}}}}\zeta (p)\), \(W = {\max_{p \in {{\chi :[b_{1},b_{2}]}_{\mathbb{T}}}}}\zeta (p)\), then

$$ \chi \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \zeta (p)\diamond p} }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \diamond p} }}} \biggr) \le \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \chi (\zeta (p))\diamond p} }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert \hbar (p) \vert \diamond p} }}. $$
(16)

Under the suppositions of Theorem 4, inequality (16) produces the linear functional

$$ J(\chi ) = \frac{{\int _{{b _{1}}}^{{b _{2}}} { { \vert \hbar (p) \vert } \chi (\zeta (p))\diamond p} }}{{\int _{{b _{1}}}^{{b _{2}}} { {\hbar (p)} \diamond p} }} - \chi \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} { {\hbar (p)} \zeta (p) \diamond p} }}{{\int _{{b _{1}}}^{{b _{2}}} { {\hbar (p)} \diamond p} }}} \biggr). $$
(17)

Remark 4

By Theorem 4, we conclude that \(J(\chi ) = 0\) if χ is a constant function or an identity function and \(J(\chi ) \ge 0\) for the group of convex functions.

In the same manner, the linear functionals for (14) and (15) can be obtained.

Generalized Jensen-type functional by extended Montgomery identity via Taylor’s formula I

Theorem 5

Let \(n \in N\) such that \(n\geq 2\) and all the suppositions of Theorem 4be satisfied. If χ is a convex function defined on \([b_{1},b_{2}]\) such that \(\chi ^{(n-1)}\) is absolutely continuous, then \(\forall t \in [b_{1},b_{2}]\) we have

$$\begin{aligned} J(\chi ) =& \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{(\mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{ \mathfrak{z} + 2}} - {\chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{ \mathfrak{z} + 2}}} \bigr)} \\ & {}+ \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {J\bigl({R_{n}}(t,s)\bigr){ \chi ^{n}}(s)\,ds,} \end{aligned}$$
(18)

where \({R_{n}}(t,s)\) is defined in (6) and

$$ J\bigl({R_{n}}(t,s)\bigr) = \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert {R_{n}}(\chi (\tau ),s)\diamond \tau } }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \diamond \tau } }} - {R_{n}} \biggl( { \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \chi (\tau )\diamond \tau } }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \diamond \tau } }}} \biggr),s} \biggr). $$
(19)

Proof

Putting (5) in (17), we obtain

$$\begin{aligned} J(\chi ) =& J \Biggl( {\frac{1}{{{b _{2}} - {b _{1}}}} \int _{{b _{1}}}^{{b _{2}}} {\chi (p)} \,dp + \sum _{\mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b _{1}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b _{1}})}^{\mathfrak{z} + 2}}}}{{{b _{2}} - {b _{1}}}}} } \\ &{} - \sum_{\mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b _{2}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b _{2}})}^{\mathfrak{z} + 2}}}}{{{b _{2}} - {b _{1}}}}} + \frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {{R_{n}}(t ,s){ \chi ^{(n)}}(s)\,ds} \Biggr). \end{aligned}$$
(20)

The linearity of the functional \(J(\cdot )\) gives us

$$\begin{aligned} J(\chi ) &= J \biggl( {\frac{1}{{{b _{2}} - {b _{1}}}} \int _{{b _{1}}}^{{b _{2}}} {\chi (p)} \,dp} \biggr) + J \Biggl( {\sum_{ \mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b _{1}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b _{1}})}^{\mathfrak{z} + 2}}}}{{{b _{2}} - {b _{1}}}}} } \Biggr) \\ &\quad {} - J \Biggl( {\sum_{\mathfrak{z} = 0}^{n - 2} { \frac{{{\chi ^{(\mathfrak{z} + 1)}}({b _{2}})}}{{\mathfrak{z}!(\mathfrak{z} + 2)}} \frac{{{{(t - {b _{2}})}^{\mathfrak{z} + 2}}}}{{{b _{2}} - {b _{1}}}}} } \Biggr) + J \biggl( {\frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {{R_{n}}(t ,s){\chi ^{(n)}}(s)\,ds} } \biggr) \\ & = \sum_{\mathfrak{z} = 0}^{n - 2} { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)({b _{2}} - {b _{1}})}} } \bigl( {{\chi ^{(\mathfrak{z} + 1)}}({b _{1}})J{{(t - {b _{1}})}^{ \mathfrak{z} + 2}} - {\chi ^{(\mathfrak{z} + 1)}}({b _{2}})J{{(t - {b _{2}})}^{ \mathfrak{z} + 2}}} \bigr) \\ &\quad {} + \frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {J\bigl({R_{n}}(t ,s) \bigr)} {\chi ^{(n)}}(s)\,ds. \end{aligned}$$

 □

Theorem 6

Let all the suppositions of Theorem 5be satisfied and

$$ J\bigl({R_{n}}(t,s)\bigr) \ge 0 $$
(21)

for all \(s \in [b_{1},b_{2}]\). If χ is n-convex such that \(\chi ^{(n-1)}\) is absolutely continuous, then we have that

$$ J(\chi ) \ge J(U) $$
(22)

holds, where

$$ U(t) = \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{(\mathfrak{z} + 1)}}({b_{1}}){{(t - {b_{1}})}^{ \mathfrak{z} + 2}} - {\chi ^{(\mathfrak{z} + 1)}}({b_{2}}){{(t - {b_{2}})}^{ \mathfrak{z} + 2}}} \bigr)} . $$
(23)

The reverse inequality in (21) gives rise to the reverse inequality in (22).

Proof

Since \(\chi ^{(n-1)}\) is absolutely continuous on \([b_{1},b_{2}]\), therefore \(\chi ^{n}\) exists almost everywhere. Now the n-convexity of χ implies that \(\chi ^{n}(t)\geq 0\), \(\forall t \in [b_{1},b_{2}]\); this fact together with (21) implies that

$$ J\bigl({R_{n}}(t,s)\bigr){\chi ^{n}}(s) \ge 0 $$
(24)

for all \(s \in [b_{1},b_{2}]\). Thus

$$ \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} J \bigl({R_{n}}(t,s)\bigr){ \chi ^{n}}(s)\,ds \ge 0. $$
(25)

Using (25) in (18), we get

$$ J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{(\mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{ \mathfrak{z} + 2}} - {\chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{ \mathfrak{z} + 2}}} \bigr)} \ge 0. $$
(26)

The linearity of \(J(\cdot )\) yields

$$ \begin{aligned}[b] &J(\chi ) - J \Biggl( {\frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{( \mathfrak{z} + 1)}}({b_{1}}){{(t - {b_{1}})}^{\mathfrak{z} + 2}} - { \chi ^{(\mathfrak{z} + 1)}}({b_{2}}){{(t - {b_{2}})}^{\mathfrak{z} + 2}}} \bigr)} } \Biggr)\\ &\quad \ge 0, \end{aligned} $$
(27)

which is the required result. The reverse inequality in (21) gives rise to the reverse inequality in (24); therefore, the reverse inequality in (22) is obtained. □

Theorem 7

Let all the suppositions of Theorem 6be valid, and let U, defined in (23), be convex on \([b_{1},b_{2}]\). Then we have

$$ J\bigl({\chi }(t)\bigr) \ge 0 $$
(28)

for all \(t \in [b_{1}, b_{2}]\).

Proof

Since \(U(t)\) is convex \(\forall t \in [b_{1}, b_{2}]\), thus by means of Remark 4, we get \(J(U(t)) \ge 0\). As a result, (22) implies \(J(\chi (t)) \ge 0\). □

Remark 5

The generalized form of Theorem 4 is Theorem 7.

Generalized Jensen-type functional by extended Montgomery identity via Taylor’s formula II

Theorem 8

Let \(n \in N\) such that \(n\geq 2\) and all the suppositions of Theorem 4be satisfied. If χ is a convex function defined on \([b_{1},b_{2}]\) such that \(\chi ^{(n-1)}\) is absolutely continuous, then \(\forall t \in [b_{1},b_{2}]\) we have \(\forall t \in [b_{1}, b_{2}]\)

$$\begin{aligned} J(\chi ) =& \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{(\mathfrak{z} + 2)!}}} \biggr) \bigl( {J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{\mathfrak{z} + 2}}\bigr) - J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}\bigr)} \bigr)} \\ & {}+ \frac{1}{{(n - 1)!}} \int _{p}^{q} {J\bigl({{\hat{R}}_{n}}(t,s) \bigr){\chi ^{n}}(s)\,ds}, \end{aligned}$$
(29)

where \({\hat{R}_{n}}\) is defined in (8) and

$$ J\bigl({\hat{R}_{n}}(t,s)\bigr) = \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert {\hat{R}_{n}}(\chi (\tau ),s)\diamond \tau } }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \diamond \tau } }} - {\hat{R}_{n}} \biggl( { \biggl( { \frac{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \chi (\tau )\diamond \tau } }}{{\int _{{b _{1}}}^{{b _{2}}} { \vert {\hbar (\tau )} \vert \diamond \tau } }}} \biggr),s} \biggr). $$
(30)

Proof

Putting (7) in (17), we obtain

$$\begin{aligned} J(\chi ) =& J \Biggl( {\frac{1}{{{b _{2}} - {b _{1}}}} \int _{{b _{1}}}^{{b _{2}}} {\chi (p)\,dp + \sum _{\mathfrak{z} = 0}^{n - 2} {{\chi ^{(\mathfrak{z} + 1)}}(t ) \frac{{{{({b _{1}} - t )}^{\mathfrak{z} + 2}} - {{({b _{2}} - t )}^{\mathfrak{z} + 2}}}}{{(\mathfrak{z} + 2)!({b _{2}} - {b _{1}})}}} } } \\ &{} + \frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {{{ \hat{R}}_{n}}(t ,s){ \chi ^{(n)}}(s)\,ds} \Biggr). \end{aligned}$$
(31)

By using the linearity of the functional \(J(\cdot )\), we get

$$\begin{aligned} J(\chi ) &= J \biggl( {\frac{1}{{{b _{2}} - {b _{1}}}} \int _{{b _{1}}}^{{b _{2}}} {\chi (p)\,dp} } \biggr) + J \Biggl( {\sum_{ \mathfrak{z} = 0}^{n - 2} {{\chi ^{(\mathfrak{z} + 1)}}(t ) \frac{{{{({b _{1}} - t )}^{\mathfrak{z} + 2}} - {{({b _{2}} - t )}^{\mathfrak{z} + 2}}}}{{(\mathfrak{z} + 2)!({b _{2}} - {b _{1}})}}} } \Biggr) \\ &\quad {} + J \biggl( {\frac{1}{{(n - 1)!}} \int _{{b _{1}}}^{{b _{2}}} {{{\hat{R}}_{n}}(t ,s){ \chi ^{(n)}}(s)\,ds} } \biggr) \\ & = \sum_{\mathfrak{z} = 0}^{n - 2} { \frac{1}{{(\mathfrak{z} + 2)!({b_{2}} - {b_{1}})}}} \times J \bigl( { \bigl( {{\chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{ \mathfrak{z} + 2}}} \bigr) - \bigl( {{\chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}} \bigr)} \bigr) \\ &\quad{} + \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {J \bigl( {{{\hat{R}}_{n}}(t,s)} \bigr){\chi ^{(n)}}(s)\,ds}. \end{aligned}$$

 □

Theorem 9

Let all the suppositions of Theorem 8be satisfied and

$$ J\bigl({\hat{R}_{n}}(t,s)\bigr) \ge 0 $$
(32)

for all \(s\in [b_{1},b_{2}]\). If χ is n-convex such that \(\chi ^{(n-1)}\) is absolutely continuous, then we have

$$ J(\chi ) \ge J(U) $$
(33)

holds, where

$$ U(t) = \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{(\mathfrak{z} + 2)!}}} \biggr) \bigl( {\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{\mathfrak{z} + 2}}\bigr) - \bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}\bigr)} \bigr)}. $$
(34)

Proof

Since \(\chi ^{(n-1)}\) is absolutely continuous on \([b_{1},b_{2}]\), therefore \(\chi ^{n}\) exists almost everywhere. Now the n-convexity of χ implies that \(\chi ^{n}(t)\geq 0\), \(\forall t \in [b_{1},b_{2}]\); this fact together with (21) implies that

$$ J\bigl({\hat{R}_{n}}(t,s)\bigr){\chi ^{n}}(s) \ge 0 $$
(35)

for all \(s \in [b_{1},b_{2}]\). Thus

$$ \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} J \bigl({\hat{R}_{n}}(t,s) \bigr){ \chi ^{n}}(s)\,ds \ge 0. $$
(36)

Using (36) in (29), we get

$$ J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{(\mathfrak{z} + 2)!}}} \biggr) \bigl( {J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{\mathfrak{z} + 2}}\bigr) - J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}\bigr)} \bigr)} \ge 0. $$
(37)

By the linearity of \(J(\cdot )\), we get

$$ J(\chi ) - J \Biggl( {\frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{(\mathfrak{z} + 2)!}}} \biggr) \bigl( {{\chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{ \mathfrak{z} + 2}} - { \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{ \mathfrak{z} + 2}}} \bigr)} } \Biggr) \ge 0, $$
(38)

which is the required result.

The reverse inequality in (32) gives rise to the reverse inequality in (35); therefore, the reverse inequality in (33) is obtained. □

Theorem 10

Let all the supposition of Theorem 9be valid, and let U, defined in (34), be convex on \([b_{1},b_{2}]\). Then we have

$$ J\bigl({\chi }(t)\bigr) \ge 0 $$
(39)

for all \(t \in [b_{1}, b_{2}]\).

Proof

Since \(U(t)\) is convex \(\forall t \in [b_{1},b_{2}]\), thus by means of Remark 4, we get \(J(U(t)) \ge 0\). As a result, (33) implies \(J(\chi (t)) \ge 0\). □

Remark 6

The generalized form of Theorem 4 is Theorem 10.

Bounds for identities associated with generalization of Jensen-type functional

Assume \(\mathfrak{f}, \nu :[b_{1}, b_{2}] \rightarrow \mathbb{R}\) as Lebesgue integrable functions, then the Čebyšev functional is

$$ \Omega (\mathfrak{f}, \nu ) = \frac{1}{{b_{2} - b_{1}}} \int _{{b _{1}}}^{{b _{2}}} {\mathfrak{f}(t)\nu (t)\,dt} - \frac{1}{{b_{2} - b_{1}}} \int _{{b _{1}}}^{{b _{2}}} {\mathfrak{f}(t)\,dt} \frac{1}{{b_{2} - b_{1}}} \int _{{b _{1}}}^{{b _{2}}} {\nu (t)\,dt.} $$
(40)

The following theorems were proved by Cerone and Dragomir in [10].

Theorem 11

Consider a Lebesgue integrable function \(\mathfrak{f}:[b_{1}, b_{2}] \rightarrow \mathbb{R}\) and an absolutely continuous function \(\nu :[b_{1}, b_{2}] \rightarrow \mathbb{R}\) with \((\cdot - b_{1})(b_{2} - \cdot )[\nu ']^{2} \in L[b_{1}, b_{2}]\). Then

$$ \bigl\vert {\Omega (\mathfrak{f},\nu )} \bigr\vert \le \frac{1}{{\sqrt{2} }}{ \bigl[ {\Omega (\mathfrak{f}, \mathfrak{f})} \bigr]^{\frac{1}{2}}}\frac{1}{{\sqrt{b_{2} - b_{1}} }}{ \biggl( { \int _{{b _{1}}}^{{b _{2}}} {(y - b_{1}) (b_{2} - y){{ \bigl[ {\nu '(y)} \bigr]}^{2}} \,dy} } \biggr)^{\frac{1}{2}}}. $$
(41)

The constant \(\frac{1}{\sqrt{2}}\) in (41) is the best possible.

Theorem 12

If \(\nu :[b_{1}, b_{2}] \rightarrow \mathbb{R}\) is monotonic nondecreasing on \([b_{1}, b_{2}]\) and \(\mathfrak{f}:[b_{1}, b_{2}] \rightarrow \mathbb{R}\) is absolutely continuous with \(\mathfrak{f}'\in L_{\infty }[b_{1}, b_{2}]\). Then we have

$$ \bigl\vert {\Omega (\mathfrak{f},\nu )} \bigr\vert \le \frac{1}{{2(b_{2} - b_{1})}}{ \bigl\Vert {\mathfrak{f}'} \bigr\Vert _{\infty }} \int _{{b _{1}}}^{{b _{2}}} {(y - b_{1}) (b_{2} - y)\,dh(y).} $$
(42)

The constant \(\frac{1}{2}\) in (42) is the best possible.

Let us denote

$$ \varrho (s) = J\bigl({R_{n}}(t, s)\bigr), \quad s \in [ {{b_{1}}, {b_{2}}} ], $$
(43)

and

$$ \hat{\varrho } (s) = J\bigl({\hat{R}_{n}}(t, s)\bigr), \quad s \in [ {{b_{1}}, {b_{2}}} ]. $$
(44)

Now, the Čebyšev functionals defined as

$$\begin{aligned}& \Omega (\varrho ,\varrho ) = \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {{\varrho ^{2}}(s)\,ds - {{ \biggl( {\frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} } \biggr)}^{2}}.} \end{aligned}$$
(45)
$$\begin{aligned}& \Omega (\hat{\varrho },\hat{\varrho }) = \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {{{\hat{\varrho }}^{2}}(s) \,ds - {{ \biggl( { \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {\hat{\varrho }(s)\,ds} } \biggr)}^{2}}.} \end{aligned}$$
(46)

Theorem 13

Let \(n \in N\) such that \(n\geq 2\) and all the suppositions of Theorem 4be satisfied. If χ is a convex function defined on \([b_{1},b_{2}]\) such that \(\chi ^{(n-1)}\) is absolutely continuous, \((\cdot - b_{1})(b_{2} - \cdot )[\chi ^{(n+1)}]^{2} \in L[b_{1}, b_{2}]\), and \(R_{n}\), \(\hat{R}_{n}\), ϱ, ϱ̂, and Ω be defined in (6), (8), (43), (44), (45), and (46), respectively, then \(\forall t \in [b_{1},b_{2}]\) we have

$$\begin{aligned} J(\chi ) =& \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{(\mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{ \mathfrak{z} + 2}} - {\chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{ \mathfrak{z} + 2}}} \bigr)} \\ & {}+ \frac{{{\chi ^{(\mathfrak{z} - 1)}}({b_{2}}) - {\chi ^{(\mathfrak{z} - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds + H_{n}( \chi ;{b_{1}},{b_{2}})} , \end{aligned}$$
(47)

and

$$\begin{aligned} J(\chi ) =& \frac{1}{{{b_{2}} - {b_{1}}}}\sum_{\mathfrak{z} = 0}^{n - 2} { \biggl( {\frac{1}{{(\mathfrak{z} + 2)!}}} \biggr) \bigl( {J\bigl({\chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{\mathfrak{z} + 2}}\bigr) - J\bigl({\chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}\bigr)} \bigr)} \\ &{} + \frac{{{\chi ^{(\mathfrak{z} - 1)}}({b_{2}}) - {\chi ^{(\mathfrak{z} - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\hat{\varrho }(s)\,ds + \hat{H}_{n}(\chi ;{b_{1}},{b_{2}})}. \end{aligned}$$
(48)

The remainders \(H_{n}(\chi ;b_{1}, b_{2})\) and \(\hat{H}_{n}(\chi ;b_{1}, b_{2})\) satisfy the approximation

$$\begin{aligned}& \bigl\vert {H_{n}(\chi ;{b_{1}},{b_{2}})} \bigr\vert \le \frac{{{{ [ {\Omega (\varrho ,\varrho )} ]}^{\frac{1}{2}}}}}{{(n - 1)!}} \sqrt{\frac{{{b_{2}} - {b_{1}}}}{2}} { \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {(s - {b_{1}}) ({b_{2}} - s){{ \bigl[ {{\chi ^{(n + 1)}}(s)} \bigr]}^{2}}\,ds} } \biggr\vert ^{ \frac{1}{2}}}, \end{aligned}$$
(49)
$$\begin{aligned}& \bigl\vert {\hat{H}_{n}(\chi ;{b_{1}},{b_{2}})} \bigr\vert \le \frac{{{{ [ {\Omega (\hat{\varrho },\hat{\varrho })} ]}^{\frac{1}{2}}}}}{{(n - 1)!}} \sqrt{\frac{{{b_{2}} - {b_{1}}}}{2}} { \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {(s - {b_{1}}) ({b_{2}} - s){{ \bigl[ {{\chi ^{(n + 1)}}(s)} \bigr]}^{2}}\,ds} } \biggr\vert ^{\frac{1}{2}}}. \end{aligned}$$
(50)

Proof

Applying Theorem 11 for \(\mathfrak{f} \rightarrow \varrho \) and \(\nu \rightarrow \chi ^{(n)}\), we obtain

$$\begin{aligned}& \biggl\vert {\frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} { \varrho (s){\chi ^{(n)}}(s) \,ds} - \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {{\chi ^{(n)}}(s)\,ds} } \biggr\vert \\& \quad \le \frac{1}{{\sqrt{2} }}{ \bigl[ {\Omega (\varrho ,\varrho )} \bigr]^{ \frac{1}{2}}}\frac{1}{{\sqrt{{b_{2}} - {b_{1}}} }}{ \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {(s - {b_{1}}) ({b_{2}} - s){{ \bigl[ {{\chi ^{(n + 1)}}(s)} \bigr]}^{2}}\,ds} } \biggr\vert ^{\frac{1}{2}}}. \end{aligned}$$
(51)

Now, dividing both sides of (51) by \((n-1)!\) and multiplying by \((b_{2}-b_{1})\), we get

$$\begin{aligned}& \biggl\vert {\frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){ \chi ^{(n)}}(s) \,ds} - \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} { \varrho (s)\,ds} \frac{{{\chi ^{(n - 1)}}({b_{2}}) - {\chi ^{(n - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})}}} \biggr\vert \\& \quad \le \sqrt{\frac{{{b_{2}} - {b_{1}}}}{2}} \frac{{{{ [ {\Omega (\varrho ,\varrho )} ]}^{\frac{1}{2}}}}}{{(n - 1)!}}{ \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {(s - {b_{1}}) ({b_{2}} - s){{ \bigl[ {{\chi ^{(n + 1)}}(s)} \bigr]}^{2}}\,ds} } \biggr\vert ^{ \frac{1}{2}}}. \end{aligned}$$
(52)

By denoting

$$\begin{aligned} H_{n}(\chi ;{b_{1}},{b_{2}}) =& \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){\chi ^{(n)}}(s) \,ds} \\ &{}- \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} \frac{{{\chi ^{(n - 1)}}({b_{2}}) - {\chi ^{(n - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})}} \end{aligned}$$
(53)

in (52), we have (49). Hence, we obtain

$$\begin{aligned} \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){\chi ^{(n)}}(s) \,ds} = \frac{{{\chi ^{(n - 1)}}({b_{2}}) - {\chi ^{(n - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} + H_{n}( \chi ;{b_{1}},{b_{2}}), \end{aligned}$$
(54)

where the remainder \(H_{n}(\chi ;b_{1},b_{2})\) satisfies approximation (49). Now from (18) we obtain (47).

Similarly, from identity (29), we get (50). □

Using Theorem 12, the Grüss-type inequality is obtained in the following theorem.

Theorem 14

Let \(n \in N\) such that \(n\geq 2\) and all the suppositions of Theorem 4be satisfied. If χ is a convex function defined on \([b_{1},b_{2}]\) such that \(\chi ^{(n)}\) is absolutely continuous, \(\chi ^{(n+1)} \geq 0\) on \([b_{1},b_{2}]\), and the functions ϱ, ϱ̂, and Ω are defined by (43), (44), (45), and (46), respectively. Then we have (47), and the remainder \(H_{n}(\chi ;b_{1}, b_{2})\) satisfies the bound

$$\begin{aligned} \bigl\vert {H_{n}(\chi ;{b_{1}},{b_{2}})} \bigr\vert \le& \frac{{({b_{2}} - {b_{1}})}}{{(n - 1)!}}{ \bigl\Vert {\varrho '} \bigr\Vert _{\infty }} \\ &{}\times \biggl\{ { \frac{{{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}}({b_{1}})}}{2} - \frac{{{\chi ^{(n - 2)}}({b_{2}}) - {\chi ^{(n - 2)}}({b_{1}})}}{{{b_{2}} - {b_{1}}}}} \biggr\} , \end{aligned}$$
(55)

where as representation (48) and the remainder \(\hat{H}_{n}(\chi ;b_{1}, b_{2})\) satisfies the approximation

$$\begin{aligned} \bigl\vert {\hat{H}_{n}(\chi ;{b_{1}},{b_{2}})} \bigr\vert \le& \frac{{({b_{2}} - {b_{1}})}}{{(n - 1)!}}{ \bigl\Vert {\varrho '} \bigr\Vert _{\infty }} \\ &{} \times \biggl\{ { \frac{{{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}}({b_{1}})}}{2} - \frac{{{\chi ^{(n - 2)}}({b_{2}}) - {\chi ^{(n - 2)}}({b_{1}})}}{{{b_{2}} - {b_{1}}}}} \biggr\} . \end{aligned}$$
(56)

Proof

Using Theorem 12 for \(f \rightarrow \varrho \) and \(\nu \rightarrow \chi ^{(n)}\), we get

$$\begin{aligned}& { \biggl\vert {\frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} { \varrho (s){\chi ^{(n)}}(s) \,ds} - \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} \frac{1}{{{b_{2}} - {b_{1}}}} \int _{{b_{1}}}^{{b_{2}}} {{\chi ^{(n)}}(s)\,ds} } \biggr\vert } \\& \quad { \le \frac{1}{{2({b_{2}} - {b_{1}})}}{{ \bigl\Vert {\varrho '} \bigr\Vert }_{\infty }} \int _{{b_{1}}}^{{b_{2}}} {(s - {b_{1}}) ({b_{2}} - s){ \chi ^{(n + 1)}}(s)\,ds.} } \end{aligned}$$
(57)

Since

$$\begin{aligned}& \int _{{b_{1}}}^{{b_{2}}} (s - {b_{1}}) ({b_{2}} - s){\chi ^{(n + 1)}}(s)\,ds \\& \quad = \int _{{b_{1}}}^{{b_{2}}} { \bigl[ {2s - ({b_{1}} + {b_{2}})} \bigr]} {\chi ^{(n)}}(s)\,ds \\& \quad = ({b_{2}} - {b_{1}}) \bigl[ {{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}}({b_{1}})} \bigr] - 2 \bigl( {{\chi ^{(n - 2)}}({b_{2}}) - { \chi ^{(n - 2)}}({b_{1}})} \bigr), \end{aligned}$$

dividing both sides of (57) by \((n-1)!\) and multiplying by \((b_{2}-b_{1})\), we have

$$\begin{aligned}& { \biggl\vert {\frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){ \chi ^{(n)}}(s) \,ds} - \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} \frac{{{\chi ^{(n - 1)}}({b_{2}}) - {\chi ^{(n - 1)}}({b_{1}})}}{{({b_{2}} - {b_{1}})(n - 1)!}}} \biggr\vert } \\& \quad { \le \frac{({b_{2}} - {b_{1}})}{{(n - 1)!}}{{ \bigl\Vert {\varrho '} \bigr\Vert }_{\infty }} \biggl[ { \frac{{ [ {{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}}({b_{1}})} ]}}{2} - \frac{{ ( {{\chi ^{(n - 2)}}({b_{2}}) - {\chi ^{(n - 2)}}({b_{1}})} )}}{{{b_{2}} - {b_{1}}}}} \biggr].} \end{aligned}$$
(58)

Using (18), we get

$$\begin{aligned}& \Biggl\vert {J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{( \mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{\mathfrak{z} + 2}} - { \chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{\mathfrak{z} + 2}}} \bigr)} } \\& \qquad {} - \frac{1}{{(n - 1)!}} \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)\,ds} \frac{{{\chi ^{(n - 1)}}({b_{2}}) - {\chi ^{(n - 1)}}({b_{1}})}}{{{b_{2}} - {b_{1}}}} \Biggr\vert \\& \quad \le \frac{({b_{2}} - {b_{1}})}{{(n - 1)!}}{ \bigl\Vert {\varrho '} \bigr\Vert _{\infty }} \biggl[ { \frac{{ [ {{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}} ({b_{1}})} ]}}{2} - \frac{{ ( {{\chi ^{(n - 2)}}({b_{2}}) - {\chi ^{(n - 2)}}({b_{1}})} )}}{{{b_{2}} - {b_{1}}}}} \biggr]. \end{aligned}$$
(59)

Now, using (47), we deduce (55)

$$\begin{aligned} \bigl\vert {\hat{H}_{n}(\chi ;{b_{1}},{b_{2}})} \bigr\vert \le& \frac{({b_{2}} - {b_{1}})}{{(n - 1)!}}{ \bigl\Vert {\varrho '} \bigr\Vert _{\infty }} \\ &{}\times\biggl[ { \frac{{ [ {{\chi ^{(n - 1)}}({b_{2}}) + {\chi ^{(n - 1)}}({b_{1}})} ]}}{2} - \frac{{ [ {{\chi ^{(n - 2)}}({b_{2}}) - {\chi ^{(n - 2)}}({b_{1}})} ]}}{{{b_{2}} - {b_{1}}}}} \biggr]. \end{aligned}$$
(60)

Similarly, using (29) instead of (18), we get (56). □

The Ostrowski-type inequality allied to generalized Jensen’s inequality is specified by the following theorem.

Theorem 15

Assume that the conditions of Theorems 5and 8are satisfied. Consider a specific pair \((r_{1},r_{2})\) of conjugate exponents, i.e., \(1 \leq r_{1}, r_{2}\leq \infty \), \(\frac{1}{r_{1}}+\frac{1}{r_{2}}=1\). For n N, assume that the function \({ \vert {{\chi ^{(n)}}} \vert ^{{r_{1}}}}:[{b_{1}}, {b_{2}}] \rightarrow \mathbb{R}\) is Riemann integrable for \(n\geq 2\). Then

$$\begin{aligned} (i)\quad &\Biggl\vert {J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{( \mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{\mathfrak{z} + 2}} - { \chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{\mathfrak{z} + 2}}} \bigr)} } \Biggr\vert \\ &\quad \le \frac{1}{{(n - 1)!}}{ \bigl\Vert {{\chi ^{(n)}}} \bigr\Vert _{{r_{1}}}} { \biggl( { \int _{{b_{1}}}^{{b_{2}}} {{{ \bigl\vert {J \bigl({R_{n}}(t,s)\bigr)} \bigr\vert }^{{r_{2}}}}\,ds} } \biggr)^{\frac{1}{{{r_{2}}}}}}. \end{aligned}$$
(61)

The constant on the right-hand side of (61) is sharp for \(1< r_{1} \leq \infty \) and best possible for \(r_{1}=1\).

$$\begin{aligned} (ii)\quad &\Biggl\vert {J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{1}} - t)}^{\mathfrak{z} + 2}}\bigr) - J\bigl({ \chi ^{(\mathfrak{z} + 1)}}(t){{({b_{2}} - t)}^{\mathfrak{z} + 2}}\bigr)} \bigr)} } \Biggr\vert \\ &\quad \le \frac{1}{{(n - 1)!}}{ \bigl\Vert {{\chi ^{(n)}}} \bigr\Vert _{{r_{1}}}} { \biggl( { \int _{{b_{1}}}^{{b_{2}}} {{{ \bigl\vert {J\bigl({{ \hat{R}}_{n}}(t,s)\bigr)} \bigr\vert }^{{r_{2}}}}\,ds} } \biggr)^{\frac{1}{{{r_{2}}}}}}. \end{aligned}$$
(62)

The constant on the right-hand side of (62) is sharp for \(1< r_{1} \leq \infty \) and best possible for \(r_{1}=1\).

Proof

(i) Let us take

$$ \varrho (s) = \frac{1}{{(n - 1)!}}J\bigl({R_{n}}(t,s)\bigr). $$

Using (18) and Hölder’s inequality, we get

$$\begin{aligned}& \Biggl\vert {J(\chi ) - \frac{1}{{{b_{2}} - {b_{1}}}}\sum _{ \mathfrak{z} = 0}^{n - 2} { \biggl( { \frac{1}{{\mathfrak{z}!(\mathfrak{z} + 2)}}} \biggr) \bigl( {{\chi ^{( \mathfrak{z} + 1)}}({b_{1}})J{{(t - {b_{1}})}^{\mathfrak{z} + 2}} - { \chi ^{(\mathfrak{z} + 1)}}({b_{2}})J{{(t - {b_{2}})}^{\mathfrak{z} + 2}}} \bigr)} } \Biggr\vert \\& \quad = \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){\chi ^{n}}(s) \,ds} } \biggr\vert \le { \bigl\Vert {{\chi ^{(n)}}} \bigr\Vert _{{r_{1}}}} { \biggl( { \int _{{b_{1}}}^{{b_{2}}} {{{ \bigl\vert {\varrho (s)} \bigr\vert }^{{r_{2}}}}\,ds} } \biggr)^{\frac{1}{{{r_{2}}}}}}. \end{aligned}$$
(63)

The sharpness of the constant \({ ( {\int _{{b_{1}}}^{{b_{2}}} {{{ \vert {\varrho (s)} \vert }^{{r_{2}}}}d} s} )^{\frac{1}{{{r_{2}}}}}}\) is proved by considering a function χ for which the inequality in (61) is obtained.

For \(1< r_{1} < \infty \), we take

$$ {\chi ^{(n)}}(s) = \operatorname{sgn}\varrho (s){ \bigl\vert {\varrho (s)} \bigr\vert ^{ \frac{1}{{r_{1} - 1}}}}. $$

For \(r_{1}=\infty \), take \({\chi ^{(n)}}(s) = \operatorname{sgn} \varrho (s)\).

For \(r_{1}=1\), we prove that

$$ \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {\varrho (s){\chi ^{n}}(s)} } \biggr\vert \le \max_{s \in [{b_{1}},{b_{2}}]} \bigl\vert {\varrho (s)} \bigr\vert \biggl( { \int _{{b_{1}}}^{{b_{2}}} { \bigl\vert {{ \chi ^{(n)}}(s)} \bigr\vert \,ds} } \biggr) $$
(64)

is the best possible inequality. Let \(\vert {\varrho (s)} \vert \) attain its maximum at \(k_{0} \in [b_{1}, b_{2}]\). First, we suppose that \(\varrho (k_{0})>0\). Taking ϵ to be small enough, we define \(\chi _{\epsilon }(s)\) by

$$ {\chi _{\epsilon }}(s): = \textstyle\begin{cases} 0,& {b_{1}} \le s \le {k_{0}}, \\ \frac{1}{{ \epsilon n!}}{{(s - {k_{0}})}^{n}}, & {k_{0}} \le s \le {k_{0}} + \epsilon , \\ \frac{1}{{n!}}{{(s - {k_{0}})}^{n - 1}}, & {k_{0}} + \epsilon \le s \le {b_{2}}. \end{cases} $$
(65)

Then, for ϵ small enough,

$$ \biggl\vert { \int _{{b_{1}}}^{{b_{2}}} {\varrho (s)} {\chi ^{(n)}}(s)} \biggr\vert = \biggl\vert { \int _{{k_{0}}}^{{k_{0}} + \epsilon } {\varrho (s) \frac{1}{ \epsilon }\,ds} } \biggr\vert = \frac{1}{ \epsilon } \int _{{k_{0}}}^{{k_{0}} + \epsilon } {\varrho (s)\,ds.} $$
(66)

By inequality (64), we have

$$ \frac{1}{ \epsilon } \int _{{k_{0}}}^{{k_{0}} + \epsilon } {\varrho (s)\,ds} \le \varrho ({k_{0}}) \int _{{k_{0}}}^{{k_{0}} + \epsilon } { \frac{1}{ \epsilon }\,ds = \varrho ({k_{0}}).} $$
(67)

Since

$$ \lim_{ \epsilon \to 0} \frac{1}{ \epsilon } \int _{{k_{0}}}^{{k_{0}} + \epsilon } {\varrho (s)\,ds} = \varrho ({k_{0}}), $$
(68)

the statement follows.

If \(\varrho (k_{0})<0\), then \(\chi _{\epsilon }(s)\) is

$$ {\chi _{\epsilon }}(s): = \textstyle\begin{cases} \frac{1}{{n!}}{{(s - {k_{0}} - \epsilon )}^{n - 1}}, & {b_{1}} \le s \le {k_{0}}, \\ - \frac{1}{{ \epsilon n!}}{{(s - {k_{0}} - \epsilon )}^{n}}, & {k_{0}} \le s \le {k_{0}} + \epsilon , \\ 0, & {k_{0}} + \epsilon \le s \le {b_{2}}, \end{cases} $$
(69)

and the remaining proof is identical as above.

(ii) is the same as (i). □

Remark 7

By analogy to Sects. 6 and 7 of Aras-Gazič et al. [22], the n-exponential convexity, exponential convexity, and applications to Stolarsky-type means can be established for the functionals defined as the difference between the R.H.S and the L.H.S of generalized inequalities (22) and (33), where U is defined in (23) and (34), respectively.

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Information sharing isn’t appropriate to this paper as no datasets were produced or analyzed during the present study.

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Acknowledgements

The writers wish to thank the mysterious refs in advance for their extremely cautious perusing of the original copy and productive remarks and recommendations.

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Ramzan, S., Nosheen, A., Bibi, R. et al. Generalized Jensen’s functional on time scales via extended Montgomery identity. J Inequal Appl 2021, 101 (2021). https://doi.org/10.1186/s13660-021-02629-4

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Keywords

  • Jensen’s inequality
  • Convex functions
  • Diamond integrals
  • Time scales calculus