Generalizations of Levinson-type inequalities via new Green functions and Hermite interpolating polynomial

Abstract

In this study, Levinson-type inequalities for the class of n-convex ($n \geq 3$) functions are generalized using new Green functions and the Hermite interpolating polynomial involving two types of data points. Some estimations for novel functionals are derived using f-divergence. Furthermore, different inequalities involving Shannon entropies are presented.

1 Introduction and preliminaries

The theory of inequalities and convex functions have a strong connection. Convex functions are important to a number of fields of mathematics and play an essential part in the research of optimization problems and modern analysis. Numerous physicists and mathematicians have used higher-order convexity to exploit inequalities and solve problems requiring greater dimensions.

The divided difference is given in the following definition:

Divided Difference

([1, p.14])

For a function $h:[\hat{d}_{1},\hat{d}_{2}] \rightarrow \mathbb{R}$, the nth order divided difference, at mutually exclusive points $u_{0},\ldots,u_{n}\in [\hat{d}_{1},\hat{d}_{2}]$, is recursively defined by

$$\begin{aligned}{} [u_{\sigma}; h ] =& h (u_{\sigma} ),\quad \sigma = 0,\ldots ,n, \\ {}[ u_{0},\dots ,u_{n};h ] =& \frac{ [ u_{1},\ldots ,u_{n};h ] - [ u_{0},\ldots ,u_{n-1};h ] }{u_{n}-u_{0}}. \end{aligned}$$

(1)

It is known that (1) is equivalent to

$$ [ u_{0},\ldots , u_{n};h ]=\sum _{\sigma =0}^{n} \frac{h (u_{\sigma} )}{l^{\prime } (h_{\sigma} )},\quad \text{where}\ l (u )=\prod_{e=0}^{n} (u-u_{e} ). $$

In the following formulation (see [1, p. 15]), nth-order divided difference is used to define an n-convex function.

n-Convex function

For $(n+1 )$ different points $u_{0},\ldots ,u_{n} \in [\hat{d}_{1},\hat{d}_{2}]$, a function $f : [\hat{d}_{1},\hat{d}_{2}] \rightarrow \mathbb{R}$ is said n-convex $(n\geq 0 )$ if and only if

$$ [u_{0},\ldots ,u_{n};f ]\geq 0 $$

holds.

$$\text{If } [u_{0},\ldots ,u_{n};f ] \leq 0,\quad \text{then }f\text{ is }n\text{-concave.} $$

The following criteria for an n-convex function is provided in [1, p. 16].

Theorem I

f is n-convex if and only if $f^{(n)}\geq 0$, given that $f^{(n)}$ exists.

Levinson [2] extended Ky Fan’s inequality for 3-convex functions as given:

Theorem A

Consider $f :\mathbb{I}_{2}=(0, 2\gamma ) \rightarrow \mathbb{R}$ with $\frac{d^{3}}{dz^{3}}f(z)\geq 0$. Consider $p_{\sigma}>0$ with $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=Q$ and $x_{\sigma} \in (0,\gamma )$. Then

$$\begin{aligned} \frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}f(x_{ \sigma})- f \Biggl(\frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}x_{\sigma} \Biggr) \leq & \frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma}f(2 \gamma -x_{\sigma}) \\ &{}-f \Biggl(\frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}(2 \gamma -x_{\sigma}) \Biggr). \end{aligned}$$

(2)

Popoviciu [3] noted that Levinson’s inequality (2) has a significant role on $(0, 2\gamma )$, while in [4], Bullen provided distinctive conformation of Popoviciu’s results also gave converse of (2).

Theorem B

(i) Assume that $f:\mathrm{T}=[\hat{d}_{1}, \hat{d}_{2}] \rightarrow \mathbb{R}$ is a convex function of the third order and $x_{\sigma}, y_{\sigma} \in \mathrm{T}$ for $\sigma =1, 2, \ldots , {\hat{m}}$, $p_{\sigma}>0$ such that

$$ \min \{y_{1} \ldots y_{\hat{m}}\}\geq \max \{x_{1} \ldots x_{\hat{m}} \} ,\qquad y_{1}+x_{1}= \cdots =y_{{\hat{m}}}+x_{{\hat{m}}} $$

(3)

then

$$ \frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}} p_{\sigma}f(_{ \sigma})-f \Biggl(\frac{1}{Q_{{\hat{m}}}}\sum _{\sigma =1}^{{\hat{m}}}p_{ \sigma}x_{\sigma} \Biggr)\leq \frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma}f(y_{\sigma})-f \Biggl(\frac{1}{Q_{{\hat{m}}}}\sum_{ \sigma =1}^{{\hat{m}}} p_{\sigma}y_{\sigma} \Biggr). $$

(4)

(ii) For $p_{\sigma}>0$, f is 3-convex if f is continuous and (3) and (4) hold.

From (4), we have the following functional:

$$\begin{aligned} \mathbb{D}\bigl(f(\cdot)\bigr) =&\frac{1}{Q_{{\hat{m}}}}\sum _{\sigma =1}^{{\hat{m}}}p_{ \sigma}f(y_{\sigma})-f \Biggl(\frac{1}{ Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}y_{\sigma} \Biggr) -\frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}} p_{ \sigma}f(x_{\sigma}) \\ &{}+f \Biggl(\frac{1}{Q_{{\hat{m}}}}\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{ \sigma} \Biggr). \end{aligned}$$

(5)

In the next result, Pečarić [5] used weakening condition (3) to derive inequality (4).

Theorem C

Let $f:\mathrm{T} \rightarrow \mathbb{R}$ be such that $f^{3}(t)\geq 0$ and $0< p_{\sigma}$. Let $x_{\sigma}, y_{\sigma} \in \mathrm{T}$ also be such that $x_{\sigma}+y_{\sigma}=2\breve{c}$, for $\sigma =1, \ldots , {\hat{m}}$, $x_{\sigma}+x_{{\hat{m}}-\sigma +1} \leq 2\breve{c}$ and $\frac{p_{\sigma}x_{\sigma}+p_{{\hat{m}}-\sigma +1}x_{{\hat{m}}-\sigma +1}}{p_{\sigma}+p_{{\hat{m}}-\sigma +1}} \leq \breve{c}$. Then (4) is valid.

In [6], Mercer stated that (4) is true for the symmetric distribution of the points, given in following theorem.

Theorem D

Let f be a 3-convex function, defined on $\mathrm{T,}$ and $p_{\sigma}$ is such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$. Choose $x_{\sigma}$, $y_{\sigma}$ such that $\min \{y_{1}\ldots y_{\hat{m}}\}\geq \max \{x_{1} \ldots x_{\hat{m}} \}$ and

$$ \sum_{\sigma =1}^{{\hat{m}}}p_{\sigma} \Biggl(x_{\sigma}-\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2}=\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma} \Biggl(y_{\sigma}- \sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}y_{\sigma} \Biggr)^{2}. $$

(6)

Then (4) holds.

Let $\mathrm{T}=[\hat{d}_{1},\hat{d}_{2}]\subset \mathbb{R}$, $\hat{d}_{1}<\hat{d}_{2}$. In [7], Pečarić et al. proved Abel-Gontscharof-type identities by applying new type of Green functions:

$$\begin{aligned}& f(\hat{u})=f(\hat{d}_{1})+(\hat{u}- \hat{d}_{1})f'(\hat{d}_{2})- \int _{ \mathrm{T}}\mathbb{\hat{G}}_{1} (\hat{u}, \hat{z})f''(\hat{z})\,d\hat{z}, \end{aligned}$$

(7)

$$\begin{aligned}& f(\hat{u})=f(\hat{d}_{2})-(\hat{d}_{2}- \hat{u})f'(\hat{d}_{1})+ \int _{ \mathrm{T}}\mathbb{\hat{G}}_{2} (\hat{u}, \hat{z})f''(\hat{z})\,dw, \end{aligned}$$

(8)

$$\begin{aligned}& f(\hat{u})=f(\hat{d}_{2})+(\hat{u}- \hat{d}_{1})f'(\hat{d}_{1})-( \hat{d}_{2}-\hat{d}_{1})f'( \hat{d}_{2}) + \int _{\mathrm{T}} \mathbb{\hat{G}}_{3}(\hat{u}, \hat{z})f''(\hat{z})\,d\hat{z}, \end{aligned}$$

(9)

$$\begin{aligned}& f(\hat{u})=f(\hat{d}_{1})+(\hat{d}_{2}- \hat{d}_{1})f'(\hat{d}_{1})-( \hat{d}_{2}-\hat{u})f'(\hat{d}_{2}) - \int _{\mathrm{T}} \mathbb{\hat{G}}_{4}(\hat{u}, \hat{z})f''(\hat{z})\,d\hat{z,} \end{aligned}$$

(10)

where $f:\mathrm{T}\rightarrow \mathbb{R}$ and for $\alpha =1,\ldots ,4$, $\mathbb{\hat{G}}_{\alpha}:\mathrm{T}\times \mathrm{T}\rightarrow \mathbb{R}$ are given as:

$$\begin{aligned}& \mathbb{\hat{G}}_{1}(\hat{u},\hat{z})=\textstyle\begin{cases} \hat{z}-\hat{d}_{1}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \hat{u}-\hat{d}_{1}, & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(11)

$$\begin{aligned}& \mathbb{\hat{G}}_{2}(\hat{u},\hat{z})=\textstyle\begin{cases} \hat{u}-\hat{d}_{2}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \hat{z}-\hat{d}_{2}, & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(12)

$$\begin{aligned}& \mathbb{\hat{G}}_{3}(\hat{u},\hat{z})=\textstyle\begin{cases} \hat{u}-\hat{d}_{1}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \hat{z}-\hat{d}_{1}, & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(13)

$$\begin{aligned}& \mathbb{\hat{G}}_{4}(\hat{u},\hat{z})=\textstyle\begin{cases} \hat{z}-\hat{d}_{2}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \hat{u}-\hat{d}_{2}, & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(14)

In [8], the Hermite interpolating polynomial is defined as follows:

Let $\hat{d}_{1}, \hat{d}_{2} \in \mathbb{R}$ with $\hat{d}_{1} < \hat{d}_{2}$, and $\hat{d}_{1} = \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a}= \hat{d}_{2}$ ($2 \leq a$) be the points. If $f \in C^{n}[\hat{d}_{1}, \hat{d}_{2}]$ and $\hat{\gamma}_{H}^{(i)}(s)$ exist, then the following Hermite conditions hold:

Hermite Conditions

$$\begin{aligned} \hat{\gamma}_{H}^{(i)}(\hat{g}_{e}) = f^{(i)}(\hat{g}_{e}); \quad 0 \leq i \leq k_{e}, 1 \leq e \leq a, \sum _{e=1}^{a}k_{e} + a = n. \end{aligned}$$

Theorem H

([8])

Let $-\infty < \hat{d}_{1} < \hat{d}_{2} < \infty$ and $\hat{d}_{1} < \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} \leq \hat{d}_{2}$ $(a \geq 2)$ and $f \in C^{n}([\hat{d}_{1}, \hat{d}_{2}])$. Then

$$\begin{aligned} f(\check{v})= \hat{\gamma}_{H}(\check{v}) + R_{H}(f, \check{v}), \end{aligned}$$

(15)

where

$$\begin{aligned} \hat{\gamma}_{H}(\check{v}) = \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}H_{i_{e}}( \check{v})f^{(i)}(\hat{g}_{e}) \end{aligned}$$

is the Hermite interpolating polynomial, and $H_{i_{e}}$ are the polynomials for the Hermite basis defined as

$$\begin{aligned} H_{i_{e}}(\check{v})=\frac{1}{i!} \frac{\mathrm{T}(\check{v})}{(\check{v} - \hat{g}_{e})^{k_{e} + 1-i}} \sum_{k=0}^{k_{e} - i} \frac{1}{k!}\frac{d^{k}}{d\check{v}^{k}} \biggl( \frac{(\check{v} -\hat{g}_{e})^{k_{e} + 1}}{\mathrm{T}(\check{v})} \biggr) \biggm|_{\check{v} = \hat{g}_{e}}(\check{v} - \hat{g}_{e})^{k}, \end{aligned}$$

(16)

with

$$\begin{aligned} \mathrm{T}(\check{v}) = \Pi _{e=1}^{a}(\check{v}- \hat{g}_{e})^{k_{e} + 1}, \end{aligned}$$

and

$$\begin{aligned} R_{H}(f, \check{v}) = \int _{\mathrm{T}}f^{(n)}(s)\mathbb{G}_{H, n}( \check{v}, s)\,ds \end{aligned}$$

is remainder, where $\mathbb{G}_{H, n}(\check{v}, s)$ is given by

$$\begin{aligned} \mathbb{G}_{H, n}(\check{v}, s)= \textstyle\begin{cases} \sum_{e=1}^{a}\sum_{i=0}^{k_{e}} \frac{(\hat{g}_{e} - s)^{n - i -1}}{(n - i - 1)!}H_{i_{e}}(\check{v}) , & \textit{$s \leq \check{v}$;} \\ -\sum_{e=r+1}^{a}\sum_{i=0}^{k_{e}} \frac{(\hat{g}_{e} - s)^{n - i -1}}{(n - i - 1)!}H_{i_{e}}(\check{v}) , & \textit{$ s \geq \check{v}$,} \end{cases}\displaystyle \end{aligned}$$

(17)

for all $\hat{g}_{r} \leq s \leq \hat{g}_{r+1}$; $r = 0, 1, \ldots , a$, with $\hat{g}_{0} = \hat{d}_{1}$ and $\hat{g}_{a+1}= \hat{d}_{2}$.

We observe that $0\leq \mathbb{G}_{H, n-3}(\check{v}, s)$, and $\mathbb{G}_{H, n-3}$ represents derivative of order three with respect to the first variable.

The positivity of $\mathbb{G}_{H, n}(\check{v}, s)$ is described in [9] and [10], as follows:

Lemma 1

The following statements are true for the Green function $\mathbb{G}_{H, n}(\check{v}, s)$ as given in (17).

• (i) $\frac{\mathbb{G}_{H, n}(\check{v}, s)}{\mathrm{T}(\check{v})} > 0$ $\hat{g}_{1} \leq \check{v} \leq \hat{g}_{a}$, $\hat{g}_{1} \leq s \leq \hat{g}_{a}$;

• (ii) $\mathbb{G}_{H, n}(\check{v}, s) \leq \frac{1}{(n-1)!(\hat{d}_{2} - \hat{d}_{1})} \vert \mathrm{T}(\check{v}) \vert$;

• (iii) $\int _{\mathrm{T}}\mathbb{G}_{H, n}(\check{v}, s)\,ds = \frac{\mathrm{T}(\check{v})}{n!}$.

In 2023, Rasheed et al. [11] defined a novel class of 3-convex Green functions and utilized them to state the following fruitful lemma.

Lemma 2

Let f be defined on T such that $f'''$ exists and $\mathbb{G}_{\alpha}$ ($\alpha ={1,\dots ,4}$) are the two-point right focal problem-type Green functions given by (22)–(25). Then

$$\begin{aligned} f(\hat{u}) =&f(\hat{d}_{1})+(\hat{u}- \hat{d}_{1})f'(\hat{d}_{2})+( \hat{u}- \hat{d}_{1}) (\hat{u}-\hat{d}_{2}) f''( \hat{d}_{1})- \frac{(\hat{u}-\hat{d}_{1})^{2}}{2}f''( \hat{d}_{2}) \\ &{}+ \int _{\mathrm{T}}G_{1} (\hat{u},\hat{z})f'''( \hat{z})\,d\hat{z}, \end{aligned}$$

(18)

$$\begin{aligned} f(\hat{u}) =&f(\hat{d}_{2})-(\hat{d}_{2}- \hat{u})f'(\hat{d}_{1}) -f''( \hat{d}_{1})\frac{(\hat{d}_{2}-\hat{u})^{2}}{2}+(\hat{u}-\hat{d}_{1}) ( \hat{u}-\hat{d}_{2})f''( \hat{d}_{2}) \\ &{}- \int _{\mathrm{T}}G_{2} (\hat{u},\hat{z})f'''( \hat{z})\,d\hat{z}, \end{aligned}$$

(19)

$$\begin{aligned} f(\hat{u}) =&f(\hat{d}_{2})+(\hat{u}- \hat{d}_{1})f'(\hat{d}_{1})-( \hat{d}_{2}-\hat{d}_{1})f'( \hat{d}_{2}) \\ &{}-f''(\hat{d}_{1}) \biggl[ \frac{(\hat{u}-\hat{d}_{1})^{2}}{2}+(\hat{u}- \hat{d}_{1}) (\hat{d}_{1}- \hat{d}_{2}) \biggr] \\ &{}+ f''(\hat{d}_{2}) \biggl[ \frac{(\hat{d}_{2}-\hat{d}_{1})^{2}}{2}+( \hat{u}-\hat{d}_{1}) (\hat{u}- \hat{d}_{2}) \biggr] - \int _{\mathrm{T}}G_{3} (\hat{u},\hat{z})f'''( \hat{z})\,d\hat{z}, \end{aligned}$$

(20)

$$\begin{aligned} f(\hat{u}) =&f(\hat{d}_{1})+(\hat{d}_{2}- \hat{d}_{1})f'(\hat{d}_{1})-( \hat{d}_{2}-\hat{u})f'(\hat{d}_{2}) \\ &{}+f''(\hat{d}_{1}) \biggl[(\hat{u}- \hat{d}_{2}) (\hat{u}-\hat{d}_{1})+ \frac{(\hat{d}_{2}-\hat{d}_{1})^{2}}{2} \biggr] \\ &{}- \biggl[(\hat{u}-\hat{d}_{2}) (\hat{d}_{2}- \hat{d}_{1})+ \frac{(\hat{u}-\hat{d}_{2})^{2}}{2} \biggr]f''( \hat{d}_{2}) + \int _{ \mathrm{T}}f'''(w)G_{4} (\hat{u},\hat{z})\,d\hat{z}, \end{aligned}$$

(21)

where $\mathbb{G}_{\alpha}:\mathrm{T}\times \mathrm{T}\rightarrow \mathbb{R}$, $\alpha \in \{1,2,3,4\}$ given as:

$$\begin{aligned}& \mathbb{G}_{1}(\hat{u},\hat{z})=\textstyle\begin{cases} \frac{1}{2}(\hat{z}-\hat{d}_{1})^{2}+(\hat{u}-\hat{d}_{1})(\hat{u}- \hat{d}_{2}), & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ (\hat{u}-\hat{d}_{1})(\hat{z}-\hat{d}_{2})+ \frac{(\hat{u}-\hat{d}_{1})^{2}}{2}, &\hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(22)

$$\begin{aligned}& \mathbb{G}_{2}(\hat{u},\hat{z})=\textstyle\begin{cases} (\hat{u}-\hat{d}_{2})(\hat{z}-\hat{d}_{1})+\frac{1}{2}(\hat{u}- \hat{d}_{2})^{2}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \frac{(\hat{z}-\hat{d}_{2})^{2}}{2}+(\hat{u}-\hat{d}_{1})(\hat{u}- \hat{d}_{2}), & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(23)

$$\begin{aligned}& \mathbb{G}_{3}(\hat{u},\hat{z})=\textstyle\begin{cases} (\hat{u}-\hat{d}_{1})(\hat{z}-\hat{d}_{2})+ \frac{(\hat{u}-\hat{d}_{1})^{2}}{2}, & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ \frac{1}{2}(\hat{z}-\hat{d}_{1})^{2}+(\hat{u}-\hat{d}_{1})(\hat{u}- \hat{d}_{2}), & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(24)

$$\begin{aligned}& \mathbb{G}_{4}(\hat{u},\hat{z})=\textstyle\begin{cases} \frac{(\hat{z}-\hat{d}_{2})^{2}}{2}+(\hat{u}-\hat{d}_{2})(\hat{u}- \hat{d}_{1}), & \hat{d}_{1}\leq \hat{z}\leq \hat{u}, \\ (\hat{u}-\hat{d}_{2})(\hat{z}-\hat{d}_{1})+\frac{1}{2}(\hat{u}- \hat{d}_{2})^{2}, & \hat{u}\leq \hat{z} \leq \hat{d}_{2}. \end{cases}\displaystyle \end{aligned}$$

(25)

Remark 1

If integration by parts is applied to integral part of (7)–(10) by selecting $f''(\hat{z})$ as the first function and $\mathbb{\hat{G}}_{\alpha}(\hat{u},\hat{z})(\alpha =1,2,3,4)$ as the second function, then (18)–(21) are derived. Graphical depection of $\mathbb{\hat{G}}_{\alpha}(\hat{u},\hat{z})(\alpha =1,2,3,4)$ is given in figure (1).

Figure 1

Graph of $\mathbb{G}_{\alpha}(\alpha =1,\ldots ,4)$ for different ranges of û and ẑ

In [12], Adeel et al. established the Levinson inequality for the class of three convex functions using two Green’s functions. They also provided valuable findings in information theory. For the class of higher-order convex functions, Adeel et al. [13] generalized Levinson-type inequalities using the Abel-Gontscharoff interpolation. Using Lidstone polynomials and Green functions in combination with Levinson-type inequalities, Adeel et al. [14] computed the Shannon entropy and calculated f-divergence.

In recent decades, several scholars have used the Hermite interpolation to modify the inequalities for higher-order convex functions. Butt et al. [15] generalized the Popoviciu inequality for higher-order convex functions using Hermite interpolation and also constructed some results relating to the Grüss- and Ostrowski-type inequalities. In another analysis, generalizations of Levinson-type inequalities are stated by Adeel et al. [16] via Hermite interpolating polynomial for n-convex functions and estimated bounds for the Shannon entropy and f-divergence. For n-convex functions, Adeel et al. [17] proved Levinson-type inequalities by applying Hermite interpolating polynomial and Green functions and also derived inequalities for the Shannon entropy and f-divergence. In [18], Mehmood et al. explored discrete and continuous cyclic refinements of Jensen’s inequality and extended them from convex function to higher-order convex function through the use of various new Green functions by employing Hermite interpolating polynomial whose error term is approximated using Peano’s kernel. In 2021, Ansari et al. [19] utilized Hermite’s interpolation to derive a new generalization of an inequality for higher-order convex functions containing Csiszár divergence on time scales.

In [20], Adeel et al. employed Fink’s identity to obtain new generalizations of Levinson-type inequalities for n-convex functions. Furthermore, they applied their results to evaluate different entropies. In [21], authors applied the Lidstone interpolating polynomial for 2n-convex functions to derive different generalizations of Levinson-type inequalities. In [22], Bilal et al. generalized Shannon-type inequalities via diamond integrals. In [23], Bilal et al. defined Csiszár’s f-divergence for diamond integrals and proved inequalities for different divergences.

2 Mian results

This section is divided in two subsections. First, we present results associated with Bullen-type inequalities associated with Green functions (22)–(25). Second, 3-convex Green functions (22)–(25) are used to generalize the Levinson-type inequalities for higher-order convex functions via Hermite interpolation.

2.1 Generalization of Bullen-type inequalities for higher-order convex function

We begin by defining the subsequent functional:

F: Suppose a function $f: \mathrm{T}=[\hat{d}_{1}, \hat{d}_{2}]\rightarrow \mathbb{R}$. Let $(p_{1}, \ldots , p_{{\hat{m}}}) \in \mathbb{R}^{{\hat{m}}}$ and $(q_{1}, \ldots , q_{\hat{\omega}}) \in \mathbb{R}^{\hat{\omega}}$ be such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$, $\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}=1$ and $x_{\sigma}$, $y_{\epsilon}$, $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{ \sigma}$, $\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}y_{\epsilon} \in \mathrm{T}$. Then

$$ \mathbf{D}\bigl(f(\cdot )\bigr)=\sum _{\epsilon =1}^{\hat{\omega}}q_{\epsilon}f(y_{ \epsilon}) -f \Biggl(\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}y_{ \epsilon} \Biggr) -\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}f(x_{\sigma})+f \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr). $$

(26)

B: Let $H_{i_{e}}$, $\mathbb{G}_{\mathcal{H}, n}$ be defined in (16) and (17) and $\mathbb{G}_{\alpha}(\cdot , \hat{z})$ for $\alpha =1,\ldots ,4$ be defined in (22)–(25).

Impelled from identity (5), the following results are constructed:

Theorem 1

Assume that F and B hold. Consider the points $\hat{d}_{1} =\hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} = \hat{d}_{2}$ $(a \geq 2)$ and $f \in C^{n}[\hat{d}_{1}, \hat{d}_{2}]$. Then, for $\alpha =1,4$,

$$\begin{aligned} \mathbf{D}\bigl(f(\cdot )\bigr) =&\frac{1}{2} \Biggl[ \sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon}^{2}- \Biggl(\sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon} \Biggr)^{2}-\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma} x_{\sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{1})-f^{(2)}( \hat{d}_{2})\bigr)+\sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{ \mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}+ \int _{\mathrm{T}} \int _{\mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{ \alpha}(\cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z} \end{aligned}$$

(27)

and for $\alpha =2, 3$,

$$\begin{aligned} \mathbf{D}\bigl(f(\cdot )\bigr) =&\frac{1}{2} \Biggl[ \sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon}^{2}- \Biggl(\sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon} \Biggr)^{2}-\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma} x_{\sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{2})-f^{(2)}( \hat{d}_{1})\bigr) -\sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{ \mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}- \int _{\mathrm{T}} \int _{\mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{ \alpha}(\cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z} \end{aligned}$$

(28)

where

$$\begin{aligned} \mathbf{D}\bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr) =& \sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon} \mathbb{G}_{\alpha}(y_{\epsilon}, \hat{z})- \mathbb{G}_{\alpha} \Biggl(\sum_{\epsilon =1}^{\hat{\omega}}q_{ \epsilon}y_{\epsilon}, \hat{z} \Biggr)-\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma} \mathbb{G}_{\alpha}(x_{\sigma}, \hat{z}) \\ &{}+\mathbb{G}_{\alpha} \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{ \sigma}, \hat{z} \Biggr). \end{aligned}$$

(29)

Proof

Let $\alpha =1,4$, then applying (26) to the identities (18) and (21) and using linearity of $\mathbf{D}(f(\cdot ))$, we have

$$\begin{aligned} \mathbf{D}\bigl(f(\cdot )\bigr) =&\frac{1}{2} \Biggl[ \sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon}^{2}- \Biggl(\sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}y_{\epsilon} \Biggr)^{2}-\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma} x_{\sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{ \sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{1})-f^{(2)}( \hat{d}_{2})\bigr) + \int _{ \mathrm{T}}\mathbf{D}\bigl(\mathbb{G}_{\alpha}(., \hat{z})\bigr)f^{(3)}(\hat{z})\,d \hat{z}. \end{aligned}$$

(30)

From Theorem H, $f^{(3)}(\hat{z})$ becomes

$$\begin{aligned} f^{(3)}(\hat{z}) = \sum _{e=1}^{a}\sum_{i=0}^{k_{e}}H_{i_{e}}( \hat{z})f^{(i+3)}(\hat{g}_{e})+ \int _{\mathrm{T}}\mathbb{G}_{ \mathcal{H}, n-3}(\hat{z}, s)f^{(n)}(s)\,ds. \end{aligned}$$

(31)

Using (31) in (30), we get (27).

Following the same steps, we get (28) for $\alpha =2,3$. □

Under the condition defined by (6), the generalized form of the Bullen-type inequality (for positive weights) is presented.

Corollary 1

Assume B. Let $(p_{1}, \ldots , p_{\hat{m}}) \in \mathbb{R}^{\hat{m}}$ be such that $\sum_{\sigma =1}^{\hat{m}}p_{\sigma}=1$ and $x_{\sigma}$, $y_{\hat{\omega}}$ satisfy (6) and $\max \{x_{1} \ldots x_{\hat{m}}\} \leq \min \{y_{1} \ldots y_{ \hat{m}}\}$ and $\hat{d}_{1} = \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} = \hat{d}_{2}$ $(a\geq 2)$ be the points. If $f \in C^{n}[\hat{d}_{1},\hat{d}_{2}]$, then (27) and (28) hold.

For n-convex functions, the following form of identities (27) and (28) are given.

Theorem 2

Assume all the suppositions of Theorem 1for the n-convex function f.

If

$$ \int _{\mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z}\geq 0,\quad s \in \mathrm{T}, $$

(32)

then for $\alpha =1,4$,

$$\begin{aligned} \mathbf{D} \bigl(f(\cdot ) \bigr) \geq& \frac{1}{2} \Biggl[\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}y_{\epsilon}^{2}- \Biggl(\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon} y_{\epsilon}\Biggr)^{2}-\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma} x_{\sigma}^{2}+\Biggl(\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{1})-f^{(2)}( \hat{d}_{2})\bigr) +\sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{ \mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d\hat{z} \end{aligned}$$

(33)

and for $\alpha =2,3$,

$$\begin{aligned} \mathbf{D} \bigl(f(\cdot ) \bigr) \leq& \frac{1}{2} \Biggl[\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}y_{\epsilon}^{2} -\Biggl(\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon} y_{\epsilon}\Biggr)^{2}-\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma} x_{\sigma}^{2}+\Biggl(\sum_{\sigma =1}^{{ \hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{2})-f^{(2)}( \hat{d}_{1})\bigr) -\sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{ \mathrm{T}}\mathbf{D} \bigl(\mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z}. \end{aligned}$$

(34)

Proof

As f is n-convex ($n\geq 3$), then by Theorem I, we have

$$ f^{(n)}(\hat{z}) \geq 0,\quad \forall ~\hat{z} \in \mathrm{T}. $$

Thus, using (32) in (27) and (28), we get (33) and (34), respectively. □

Remark 2

• (i) According to Theorem 2, if the inequality in (32) is reversed, then inequalities in (33) and (34) hold conversely.

• (ii) If f is n-concave, then inequalities (33) and (34) hold in the opposite direction.

Remark 3

$\mathbf{D}(\cdot )$ is reduced in $\mathbb{D}(\cdot )$ if $\hat{\omega}={\hat{m}}$, $p_{\hat{m}}=q_{\hat{\omega}}$ and weights are positive. Then for $\alpha =1,4$, (27), (32) and (33) become

$$\begin{aligned} \mathbb{D} \bigl(f(\cdot ) \bigr) =& \sum _{e=1}^{a}\sum_{i=0}^{k_{e}}f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{D} \bigl( \mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}+ \int _{\mathrm{T}} \int _{\mathrm{T}} \mathbb{D} \bigl(\mathbb{G}_{ \alpha}(\cdot ,\hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z}, \end{aligned}$$

(35)

$$ \int _{\mathrm{T}}\mathbb{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z} \geq 0,\quad s \in \mathrm{T} $$

(36)

and

$$ \mathbb{D} \bigl(f(\cdot ) \bigr) \geq \sum _{e=1}^{a}\sum_{i=0}^{k_{e}} f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z}, $$

(37)

respectively.

For $\alpha =2,3$, (28), (32) and (34) become

$$\begin{aligned} \mathbb{D} \bigl(f(\cdot ) \bigr) =& -\sum _{e=1}^{a}\sum_{i=0}^{k_{e}}f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{D} \bigl( \mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d \hat{z} \\ &{}- \int _{\mathrm{T}} \int _{\mathrm{T}} \mathbb{D} \bigl(\mathbb{G}_{ \alpha}(\cdot ,\hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z}, \end{aligned}$$

(38)

$$ \int _{\mathrm{T}}\mathbb{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z}, \geq 0\quad s \in \mathrm{T} $$

(39)

and

$$ \mathbb{D} \bigl(f(\cdot ) \bigr) \leq -\sum _{e=1}^{a}\sum_{i=0}^{k_{e}} f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z}, $$

(40)

respectively.

Theorem 3

Assume B. Let $\hat{d}_{1} = \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} = \hat{d}_{2}$ $(a\geq 2)$ be the points and $f \in C^{n}[\hat{d}_{1},\hat{d}_{2}]$. Choose positive weights $(p_{1}, \ldots , p_{\hat{m}})$ such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$. Then

• (i) For every $e= 2, \ldots , a$, if $k_{e}$ is odd, then (37) and (40) hold.

• (ii) Let (37) and (40) be fulfilled and the function

  $$\begin{aligned} F(\hat{z})= \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)}( \hat{g}_{e})H_{i_{e}}( \hat{z}). \end{aligned}$$

  (41)

  If $F(\hat{z})\geq 0$, then (37) and (40) become

  $$\begin{aligned} \mathbb{D}\bigl(f(\cdot )\bigr) \geq 0 \end{aligned}$$

  (42)

  and

  $$\begin{aligned} \mathbb{D}\bigl(f(\cdot )\bigr) \leq 0, \end{aligned}$$

  (43)

  respectively.

Proof

• (i) Since the weights are positive and the Green functions $\mathbb{G}_{\alpha}(\cdot , \hat{z})$ are 3-convex, $\mathbb{D} (\mathbb{G}_{\alpha}(\cdot , \hat{z}) )\geq 0$, for fixed α. Additionally, for each $e=2,\ldots , a$, $k_{e}$ is odd this implies $\mathrm{T}(\cdot ) \geq 0$ and by part (i) of Lemma 1

  $$ \mathbb{G}_{\mathcal{H}, n-3}(\cdot , s)\geq 0, $$

  hence (32) holds. Thus, applying Theorem 2 for the n-convex function f, we obtain (37) and (40).

• (ii) Using (41) in (37) and (40) gives (42) and (43), respectively. □

Theorem 4

Let positive real numbers $p_{1},\ldots ,p_{{\hat{m}}}$ be such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$. Let also $x_{\sigma}, y_{\sigma}\in \mathrm{T}$ be such that $x_{\sigma}+y_{\sigma}=2\breve{c}$, for $\sigma =1,\ldots , {\hat{m}}$, $x_{\sigma}+x_{{\hat{m}}-\sigma +1} \leq 2\breve{c}$ and $\frac{p_{\sigma}x_{\sigma}+p_{{\hat{m}}-\sigma +1}x_{{\hat{m}}-\sigma +1}}{p_{\sigma}+p_{{\hat{m}}-\sigma +1}} \leq \breve{c}$. Then for an n-convex function f, we have the following

• (i) For each $e= 2, \ldots , a$, if $k_{e}$ is odd, then (37) and (40) hold.

• (ii) Let (37) and (40) hold and

  $$\begin{aligned} F(\hat{z}) = \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)}( \hat{g}_{e})H_{i_{e}}( \hat{z}) \end{aligned}$$

  (44)

  be 3-convex. Then (37) and (40) become

  $$\begin{aligned} \mathbb{D}\bigl(f(\cdot )\bigr) \geq 0 \end{aligned}$$

  (45)

  and

  $$\begin{aligned} \mathbb{D}\bigl(f(\cdot )\bigr) \leq 0, \end{aligned}$$

  (46)

  respectively.

Proof

Proof is similar to Theorem 3. □

2.2 Levinson-type inequality for n-convex ($n \geq 3$) functions

In this section, results are given for the generalization of Levinson-type inequality using new green functions $\mathbb{G}_{\alpha}(\alpha =1,\ldots ,4)$ and interpolating Hermite polynomial. For this, first, we have

$\boldsymbol{\mathcal{H}}$: Consider $x_{1}, \ldots , x_{\hat{m}} \in (0, \gamma )$ and $f: \mathbb{I}_{2}= [0, 2\gamma ] \rightarrow \mathbb{R}$. Choose $(p_{1}, \ldots , p_{\hat{m}})\in \mathbb{R}^{\hat{m}}$ and $(q_{1}, \ldots , q_{\hat{\omega}})\in \mathbb{R}^{\hat{\omega}}$ be such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$ and $\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}=1$. Also, let $x_{\sigma}$, $\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}(2\gamma -x_{\epsilon})$ and $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma} \in \mathbb{I}_{2}$. Then

$$\begin{aligned} \breve{\mathbf{D}}\bigl(f(\cdot )\bigr) =&\sum _{\epsilon =1}^{\hat{\omega}}q_{ \epsilon}f(2\gamma -x_{\epsilon}) -f \Biggl(\sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon} (2\gamma -x_{\epsilon}) \Biggr) -\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma}f(x_{\sigma}) \\ &{}+f \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr). \end{aligned}$$

(47)

For the next results, we construct the following identities:

Theorem 5

Assume $\boldsymbol{\mathcal{H}}$ and B. Let $\hat{d}_{1} = \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} = \hat{d}_{2}$ $(a\geq 2)$ be the points and $f \in C^{n}[\hat{d}_{1}, \hat{d}_{2}]$. Then for $0 \leq \hat{d}_{1}<\hat{d}_{2}\leq 2\gamma$ and $\alpha =1,4$, we have

$$\begin{aligned} \breve{\mathbf{D}} \bigl(f(\cdot ) \bigr) =& \Biggl[\sum _{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}(2\gamma -x_{\epsilon})^{2}- \Biggl(\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}(2 \gamma -x_{\epsilon}) \Biggr)^{2} -\sum _{\sigma =1}^{{\hat{m}}}p_{\sigma} x_{\sigma}^{2}+ \Biggl(\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \frac{1}{2}\bigl(2f^{(2)}(\hat{d}_{1})-f^{(2)}( \hat{d}_{2})\bigr)+ \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}+ \int _{\mathrm{T}} \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl( \mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z} \end{aligned}$$

(48)

and for $\alpha =2,3$,

$$\begin{aligned} \breve{\mathbf{D}} \bigl(f(\cdot ) \bigr) =& \frac{1}{2} \Biggl[\sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}(2 \gamma -x_{\epsilon})^{2}- \Biggl(\sum _{\epsilon =1}^{\hat{\omega}}q_{\epsilon}(2\gamma -x_{ \epsilon}) \Biggr)^{2}-\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma} x_{ \sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times \bigl(2f^{(2)}(\hat{d}_{2})-f^{(2)}( \hat{d}_{1})\bigr)-\sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{ \mathrm{T}}\breve{\mathbf{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}- \int _{\mathrm{T}} \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl( \mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z}, \end{aligned}$$

(49)

where $\breve{\mathbf{D}}(f(\cdot ))$ is defined in (47) and

$$\begin{aligned} \breve{\mathbf{D}}\bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z})\bigr) =& \sum_{ \epsilon =1}^{\hat{\omega}}q_{\epsilon} \mathbb{G}_{k}(2\gamma -x_{ \epsilon},\hat{z}) - \mathbb{G}_{k} \Biggl(\sum_{\epsilon =1}^{ \hat{\omega}}q_{\epsilon}(2 \gamma -x_{\epsilon}, \hat{z}) \Biggr) - \sum_{\sigma =1}^{{\hat{m}}}p_{\sigma} \mathbb{G}_{k}(x_{\sigma}, \hat{z}) \\ &\ +\mathbb{G}_{k} \Biggl(\sum_{ \sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma}, \hat{z} \Biggr). \end{aligned}$$

(50)

Proof

Replace T, $\mathbf{D}(\cdot )$ and $y_{\epsilon}$ with $\mathbb{I}_{2}$, $\breve{\mathbf{D}}(\cdot )$ and $(2\gamma -x_{\epsilon})$ in Theorem 1, respectively, to get the required result. □

For n-convex functions, we give the following form of identity (48).

Theorem 6

Consider f is an n-convex function and all the conditions of Theorem 2hold.

If

$$ \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z} \geq 0, \quad s \in \mathbb{I}_{2}, $$

(51)

then for $\alpha =1,4$,

$$\begin{aligned} \breve{\mathbf{D}} \bigl(f(\cdot ) \bigr) \geq & \Biggl[\sum _{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}(2\gamma -x_{\epsilon})^{2}- \Biggl(\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}(2 \gamma -x_{ \epsilon}) \Biggr)^{2} -\sum _{\sigma =1}^{{\hat{m}}}p_{\sigma} x_{ \sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times\frac{1}{2}\bigl(2f^{(2)}(\hat{d}_{1})-f^{(2)}( \hat{d}_{2})\bigr) + \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl(\mathbb{G}_{\alpha}( \cdot ,t) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z} \end{aligned}$$

(52)

and for $\alpha =2,3$,

$$\begin{aligned} \breve{\mathbf{D}} \bigl(f(\cdot ) \bigr) \leq & \Biggl[\sum _{ \epsilon =1}^{\hat{\omega}}q_{\epsilon}(2\gamma -x_{\epsilon})^{2}- \Biggl(\sum_{\epsilon =1}^{\hat{\omega}}q_{\epsilon}(2 \gamma -x_{ \epsilon}) \Biggr)^{2}-\sum _{\sigma =1}^{{\hat{m}}}p_{\sigma} x_{ \sigma}^{2}+ \Biggl(\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}x_{\sigma} \Biggr)^{2} \Biggr] \\ &{}\times\frac{1}{2}\bigl(2f^{(2)}(\hat{d}_{2})-f^{(2)}( \hat{d}_{1})\bigr)- \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{\mathrm{T}}\breve{\mathbf{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z} \end{aligned}$$

(53)

where $0 \leq \hat{d}_{1}<\hat{d}_{2}\leq 2\gamma$.

Proof

As a consequences of conditions mentioned in the statement, the proof is similar to Theorem 2. □

Remark 4

$\breve{\mathbf{D}}(\cdot )$ is reduced in $\breve{\mathbb{D}}(\cdot )$ if $\hat{\omega}={\hat{m}}$, $p_{\hat{m}}=q_{\hat{\omega}}$ and weights are positive. Then for $\alpha =1,4$, (48), (51) and (52) become

$$\begin{aligned} \breve{\mathbb{D}} \bigl(f(\cdot ) \bigr) =& \sum _{e=1}^{a}\sum_{i=0}^{k_{e}}f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl( \mathbb{G}_{\alpha}( \cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d \hat{z} \\ &{}+ \int _{\mathrm{T}} \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl( \mathbb{G}_{\alpha}( \cdot ,\hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z}, \end{aligned}$$

(54)

$$ \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z} \geq 0, \quad s \in \mathbb{I}_{2} $$

(55)

and

$$ \breve{\mathbb{D}} \bigl(f(\cdot ) \bigr) \geq \sum _{e=1}^{a}\sum_{i=0}^{k_{e}} f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}} \breve{\mathbb{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr)H_{i_{e}}( \hat{z})\,d\hat{z}. $$

(56)

For $\alpha =2,3$, (49), (51), and (53) become

$$\begin{aligned} \breve{\mathbb{D}} \bigl(f(\cdot ) \bigr) =& -\sum _{e=1}^{a}\sum_{i=0}^{k_{e}}f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl( \mathbb{G}_{\alpha}( \cdot ,t) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z} \\ &{}- \int _{\mathrm{T}} \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl( \mathbb{G}_{\alpha}( \cdot ,\hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}( \hat{z}, s)f^{(n)}(s)\,ds\,d \hat{z}, \end{aligned}$$

(57)

$$ \int _{\mathrm{T}}\breve{\mathbb{D}} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr) \mathbb{G}_{\mathcal{H}, n-3}(\hat{z}, s)\,d\hat{z} \geq 0, \quad s \in \mathbb{I}_{2} $$

(58)

and

$$ \breve{\mathbb{D}} \bigl(f(\cdot ) \bigr) \leq -\sum _{e=1}^{a}\sum_{i=0}^{k_{e}} f^{(i+3)} (\hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{D} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z}, $$

(59)

respectively.

Theorem 7

Assume B. Let an n-convex function $f: \mathbb{I}_{2} \rightarrow \mathbb{R}$ and $(p_{1}, \ldots , p_{\hat{m}})\in \mathbb{R}^{+}$ be such that $\sum_{\sigma =1}^{{\hat{m}}}p_{\sigma}=1$. Let also $f \in C^{n}([0, 2\gamma ])$ and $\hat{d}_{1} = \hat{g}_{1} < \hat{g}_{2} < \cdots < \hat{g}_{a} = \hat{d}_{2}$ $(a \geq 2)$ be the points. Then

• (i) If $k_{e}$ is odd for each $e = 2, \ldots , a$, then (56) and (59) hold.

• (ii) Consider the function

  $$\begin{aligned} F(\hat{z}) = \sum_{e=1}^{a} \sum_{i=0}^{k_{e}}f^{(i+3)}( \hat{g}_{e})H_{i_{e}}( \hat{z}) \end{aligned}$$

  (60)

  is nonnegative, and (56) and (59) also hold. Then (56) and (59) become

  $$\begin{aligned} \breve{\mathbb{D}}\bigl(f(\cdot )\bigr) \geq 0 \end{aligned}$$

  (61)

  and

  $$\begin{aligned} \breve{\mathbb{D}}\bigl(f(\cdot )\bigr) \leq 0, \end{aligned}$$

  (62)

  respectively, where $0 \leq \hat{d}_{1}<\hat{d}_{2}\leq 2\gamma$.

Proof

Proof is same as of Theorem 3. □

3 Applications to information theory

Information theory is a branch of science that deals with data quantification, storage, and transfer. In 1948, Claude Shannon [24] gave the idea of information theory and described entropy as the fundamental unit of information. In other words, it is also possible to determine the information using the probability density function. Divergence measure is an idea in probability theory that helps solve certain problems because divergence measure is used to calculate the distance between the two probability distributions. Moreover, divergence measures are used to solve many problems in probability theory. Information and divergence measures are extremely valuable and essential in many fields, including Sensor networks [25], finance [26], economics [27], and approximation of probability distributions [28].

Levinson- type inequalities are essential for generalizing inequalities for divergence between probability distributions. The key conclusions from Sect. 2 are linked to information theory in this part, using the Shannon entropy and f-divergence.

3.1 Csiszár divergence

Csiszár [29, 30] presented the subsequent definition.

Definition 1

If $f: \mathbb{R}_{+}\to \mathbb{R}_{+}$ is a convex function, choose $\tilde{\mathbf{v}}, \tilde{\mathbf{l}} \in \mathbb{R}_{+}^{{\hat{m}}}$ such that $\sum_{\sigma =1}^{{\hat{m}}}v_{\sigma}=1$ and $\sum_{\sigma =1}^{{\hat{m}}}l_{\sigma}=1$. Then the Csiszár f-divergence is defines as follows:

$$\begin{aligned} \mathbb{I}_{f}(\tilde{\mathbf{v}}, \tilde{ \mathbf{l}}) := \sum_{ \sigma =1}^{{\hat{m}}}l_{\sigma}f \biggl(\frac{v_{\sigma}}{l_{\sigma}} \biggr). \end{aligned}$$

(63)

In [31], Horv́ath et al. generalized (63) as follows:

Definition 2

If $f: \mathbb{I} \rightarrow \mathbb{R}$ is such that $\mathbb{I} \subset \mathbb{R}$, choose $\tilde{\mathbf{v}}=(v_{1}, \ldots , v_{\hat{m}})\in \mathbb{R}^{ \hat{m}}$ and $\tilde{\mathbf{l}}=(l_{1}, \ldots , l_{\hat{m}})\in (0, \infty )^{{ \hat{m}}}$ such that

$$\begin{aligned} \frac{v_{\sigma}}{l_{\sigma}} \in \mathbb{I}, \quad \sigma = 1, \ldots , {\hat{m}}. \end{aligned}$$

Then

$$\begin{aligned} \hat{\mathbb{I}}_{f}(\tilde{\mathbf{v}}, \tilde{ \mathbf{l}}) : = \sum_{\sigma =1}^{{\hat{m}}} l_{\sigma}f \biggl( \frac{v_{\sigma}}{l_{\sigma}} \biggr). \end{aligned}$$

(64)

Theorem 8

Assume that $f\in C^{n}[\hat{d}_{1},\hat{d}_{2}]$ is an n-convex function and $\breve{\mathbf{r}}= (\breve{r}_{1}, \ldots , \breve{r}_{\hat{m}} )$, $\breve{\mathbf{k}}=(\breve{k_{1}}, \ldots ,\breve{k}_{ \hat{m}})\in (0, \infty )^{{\hat{m}}}$ and $\breve{\mathbf{w}}= (\breve{w_{1}}, \ldots , \breve{w_{\hat{\omega}}} )$, $\breve{\mathbf{t}}= ( \breve{t_{1}}, \ldots , \breve{t_{\hat{\omega}}} ) \in (0, \infty )^{\hat{\omega}}$ are such that

$$\begin{aligned} \frac{\breve{r_{\sigma}}}{\breve{k_{\sigma}}} \in \mathbb{I}, \quad \sigma = 1, \ldots , {\hat{m}}, \end{aligned}$$

and

$$\begin{aligned} \frac{\breve{w_{\epsilon}}}{\breve{t_{\epsilon}}} \in \mathbb{I}, \quad \epsilon = 1, \ldots , \hat{\omega}. \end{aligned}$$

If $k_{e}$ is odd for each $e = 2, \ldots , a$, then for $\alpha =1,4$,

$$\begin{aligned} \mathbb{J}_{\mathrm{cis}}\bigl(f(\cdot )\bigr) \geq & \frac{1}{2} \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t}_{\epsilon}} \sum _{\epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}}- \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} -\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{\sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} \\ & {}+\Biggl(\sum_{\sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \bigl(2f^{(2)}( \hat{d}_{1})-f^{(2)}(\hat{d}_{2})\bigr) \\ &{}+\sum_{e=1}^{a}\sum _{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}( \cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d\hat{z} \end{aligned}$$

(65)

and for $\alpha =2,3$,

$$\begin{aligned} \mathbb{J}_{\mathrm{cis}}\bigl(f(\cdot )\bigr) \leq & \frac{1}{2} \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t}_{\epsilon}} \sum _{\epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}}- \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} -\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{\sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} \\ & {}+\Biggl(\sum_{\sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \bigl(2f^{(2)}( \hat{d}_{1})-f^{(2)}(\hat{d}_{2})\bigr) \\ &{}-\sum_{e=1}^{a}\sum _{i=0}^{k_{e}}f^{(i+3)} ( \hat{g}_{e} ) \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}( \cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d\hat{z}, \end{aligned}$$

(66)

where

$$\begin{aligned} \mathbb{J}_{\mathrm{cis}}\bigl(\mathbf{f}(\cdot )\bigr) =& \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t}_{\epsilon}} \hat{\mathbb{I}}_{\mathbf{f}}(\breve{\mathbf{w}}, \breve{\mathbf{t}})- \mathbf{f}\Biggl(\sum_{\epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}\Biggr)- \frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \hat{\mathbb{I}}_{\mathbf{f}}( \breve{\mathbf{r}}, \breve{\mathbf{k}}) \\ &{}+\mathbf{f}\Biggl(\sum_{\sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}\Biggr) \end{aligned}$$

(67)

and

$$\begin{aligned} \mathbb{J}\bigl(\mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) =&\sum_{\epsilon =1}^{ \hat{\omega}} \frac{\breve{t_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \mathbb{G}_{k} \biggl(\frac{\breve{w_{\epsilon}}}{\breve{t_{\epsilon}}}, \hat{z} \biggr) -\mathbb{G}_{k} \Biggl(\sum _{\epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{ \sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}},\hat{z} \Biggr) \\ &{}-\sum_{\sigma =1}^{{\hat{m}}} \frac{\breve{k_{\sigma}}}{ \sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \mathbb{G}_{k} \biggl( \frac{\breve{r_{\sigma}}}{\breve{k_{\sigma}}}, \hat{z} \biggr) + \mathbb{G}_{k} \Biggl(\sum_{\sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}, \hat{z} \Biggr). \end{aligned}$$

(68)

Proof

Since the weights are positive and the Green functions $\mathbb{G}_{\alpha}(\cdot , \hat{z})$ given in (22)–(25) are 3-convex, therefore,

$$ \mathbb{J}\bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z})\bigr) \geq 0 $$

for fixed $\alpha =1, 2,3,4$.

Since $k_{e}$ is odd for each $e=2, \ldots , a$, so we have $\mathrm{T}(\cdot ) \geq 0$ and by part (i) of Lemma 1

$$ \mathbb{G}_{\mathcal{H}, n-3}(\cdot , s) \geq 0 $$

hence $\mathbb{G}_{\mathcal{H}, n}$ is 3-convex, therefore (32) holds. Thus, using $p_{\sigma} = \frac{\breve{k_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}$, $x_{\sigma} = \frac{\breve{r_{\sigma}}}{\breve{k_{\sigma}}}$, $q_{\epsilon} = \frac{\breve{t_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}$, $y_{\epsilon} = \frac{\breve{w_{\epsilon}}}{\breve{t_{\epsilon}}}$ in Theorem 2, (33) and (34) become (65) and (66), respectively. □

3.2 Shannon entropy

Definition 3

(see [31]) For positive probability distribution $\tilde{\mathbf{l}}=(l_{1}, \ldots , l_{\hat{m}})$, the Shannon entropy is given by

$$\begin{aligned} \boldsymbol{\mathbb{S}} : = - \sum_{\sigma =1}^{{\hat{m}}}l_{\sigma} \log (l_{\sigma}). \end{aligned}$$

(69)

In order to avoid many notions, we define the following functional:

$\boldsymbol{\mathbb{Q}}$: Let $\breve{\mathbf{r}}= (\breve{r}_{1}, \ldots , \breve{r}_{\hat{m}} )$, $\breve{\mathbf{k}}= (\breve{k_{1}}, \ldots ,\breve{k}_{ \hat{m}} )\in (0, \infty )^{{\hat{m}}}$ and $\breve{\mathbf{w}}= (\breve{w_{1}}, \ldots , \breve{w_{\hat{\omega}}} )$, $\breve{\mathbf{t}}= ( \breve{t_{1}}, \ldots , \breve{t_{\hat{\omega}}} ) \in (0, \infty )^{\hat{\omega}}$.

We denote b as a base of log function.

Corollary 2

Assume $\boldsymbol{\mathbb{Q}}$.

• (i) If $n=3, 5, \ldots$ and $b>1$, then for $\alpha =1,4$,

  $$\begin{aligned} \mathbb{J}_{s}(\cdot ) \geq & \biggl( \frac{\hat{d}_{1}^{2}-2\hat{d}_{2}^{2}}{(\hat{d}_{1}\hat{d}_{2})^{2}} \biggr) \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}\sum _{ \epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}}- \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} \\ &{}-\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{ \sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} + \Biggl(\sum_{ \sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{ \sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \\ &{}+ \sum_{e=1}^{a}\sum _{i=0}^{k_{e}} \frac{(-1)^{i+2}(i+2)!}{(c_{j})^{i+3}} \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z} \end{aligned}$$

  (70)

  and for $\alpha =2,3$,

  $$\begin{aligned} \mathbb{J}_{s}(\cdot ) \leq & \biggl( \frac{\hat{d}_{2}^{2}-2\hat{d}_{1}^{2}}{(\hat{d}_{1}\hat{d}_{2})^{2}} \biggr) \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}\sum _{ \epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}}- \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{ \sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} \\ &{}-\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{ \sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} + \Biggl(\sum_{ \sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{ \sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \\ &{}- \sum_{e=1}^{a}\sum _{i=0}^{k_{e}} \frac{(-1)^{i+2}(i+2)!}{(c_{j})^{i+3}} \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z}, \end{aligned}$$

  (71)

  where $\mathbb{J}(\mathbb{G}_{\alpha}(\cdot , \hat{z}))$ is defined in (68), and $\mathbb{J}_{s}(\cdot )$ is given by

  $$\begin{aligned} \mathbb{J}_{s}(\cdot ) =&\sum _{\epsilon =1}^{\hat{\omega}} \breve{t_{\epsilon}}\log ( \breve{w_{\epsilon}})+ \tilde{\boldsymbol{\mathbb{S}}} -\log \Biggl(\sum _{\epsilon =1}^{ \hat{\omega}}\breve{w_{\epsilon}} \Biggr)-\sum_{\sigma =1}^{{\hat{m}}} \breve{k_{\sigma}}\log (\breve{r_{\sigma}}) +\boldsymbol{\mathbb{S}} \\ &{}+\log \Biggl(\sum_{\sigma =1}^{{\hat{m}}} \breve{r_{\sigma}} \Biggr). \end{aligned}$$

  (72)

• (ii) If $k_{e}$ is odd and $1>b$ or $n=4, 6, \ldots$ , then inequalities in (70) and (71) are conversed.

Proof

• (i) Since $f(x) = \log (x)$ is an n-convex function for $n=3, 5, \ldots$ , and $b>1$, putting $f(x)=\log (x)$ in Theorem 8 gives (70) and (71), where $\boldsymbol{\mathbb{S}}$ is given in (69) and

  $$ \tilde{\boldsymbol{\mathbb{S}}}= - \sum _{\epsilon =1}^{\hat{\omega}} \breve{t_{\epsilon}}\log ( \breve{t_{\epsilon}}). $$

• (ii) As $k_{e}$ is odd and $f(x) = \log (x)$ is n-concave for $n=4, 6, \ldots$ , then by Remark 2(ii), inequality (33) is reversed. Hence, using $f(x)=\log (x)$ and $p_{\sigma} = \frac{\breve{k_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}$, $x_{\sigma} = \frac{\breve{r_{\sigma}}}{\breve{k_{\sigma}}}$, $q_{\epsilon} = \frac{\breve{t_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}$, $y_{\epsilon} = \frac{\breve{w_{\epsilon}}}{\breve{t_{\epsilon}}}$ in reversed inequality (33) and (34), we obtain (70) and (71) in the reverse direction. □

Corollary 3

Assume $\boldsymbol{\mathbb{Q}}$ with odd values of $k_{e}\mathrm{}$.

• (i) If $b>1$ and n = even ($n \geq 4$), then for $\alpha =1,4$,

  $$\begin{aligned} \mathbf{J}_{s}(\cdot ) \geq & \biggl( \frac{\hat{d}_{1}^{2}-2\hat{d}_{2}^{2}}{(\hat{d}_{1}\hat{d}_{2})^{2}} \biggr) \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \sum _{\epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}} - \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{ \sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} \\ &{}-\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{ \sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} + \Biggl(\sum_{ \sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{ \sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \\ &{}+\sum_{e=1}^{a}\sum _{i=0}^{k_{e}} \frac{(-1)^{i+2}(i+1)!}{(c_{j})^{i+2}} \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}(\cdot ,\hat{z}) \bigr) H_{i_{e}}(\hat{z})\,d \hat{z}, \end{aligned}$$

  (73)

  and for $\alpha =2,3$,

  $$\begin{aligned} \mathbf{J}_{s}(\cdot ) \leq & \biggl( \frac{\hat{d}_{2}^{2}-2\hat{d}_{1}^{2}}{(\hat{d}_{1}\hat{d}_{2})^{2}} \biggr) \Biggl[ \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \sum _{\epsilon =1}^{\hat{\omega}} \frac{(\breve{w_{\epsilon}})^{2}}{\breve{t_{\epsilon}}} - \Biggl(\sum _{ \epsilon =1}^{\hat{\omega}} \frac{\breve{w_{\epsilon}}}{ \sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggr)^{2} \\ &{}-\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \sum_{ \sigma =1}^{{\hat{m}}} \frac{(\breve{r_{\sigma}})^{2}}{\breve{k_{\sigma}}} + \Biggl(\sum_{ \sigma =1}^{{\hat{m}}} \frac{\breve{r_{\sigma}}}{ \sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggr)^{2} \Biggr] \\ &{}-\sum_{e=1}^{a}\sum _{i=0}^{k_{e}} \frac{(-1)^{i+2}(i+1)!}{(c_{j})^{i+2}} \int _{\mathrm{T}}\mathbb{J} \bigl(\mathbb{G}_{\alpha}(\cdot , \hat{z}) \bigr)H_{i_{e}}(\hat{z})\,d \hat{z}, \end{aligned}$$

  (74)

  where

  $$\begin{aligned} \mathbf{J}_{s}(\cdot ) =& \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}} \Biggl(\tilde{S}+\sum_{\epsilon =1}^{\hat{\omega}} \breve{w_{\epsilon}}\log (\breve{t_{\epsilon}}) \Biggr) - \frac{1}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}\log \Biggl(\sum_{\epsilon =1}^{\hat{\omega}} \breve{t_{\epsilon}} \Biggr) \\ &{}-\frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}} \Biggl(S+ \sum_{\sigma =1}^{{\hat{m}}} \breve{r_{\sigma}}\log ( \breve{k_{\sigma}}) \Biggr) + \frac{1}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}\log \Biggl( \sum_{\sigma =1}^{{\hat{m}}} \breve{k_{\sigma}} \Biggr). \end{aligned}$$

• (ii) If n = odd ($n \geq 3$) or $b<1$, then inequality in (73) is inverted.

Proof

• (i) Since $f(x) = -x\log (x)$ is n-convex for $n=4, 6, \ldots$ , and $b>1$, substituting $f(x)=-x\log (x)$ in Theorem 8 gives (73) and (74), where

  $$ \tilde{S}= - \sum_{\epsilon =1}^{\hat{\omega}} \breve{w_{\epsilon}} \log (\breve{w_{\epsilon}}) $$

  and

  $$ S=- \sum_{\sigma =1}^{{\hat{m}}} \breve{r_{\sigma}}\log ( \breve{r_{\sigma}}). $$

• (ii) Since $f(x) = -x\log (x)$ is n-concave $(n=3, 5, \ldots )$, then (33) holds in the reverse direction by Remark 2(ii). Thus, using $p_{\sigma} = \frac{\breve{k_{\sigma}}}{\sum_{\sigma =1}^{{\hat{m}}}\breve{k_{\sigma}}}$, $x_{\sigma} = \frac{\breve{r_{\sigma}}}{\breve{k_{\sigma}}}$, $q_{\epsilon} = \frac{\breve{t_{\epsilon}}}{\sum_{\epsilon =1}^{\hat{\omega}}\breve{t_{\epsilon}}}$ and $y_{\epsilon} = \frac{\breve{w_{\epsilon}}}{\breve{t_{\epsilon}}}$ in reversed inequality (33) and (34), we have (73) and (74) in the reverse direction.

 □

4 Conclusion

The purpose of this study is to generalize Levinson-type inequalities (with real weights) for two distinct types of data points that use convex functions of higher order. Newly defined 3-convex Green functions and Hermite interpolating polynomial are utilized for the class of n-convex ($n\geq 3$) functions. We are able to find applications to information theory, as well as the bounds for obtained entropies and divergences. Moreover, other interpolations, e.g., Lidstone interpolation, Taylor’s polynomial, and Montgomery identity, are also useful for exploring the related results.