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Generalization of the Levinson inequality with applications to information theory

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Abstract

In the presented paper, Levinson’s inequality for 3-convex function is generalized by using two Green’s functions. Čebyšev, Grüss, and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, the main results are applied to information theory via f-divergence, Rényi divergence, Rényi entropy, Shannon entropy, and Zipf–Mandelbrot law.

Introduction and preliminaries

In [12], Ky Fan’s inequality is generalized by Levinson for 3-convex functions as follows:

Theorem A

Let \(f :I=(0, 2\alpha ) \rightarrow \mathbb{R}\) with \(f^{(3)}(t) \geq 0\). Let \(x_{k} \in (0, \alpha )\) and \(p_{k}>0\). Then

$$ J_{1}(f) \geq 0, $$
(1)

where

$$\begin{aligned} J_{1} \bigl(f(\cdot) \bigr) =& \frac{1}{\mathbf{P}_{n}} \sum_{\rho =1}^{n}p_{\rho }f(2 \alpha -x_{\rho })-f \Biggl(\frac{1}{\mathbf{P}_{n}}\sum _{\rho =1}^{n}p_{ \rho }(2\alpha - x_{\rho }) \Biggr)-\frac{1}{\mathbf{P} _{n}}\sum _{\rho =1} ^{n}p_{\rho }f(x_{\rho }) \\ &{}+ f \Biggl(\frac{1}{\mathbf{P}_{n}}\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr). \end{aligned}$$
(2)

Working with the divided differences, the assumptions of differentiability on f can be weakened.

In [18], Popoviciu noted that (1) is valid on \((0, 2a)\) for 3-convex functions, while in [2], Bullen gave a different proof of Popoviciu’s result and also the converse of (1).

Theorem B

(a) Let \(f:I=[a, b] \rightarrow \mathbb{R}\) be a 3-convex function and \(x_{n}, y_{n} \in [a, b]\) for \(n=1, 2, \ldots , k \) such that

$$ \max \{x_{1} \ldots x_{k}\} \leq \min \{y_{1} \ldots y_{k}\}, \quad x_{1}+y_{1}= \cdots =x_{k}+y_{k} $$
(3)

and \(p_{n}>0\). Then

$$ J_{2}(f) \geq 0, $$
(4)

where

$$ J_{2} \bigl(f(\cdot) \bigr)=\frac{1}{\mathbf{P}_{k}} \sum_{\rho =1}^{k}p_{\rho }f(y _{\rho })-f \Biggl(\frac{1}{\mathbf{P}_{k}}\sum _{\rho =1}^{k}p_{\rho }y_{ \rho } \Biggr)- \frac{1}{\mathbf{P}_{k}}\sum_{\rho =1}^{k} p_{\rho }f(x_{ \rho })+ f \Biggl(\frac{1}{\mathbf{P}_{k}}\sum _{\rho =1}^{k}p_{\rho }x_{ \rho } \Biggr). $$
(5)

(b) If f is continuous and \(p_{\rho }>0\), (4) holds for all \(x_{\rho }\), \(y_{\rho }\) satisfying (3), then f is 3-convex.

In [17], Pečarić weakened assumption (3) and proved that inequality (1) still holds, i.e., the following result holds:

Theorem C

Let \(f:I=[a, b] \rightarrow \mathbb{R}\) be a 3-convex function, \(p_{k}>0\), and let for \(k=1, \ldots , n\), \(x_{k}\), \(y_{k}\) be such that \(x_{k}+y_{k}=2\breve{c}\), \(x_{k}+x_{n-k+1} \leq 2\breve{c}\) and \(\frac{p_{k}x_{k}+p_{n-k+1}x_{n-k+1}}{p_{k}+p _{n-k+1}} \leq \breve{c}\). Then (4) holds.

In [15], Mercer made a notable work by replacing the condition of symmetric distribution of points \(x_{i}\) and \(y_{i}\) with symmetric variances of points \(x_{i}\) and \(y_{i}\). The second condition is a weaker condition.

Theorem D

Let f be a 3-convex function on \([a, b]\), \(p_{k}\) be positive such that \(\sum_{k=1}^{n}p_{k}=1\). Also let \(x_{k}\), \(y_{k}\) satisfy (3) and

$$ \sum_{\rho =1}^{n}p_{\rho } \Biggl(x_{\rho }-\sum_{\rho =1}^{n}p_{ \rho }x_{\rho } \Biggr)^{2}=\sum_{\rho =1}^{n}p_{\rho } \Biggl(y_{ \rho }- \sum_{\rho =1}^{n}p_{\rho }y_{\rho } \Biggr)^{2}. $$
(6)

Then (1) holds.

On the other hand, the error function \(e_{\mathcal{F}}(t)\) can be represented in terms of the Green’s function \(G_{\mathcal{F}, n}(t, s)\) of the boundary value problem

$$\begin{aligned}& z^{(n)}(t)=0, \\& z^{(i)}(a_{1}) = 0,\quad 0 \leq i \leq p, \\& z^{(i)}(a_{2}) = 0,\quad p+1 \leq i \leq n-1, \\& e_{F}(t)= \int ^{a_{2}}_{a_{1}}G_{F, n}(t, s)f^{(n)}(s)\,ds, \quad t \in [a, b], \end{aligned}$$

where

G F , n (t,s)= 1 ( n 1 ) ! { i = 0 p ( n 1 i ) ( t a 1 ) i ( a 1 s ) n i 1 , a 1 s t ; i = p + 1 n 1 ( n 1 i ) ( t a 1 ) i ( a 1 s ) n i 1 , t s a 2 .
(7)

The following result holds in [1]:

Theorem E

Let \(f \in C^{n}[a, b]\), and let \(P_{F}\) be its ’two-point right focal’ interpolating polynomial. Then, for \(a \leq a_{1} < a_{2} \leq b\) and \(0 \leq p \leq n-2\),

$$\begin{aligned} f(t) =&P_{F}(t)+e_{F}(t) \\ =& \sum_{i=0}^{p}\frac{(t-a_{1})^{i}}{i!}f^{(i)}(a_{1}) \\ &{}+ \sum_{j=0}^{n-p-2} \Biggl(\sum _{i=0}^{j}\frac{(t-a_{1})^{p+1+i}(a _{1}-a_{2})^{j-i}}{(p+1+i)!(j-i)!} \Biggr)f^{(p+1+j)}(a_{2} ) \\ &{}+ \int ^{a_{2}}_{a_{1}}G_{F, n}(t, s)f^{(n)}(s)\,ds, \end{aligned}$$
(8)

where \(G_{F, n}(t, s)\) is the Green’s function, defined by (7).

Let \(f \in C^{n}[a, b]\), and let \(P_{F}\) be its ‘two-point right focal’ interpolating polynomial for \(a \leq a_{1} < a_{2} \leq b\). Then, for \(n=3\) and \(p=0\), (8) becomes

$$\begin{aligned} f(t) =& f(a_{1}) + (t-a_{1})f^{(1)}(a_{2})+ (t-a_{1}) (a_{1}-a_{2})f ^{(2)}(a_{2})+\frac{(t-a_{1})^{2}}{2}f^{(2)}(a_{2}) \\ &{}+ \int ^{a_{2}}_{a_{1}}G_{1}(t, s)f^{(3)}(s)\,ds, \end{aligned}$$
(9)

where

$$\begin{aligned} G_{1}(t, s) = \textstyle\begin{cases} (a_{1}-s)^{2}, & \text{$a_{1} \leq s \leq t$;} \\ - (t-a_{1})(a_{1}-s)+\frac{1}{2}(t-a_{1})^{2}, & \text{$t \leq s \leq a_{2}$.} \end{cases}\displaystyle \end{aligned}$$
(10)

For \(n=3\) and \(p=1\), (8) becomes

$$\begin{aligned} f(t) = f(a_{1}) + (t-a_{1})f^{(1)}(a_{2})+ \frac{(t-a_{1})^{2}}{2}f ^{(2)}(a_{2})+ \int ^{a_{2}}_{a_{1}}G_{2}(t, s)f^{(3)}(s)\,ds, \end{aligned}$$
(11)

where

$$\begin{aligned} G_{2}(t, s) = \textstyle\begin{cases} \frac{1}{2}(a_{1}-s)^{2}+(t-a_{1})(a_{1}-s), & \text{$a_{1} \leq s \leq t$;} \\ - \frac{1}{2} (t-a_{1})^{2}, & \text{$t \leq s \leq a_{2}$.} \end{cases}\displaystyle \end{aligned}$$
(12)

The presented work is organized as follows: In Sect. 2, Levinson’s inequality for 3-convex function is generalized by using two Green’s functions defined by (10) and (12). In Sect. 3, Čebyšev, Grüss, and Ostrowski-type new bounds are found for the functionals involving data points of two types. In Sect. 4, the main results are applied to information theory via f-divergence, Rényi divergence, Rényi entropy, Shannon entropy, and Zipf–Mandelbrot law.

Main results

First we give an identity involving Jensen’s difference of two different data points. Then we give an equivalent form of identity by using the Green’s function defined by (10) and (12).

Theorem 1

Let \(f\in C^{3}[\zeta _{1}, \zeta _{2}]\) such that \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\) such that \(\sum_{\rho =1}^{n}p_{\rho }=1\) and \(\sum_{\varrho =1}^{m}q_{\varrho }=1\). Also let \(x_{\rho }\), \(y_{\varrho }\), \(\sum_{\rho =1}^{n}p_{\rho }x _{\rho }\), \(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \in I\). Then

$$\begin{aligned} J \bigl(f(\cdot) \bigr) =&\frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y_{ \varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n}p_{\rho } x_{\rho }^{2}+ \Biggl(\sum_{\rho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \\ &{}+ \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds, \end{aligned}$$
(13)

where

$$ J \bigl(f(\cdot) \bigr)=\sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })-f \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)-\sum_{\rho =1}^{n}p _{\rho }f(x_{\rho }) +f \Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) $$
(14)

and

$$ \begin{aligned}[b] J \bigl(G_{k}(\cdot, s) \bigr)&= \sum _{\varrho =1}^{m}q_{\varrho }G_{k}(y_{\varrho }, s)-G _{k} \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }, s \Biggr)\\&\quad{}-\sum_{\rho =1} ^{n}p_{\rho }G_{k}(x_{\rho }, s) +G_{k} \Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho }, s \Biggr), \end{aligned} $$
(15)

for \(G_{k}(\cdot, s)\) (\(k=1, 2\)) defined in (10) and (12) respectively.

Proof

(i) For \(k=1\).

Using (9) in (14), we have

$$\begin{aligned}& \begin{aligned} J \bigl(f(\cdot) \bigr) &=\sum _{\varrho =1}^{m}q_{\varrho } \biggl[f(\zeta _{1})+(y_{ \varrho }-\zeta _{1})f^{(1)}( \zeta _{2})+(y_{\varrho }- \zeta _{1}) (\zeta _{1}-\zeta _{2})f^{(2)} (\zeta _{2}) \\ &\quad{}+\frac{(y_{\varrho }-\zeta _{1})^{2}}{2}f^{(2)}(\zeta _{2})+ \int _{\zeta _{1}}^{\zeta _{2}}G_{1}(y_{\varrho }, s)f^{(3)}(s)\,ds \biggr] \\ &\quad{}- \Biggl[f(\zeta _{1})+ \Biggl(\sum _{\varrho =1}^{m}q_{\varrho }y_{\varrho }- \zeta _{1} \Biggr)f^{(1)}(\zeta _{2})+ \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{ \varrho }- \zeta _{1} \Biggr) (\zeta _{1}- \zeta _{2})f^{(2)}(\zeta _{2}) \\ &\quad{}+\frac{(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }-\zeta _{1})^{2}}{2}f^{(2)}(\zeta _{2})+ \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl( \sum _{\varrho =1}^{m} q_{\varrho }y_{\varrho }, s \Biggr)f^{(3)}(s)\,ds \Biggr] \\ &\quad{}-\sum_{\rho =1}^{n}p_{\rho } \biggl[f(\zeta _{1})+(x_{\rho }-\zeta _{1})f^{(1)}( \zeta _{2})+(x_{\rho }- \zeta _{1}) (\zeta _{1}-\zeta _{2})f ^{(2)} (\zeta _{2}) \\ &\quad{}+\frac{(x_{\rho }-\zeta _{1})^{2}}{2}f^{(2)}(\zeta _{2})+ \int _{\zeta _{1}}^{\zeta _{2}}G_{1}(x_{\rho }, s)f^{(3)}(s)\,ds \biggr] \\ &\quad{}+ \Biggl[f(\zeta _{1})+ \Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr)f^{(1)}(\zeta _{2})+ \Biggl( \sum_{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr) (\zeta _{1}-\zeta _{2}) f^{(2)}(\zeta _{2}) \\ &\quad{}+\frac{(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }-\zeta _{1})^{2}}{2}f ^{(2)}(\zeta _{2})+ \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl(\sum _{\rho =1} ^{n} p_{\rho }x_{\rho }, s \Biggr)f^{(3)}(s)\,ds \Biggr]. \end{aligned} \\& \begin{aligned} J \bigl(f(\cdot) \bigr) &=f(\zeta _{1})+ \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }- \zeta _{1} \Biggr)f^{(1)}(\zeta _{2})+ \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{ \varrho }- \zeta _{1} \Biggr) (\zeta _{1}-\zeta _{2})f^{(2)}( \zeta _{2}) \\ &\quad{}+\frac{(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }^{2}-2\zeta _{1}\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }+\zeta _{1}^{2})f^{(2)}( \zeta _{2})}{2}+\sum_{i=1}^{m} q_{\varrho } \int _{\zeta _{1}}^{\zeta _{2}}G _{1}(y_{\varrho }, s)f^{(3)}(s)\,ds \\ &\quad{}-f(\zeta _{1})- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }- \zeta _{1} \Biggr)f^{(1)}(\zeta _{2})- \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }- \zeta _{1} \Biggr) (\zeta _{1}-\zeta _{2})f^{(2)}( \zeta _{2}) \\ &\quad{}-\frac{((\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho })^{2}-2\zeta _{1}\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }+\zeta _{1}^{2})f^{(2)}( \zeta _{2})}{2}\\&\quad{}- \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl(\sum _{\varrho =1} ^{m}q_{\varrho }y_{\varrho }, s \Biggr)f^{(3)}(s)\,ds \\ &\quad{}-f(\zeta _{1})- \Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr)f^{(1)}( \zeta _{2})- \Biggl( \sum_{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr) (\zeta _{1}-\zeta _{2})f^{(2)}( \zeta _{2}) \\ &\quad{}-\frac{(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }^{2}-2\zeta _{1} \sum_{\rho =1}^{n}p_{\rho }x_{\rho }+\zeta _{1}^{2})f^{(2)}(\zeta _{2})}{2}- \sum_{\rho =1}^{n}p_{\rho } \int _{\zeta _{1}}^{\zeta _{2}}G_{1}(x_{ \rho }, s)f^{(3)}(s)\,ds \\ &\quad{}+f(\zeta _{1})+ \Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr)f^{(1)}( \zeta _{2})+ \Biggl( \sum_{\rho =1}^{n}p_{\rho }x_{\rho }- \zeta _{1} \Biggr) (\zeta _{1}-\zeta _{2})f^{(2)}( \zeta _{2}) \\ &\quad{}+\frac{((\sum_{\rho =1}^{n}p_{\rho }x_{\rho })^{2}-2\zeta _{1} \sum_{\rho =1}^{n}p_{\rho }x_{\rho }+\zeta _{1}^{2})f^{(2)}(\zeta _{2})}{2} \\&\quad{}+ \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{ \rho }, s \Biggr)f^{(3)}(s)\,ds, \end{aligned} \\& \begin{aligned} J \bigl(f(\cdot) \bigr) &=\frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y_{ \varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n}p_{\rho }x_{\rho }^{2} + \Biggl(\sum_{\rho =1}^{n}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr]f^{(2)}(\zeta _{2}) \\ &\quad{}+\sum_{\varrho =1}^{m}q_{\varrho } \int _{\zeta _{1}}^{\zeta _{2}}G_{1}(y _{\varrho },s)f^{(3)}(s)\,ds- \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl( \sum _{\varrho =1}^{m}q_{\varrho }y_{\varrho }, s \Biggr)f^{(3)}(s)\,ds \\ &\quad{}-\sum_{\rho =1}^{n}p_{\rho } \int _{\zeta _{1}}^{\zeta _{2}}G_{1}(x_{ \rho }, s)f^{(3)}(s)\,ds + \int _{\zeta _{1}}^{\zeta _{2}}G_{1} \Biggl(\sum _{ \rho =1}^{n}p_{\rho }x_{\rho }, s \Biggr)f^{(3)}(s)\,ds. \end{aligned} \end{aligned}$$

After rearranging, we have (13).

(ii) For \(k=2\)

Using (11) in (14) and following similar steps as in the proof of (i), we get (13). □

Corollary 1

Let \(f\in C^{3}[0, 2\alpha ]\) such that \(f: I= [0, 2\alpha ] \rightarrow \mathbb{R}\), \(x_{1}, \ldots , x_{n} \in (0, \alpha )\), \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\) such that \(\sum_{\rho =1}^{n}p _{\rho }=1\). Also let \(x_{\rho }\), \(\sum_{\rho =1}^{n}p_{\rho }(2\alpha -x_{\rho })\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \in I\). Then

$$\begin{aligned} J \bigl(f(\cdot) \bigr) =& \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds ,\quad 0\leq \zeta _{1}< \zeta _{2} \leq 2\alpha , \end{aligned}$$
(16)

where \(J(f(\cdot))\) and \(J(G(\cdot, s))\) are defined in (14) and (15) respectively.

Proof

Choosing \(I=[0, 2\alpha ]\), \(y_{\varrho }=(2\alpha -x_{\rho })\), \(x_{1}, \ldots , x_{n} \in (0, \alpha )\), \(p_{\rho }=q_{\varrho }\), and \(m=n\) in Theorem 1, after simplification we get (16). □

Theorem 2

Let \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be a 3-convex function. Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m})\in \mathbb{R}^{m}\) be such that \(\sum_{\rho =1}^{n}p_{\rho }=1\) and \(\sum_{\varrho =1}^{m}q_{ \varrho }=1\) and \(x_{\rho }\), \(y_{\varrho }\), \(\sum_{\rho =1}^{n}p_{ \rho }x_{\rho }\), \(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \in I\).

If

$$ \Biggl[\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }^{2}- \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1}^{n}p _{\rho }x_{\rho }^{2}+ \Biggl(\sum _{\rho =1}^{n} p_{\rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \geq 0, $$
(17)

then the following statements are equivalent:

For \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\),

$$ \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })-f \Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho } \Biggr)\leq \sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })- f \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr). $$
(18)

For all \(s \in I\),

$$\begin{aligned} \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s)-G_{k} \Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s \Biggr)\leq \sum _{\varrho =1}^{m}q_{\varrho }G_{k}(y _{\varrho }, s)- G_{k} \Biggl(\sum _{\varrho =1}^{m}q_{\varrho }y_{\varrho }, s \Biggr), \end{aligned}$$
(19)

where \(G_{k}(\cdot, s)\) are defined by (10) and (12) for \(k=1, 2\) respectively.

Moreover, inequality in (18) is reversed iff inequality in (19) is reversed.

Proof

(18) (19): Let (18) be valid. Then, as the function \(G_{k}(\cdot, s)\) (\(s \in I\)) is also continuous and 3-convex, it follows that also for this function (18) holds, i.e., (19) is valid.

(19) (18): If f is 3-convex, then without loss of generality we can suppose that there exists the third derivative of f. Let \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\) be a 3-convex function and (19) hold. Then we can represent function f in the form (9). Now, by means of some simple calculations, we can write

$$\begin{aligned} &\sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })-f \Biggl(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \Biggr)- \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })+ f \Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) \\ &\quad = \frac{1}{2} \Biggl[\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } ^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }^{2}+ \Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2})\\ &\qquad {}+ \int _{\zeta _{1}}^{\zeta _{2}} \Biggl( \sum _{\varrho =1}^{m}q_{\varrho }G _{k}(y_{\varrho }, s)-G_{k} \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }(y_{\varrho }, s) \Biggr) \\ &\qquad {} - \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s)+G_{k} \Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s \Biggr) \Biggr)f^{(3)}(s)\,ds. \end{aligned}$$

By the convexity of f, we have \(f^{(3)}(s) \geq 0\) for all \(s \in I\). Hence, if for every \(s \in I\), (19) is valid, then it follows that for every 3-convex function \(f:I \rightarrow \mathbb{R}\), with \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), (18) is valid. □

Remark 1

If the expression

$$ \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }^{2}- \Biggl(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1}^{n}p_{\rho }x_{ \rho }^{2}+ \Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) ^{2} $$

and \(f^{(2)}(\zeta _{2})\) have different signs in (17), then inequalities (18) and (19) are reversed.

Next we have the results about generalization of Bullen-type inequality (for real weights) given in [2] (see also [16] and [11]).

Corollary 2

Let \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be a 3-convex function and \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), \(x_{1}, \ldots , x_{n} \), \(y_{1}, \ldots , y_{m} \in I\) such that

$$ \max \{x_{1}, \ldots , x_{n}\} \leq \min \{y_{1}, \ldots , y_{m}\} $$
(20)

and

$$ x_{1}+y_{1}= \cdots =x_{n}+y_{m}. $$
(21)

Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m})\in \mathbb{R}^{m}\) be such that \(\sum_{\rho =1}^{n}p _{\rho }=1\) and \(\sum_{\varrho =1}^{m}q_{\varrho }=1\) and \(x_{\rho }\), \(y_{\varrho }\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }\), \(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \in I\). If (17) holds, then (18) and (19) are equivalent.

Proof

By choosing \(x_{\rho }\) and \(y_{\varrho }\) such that conditions (20) and (21) hold in Theorem 2, we get the required result. □

Remark 2

If \(p_{\rho }=q_{\varrho }\) are positive and \(x_{\rho }\), \(y_{\varrho }\) satisfy (20) and (21), then inequality (18) reduces to Bullen’s inequality given in [16, p. 32, Theorem 2] for \(m=n\).

Next we have a generalized form (for real weights) of Bullen-type inequality given in [17] (see also [16]).

Corollary 3

Let \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be a 3-convex function and \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\) be such that \(\sum_{\rho =1}^{n}p_{\rho }=1\) and \(\sum_{\varrho =1}^{m}q_{\varrho }=1\). Also let \(x_{1}, \ldots , x _{n} \) and \(y_{1}, \ldots , y_{m} \in I\) be such that \(x_{\rho }+y _{\varrho }=2c\), and for \(\rho =1, \ldots , n\), \(x_{\rho }+x_{n-\rho +1} \) and \(\frac{p_{\rho }x_{\rho }+p_{n-\rho +1}x_{n-\rho +1}}{p_{\rho }+p _{n-\rho +1}} \leq c\). If (17) holds, then (18) and (19) are equivalent.

Proof

Using Theorem 2 with the conditions given in the statement, we get the required result. □

Remark 3

In Theorem 2, if \(m=n\), \(p_{\rho }=q_{\varrho }\) are positive, \(x_{\rho }+y_{\varrho }=2c\), \(x_{\rho }+x_{n-\rho +1} \) and \(\frac{p_{\rho }x_{\rho }+p_{n-\rho +1}x_{n-\rho +1}}{p_{\rho }+p_{n- \rho +1}} \leq c\). Then (18) reduces to a generalized form of Bullen’s inequality defined in [16, p. 32, Theorem 4].

In [15], Mercer made a notable work by replacing condition (21) of symmetric distribution of points \(x_{\rho }\) and \(y_{\varrho }\) with symmetric variances of points \(x_{\rho }\) and \(y_{\varrho }\) for \(\rho =1, \ldots , n\) and \(\varrho =1, \ldots , m\).

So in the next result we use Mercer’s condition (6), but for \(\rho =\varrho \) and \(m=n\).

Corollary 4

Let \(f: I= [\zeta _{1}, \zeta _{1}] \rightarrow \mathbb{R}\) be a 3-convex function and \(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), \(p_{\rho }\), \(q_{\rho }\) be positive such that \(\sum_{\rho =1}^{n}p _{\rho }=1\) and \(\sum_{\rho =1}^{n}q_{\rho }=1\). Also let \(x_{\rho }\), \(y_{\rho }\) satisfy (20) and

$$ \sum_{\rho =1}^{n}p_{\rho } \Biggl(x_{\rho }-\sum_{\rho =1}^{n}p_{ \rho }x_{\rho } \Biggr)^{2}=\sum_{\rho =1}^{n}p_{\rho } \Biggl(y_{ \rho }-\sum_{\rho =1} ^{n}q_{\rho }y_{\rho } \Biggr)^{2}. $$
(22)

If (17) holds, then (18) and (19) are equivalent.

Proof

For positive weights, using (6) and (20) in Theorem 2, we get the required result. □

Next we have the results that lean on the generalization of Levinson-type inequality given in [12] (see also [16]).

Corollary 5

Let \(f: I= [0, 2\alpha ] \rightarrow \mathbb{R}\) be a 3-convex function and \(f \in C^{3}[0, 2\alpha ]\), \(x_{1}, \ldots , x_{n} \in (0, \alpha )\), \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\) and \(\sum_{\rho =1}^{n}p_{\rho }=1\). Also let \(x_{\rho }\), \(\sum_{\rho =1}^{n}p_{\rho }(2\alpha -x_{\rho })\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \in I\). Then the following are equivalent:

$$ \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })-f \Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho } \Biggr)\leq \sum_{\rho =1}^{n}p_{\rho }f(2 \alpha -x_{\rho })- f \Biggl( \sum_{\rho =1}^{n}p_{\rho }(2 \alpha -x_{\rho }) \Biggr). $$
(23)

For all \(s \in I\),

$$\begin{aligned} \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s) -G_{k} \Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s \Biggr) \leq& \sum _{\rho =1}^{n}p_{\rho }G_{k}(2 \alpha -x_{\rho }, s) \\ &{}-G_{k} \Biggl(\sum_{\rho =1}^{n}p_{\rho }(2 \alpha -x_{\rho }), s \Biggr), \end{aligned}$$
(24)

where \(G_{k}(\cdot, s)\) is defined in (10) and (12) for \(k=1, 2\) respectively.

Proof

If \(I=[0, 2\alpha ]\), \((x_{1}, \ldots , x_{n}) \in (0, \alpha )\), \(p_{\rho }=q_{\varrho }\), \(m=n\), and \(y_{\varrho }=(2\alpha -x_{ \rho })\) in Theorem 2 with \(0 \leq \zeta _{1} < \zeta _{2} \leq 2\alpha \), we get the required result. □

Remark 4

In Corollary 5, if \(p_{\rho }\) are positive, then inequality (23) reduces to Levinson’s inequality given in [16, p. 32, Theorem 1].

New bounds for Levinson-type functionals

Consider the Čebyšev functional for two Lebesgue integrable functions \(f_{1}, f_{2}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)

$$\begin{aligned} \varTheta (f_{1}, f_{2}) =& \frac{1}{\zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}} _{\zeta _{1}}f_{1}(x)f_{2}(x) \,dx \\ &{}- \frac{1}{\zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}}_{\zeta _{1}}f_{1}(x)\,dx\times \frac{1}{ \zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}}_{\zeta _{1}}f_{2}(x)\,dx, \end{aligned}$$
(25)

where the integrals are assumed to exist.

Theorem F

([3])

Let \(f_{1} : [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be a Lebesgue integrable function and \(f_{2} : [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be an absolutely continuous function with \((\cdot, -\zeta _{1})(\cdot, -\zeta _{2})[f'_{2}]^{2} \in L[\zeta _{1}, \zeta _{2}]\). Then

$$\begin{aligned} \bigl\vert \varTheta (f_{1}, f_{2}) \bigr\vert \leq \frac{1}{\sqrt{2}} \bigl[\varTheta (f_{1}, f _{1}) \bigr]^{\frac{1}{2}}\frac{1}{\sqrt{\zeta _{2}-\zeta _{1}}} \biggl( \int ^{ \zeta _{2}}_{\zeta _{1}} (t-\zeta _{1}) ( \zeta _{2}-t) \bigl[f'_{2}(t) \bigr]^{2}\,dt \biggr) ^{\frac{1}{2}}. \end{aligned}$$
(26)

\(\frac{1}{\sqrt{2}}\) is the best possible.

Theorem G

([3])

Let \(f_{1}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be absolutely continuous with \(f^{\prime }_{1} \in L_{\infty }[ \zeta _{1}, \zeta _{2}]\), and let \(f_{2}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) be monotonic nondecreasing on \([\zeta _{1}, \zeta _{2}]\). Then

$$\begin{aligned} \bigl\vert \varTheta (f_{1}, f_{2}) \bigr\vert \leq \frac{1}{2(\zeta _{2}-\zeta _{1})} \bigl\Vert f^{\prime } \bigr\Vert _{ \infty } \int ^{\zeta _{2}}_{\zeta _{1}} (t-\zeta _{1}) ( \zeta _{2}-t) \bigl[f'_{2}(t) \bigr]\,df _{2}(t). \end{aligned}$$
(27)

\(\frac{1}{2}\) is the best possible.

In the next result we construct the Čebyšev-type bound for our functional defined in (5).

Theorem 3

Let \(f\in C^{3}[\zeta _{1}, \zeta _{2}]\) be such that \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) and \(f^{(3)}(\cdot)\) is absolutely continuous with \((\cdot-\zeta _{1})(\zeta _{2} -\cdot)[f^{(4)}]^{2} \in L[\zeta _{1}, \zeta _{2}]\). Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R} ^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\) be such that \(\sum_{\rho =1}^{n}p_{\rho }=1\), \(\sum_{\varrho =1}^{m}q_{\varrho }=1\), \(x_{\rho }\), \(y_{\varrho }\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }\), \(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \in I\). Then

$$\begin{aligned} J \bigl(f(\cdot) \bigr) =&\frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y_{ \varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n}p_{\rho }x_{\rho }^{2} + \Biggl(\sum_{\varrho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \\ &{}+\frac{f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2}-\zeta _{1})} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds +\mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f), \end{aligned}$$
(28)

where \(J(f(\cdot))\), \(J(G_{k}(\cdot, s))\) are defined in (14) and (15) respectively, and the remainder \(\mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f)\) satisfies the bound

$$\begin{aligned} \bigl\vert \mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f) \bigr\vert \leq& \frac{\zeta _{2}-\zeta _{1}}{\sqrt{2}} \bigl[\varTheta \bigl(J \bigl(G_{k}(\cdot, s) \bigr),J \bigl(G_{k}(\cdot, s) \bigr) \bigr) \bigr] ^{\frac{1}{2}}\times \\ &{}\frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s) \bigl[f^{(4)}(s) \bigr] \,ds \biggr)^{ \frac{1}{2}} \end{aligned}$$
(29)

for \(G_{k}(\cdot, s)\) (\(k=1, 2\)) defined in (10) and (12) respectively.

Proof

Setting \(f_{1} \mapsto J(G_{k}(\cdot, s))\) and \(f_{2} \mapsto f^{(3)}\) in Theorem F, we get

$$\begin{aligned}& \begin{aligned} & \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G _{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds - \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)\,ds\times \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f^{(3)}(s)\,ds \biggr\vert \\ &\quad \leq \frac{1}{\sqrt{2}} \bigl[\varTheta \bigl(J \bigl(G_{k}( \cdot, s) \bigr), J \bigl(G_{k}(\cdot, s) \bigr) \bigr) \bigr]^{ \frac{1}{2}} \frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s) \bigl[f^{(4)}(s) \bigr] \,ds \biggr) ^{\frac{1}{2}}, \end{aligned} \\& \begin{aligned} & \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G _{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds - \frac{f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2} - \zeta _{1})^{2}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G _{k}(\cdot, s) \bigr)\,ds \biggr\vert \\ &\quad \leq \frac{1}{\sqrt{2}} \bigl[\varTheta \bigl(J \bigl(G_{k}( \cdot, s) \bigr), J \bigl(G_{k}(\cdot, s) \bigr) \bigr) \bigr]^{ \frac{1}{2}} \frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s) \bigl[f^{(4)}(s) \bigr] \,ds \biggr) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

Multiplying \((\zeta _{2} - \zeta _{1})\) on both sides of the above inequality and using estimation (29), we get

$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)f^{(3)}\,ds = \frac{f^{(2)}( \zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2} - \zeta _{1})} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)\,ds + \mathcal{R}_{3}(\zeta _{1}, \zeta _{1}; f). \end{aligned}$$

Using identity (13), we get (28). □

In the next result the bounds of Grüss-type inequalities are estimated.

Theorem 4

Let \(f\in C^{3}[\zeta _{1}, \zeta _{2}]\) be such that \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), \(f^{(3)}(\cdot)\) is absolutely continuous and \(f^{(4)}(\cdot) \geq 0\) a.e. on \([\zeta _{1}, \zeta _{2}]\). Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\) be such that \(\sum_{\rho =1}^{n}p _{\rho }=1\), \(\sum_{\varrho =1}^{m}q_{\varrho }=1\), \(x_{\rho }\), \(y_{ \varrho }\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }\), \(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \in I\). Then identity (28) holds, where the remainder satisfies the estimation

$$\begin{aligned} \bigl\vert \mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f) \bigr\vert \leq (\zeta _{2} - \zeta _{1}) \bigl\Vert J \bigl(G_{k}( \cdot, s) \bigr)^{\prime } \bigr\Vert _{\infty } \biggl[ \frac{f^{(2)}(\zeta _{2})+f ^{(2)}(\zeta _{1})}{2} - \frac{f^{(2)}(\zeta _{2})- f^{(2)}(\zeta _{1})}{ \zeta _{2} - \zeta _{1}} \biggr]. \end{aligned}$$
(30)

Proof

Setting \(f_{1} \mapsto J(G_{k}(\cdot, s))\) and \(f_{2} \mapsto f^{(3)}\) in Theorem G, we get

$$\begin{aligned}& \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G _{k}(\cdot, s) \bigr))f^{(3)}(s)\,ds - \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)\,ds\cdot \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f^{(3)}(s)\,ds \biggr\vert \\& \quad \leq \frac{1}{2} \bigl\Vert J \bigl(G_{k}(\cdot, s) \bigr)^{\prime } \bigr\Vert _{\infty }\frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s) \bigl[f ^{(4)}(s) \bigr] \,ds. \end{aligned}$$
(31)

Since

$$\begin{aligned} & \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s) \bigl[f^{(4)}(s) \bigr] \,ds \\ &\quad = \int _{\zeta _{1}}^{\zeta _{2}}[2s - \zeta _{1} - \zeta _{2}]f^{3}(s)\,ds \\ & \quad = (\zeta _{2} - \zeta _{1}) \bigl[f^{(2)}( \zeta _{2}) + f^{(2)}(\zeta _{1}) \bigr] - 2 \bigl(f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1}) \bigr), \end{aligned}$$
(32)

using (13), (31), and (32), we have (28). □

Ostrowski-type bounds for a newly constructed functional defined in (5).

Theorem 5

Let \(f\in C^{3}[\zeta _{1}, \zeta _{2}]\) be such that \(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) and \(f^{(2)}(\cdot)\) is absolutely continuous. Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\) be such that \(\sum_{ \rho =1}^{n}p_{\rho }=1\), \(\sum_{\varrho =1}^{m}q_{\varrho }=1\), \(x_{\rho }\), \(y_{\varrho }\), \(\sum_{\rho =1}^{n}p_{\rho }x_{\rho }\), \(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \in I\). Also let \((r, s)\) be a pair of conjugate exponents, that is, \(1 \leq r, s, \leq \infty \), \(\frac{1}{r}+\frac{1}{s}=1\). If \(|f^{(3)}|^{r}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\) is a Riemann integrable function, then

$$\begin{aligned}& \Biggl\vert J \bigl(f(\cdot) \bigr) - \frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y _{\varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n} p_{\rho }x_{\rho }^{2} + \Biggl(\sum _{\varrho =1}^{m}p _{\rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \Biggr\vert \\& \quad \leq \bigl\Vert f^{(3)} \bigr\Vert _{r} \biggl( \int _{\zeta _{1}}^{\zeta _{2}} \bigl\vert J \bigl(G _{k}(\cdot, s) \bigr)\,ds \bigr\vert ^{s} \biggr)^{\frac{1}{s}}. \end{aligned}$$
(33)

Proof

Rearrange identity (13) in the following way:

$$\begin{aligned}& \Biggl\vert J \bigl(f(\cdot) \bigr) - \frac{1}{2} \Biggl(\sum _{\varrho =1}^{m}q_{\varrho }y _{\varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n} p_{\rho } x_{\rho }^{2}+ \Biggl(\sum _{\rho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr)f^{(2)}(\zeta _{2}) \Biggr\vert \\& \quad \leq \int _{\zeta _{1}}^{\zeta _{2}}J \bigl(G_{k}(\cdot, s) \bigr)f^{(3)}(s)\,ds. \end{aligned}$$
(34)

Employing the classical Holder’s inequality to R.H.S of (34) yields (33). □

Application to information theory

The idea of Shannon entropy is the focal job of data hypothesis once in a while alluded as measure of uncertainty. The entropy of a random variable is characterized regarding its probability distribution and can be shown to be a decent measure of randomness or uncertainty. Shannon entropy permits to evaluate the normal least number of bits expected to encode a series of images dependent on the letters in order size and the recurrence of the symbols.

Divergences between probability distributions have been acquainted with measure of the difference between them. A variety of sorts of divergences exist, for instance the f-difference (particularly, Kullback–Leibler divergence, Hellinger distance, and total variation distance), Rényi divergence, Jensen–Shannon divergence, and so forth (see [13, 21]). There are a lot of papers managing inequalities and entropies, see, e.g., [8, 10, 20] and the references therein. Jensen’s inequality assumes a crucial role in a portion of these inequalities. In any case, Jensen’s inequality deals with one sort of information focuses and Levinson’s inequality manages two types of information points.

Zipf’s law is one of the central laws in data science, and it has been utilized in linguistics. George Zipf in 1932 found that we can tally how frequently each word shows up in the content. So on the off chance that we rank (r) word as per the recurrence of word event \((f)\), at that point the result of these two numbers is steady \((C): C = r \times f\). Aside from the utilization of this law in data science and linguistics, Zipf’s law is utilized in city population, sun powered flare power, site traffic, earthquake magnitude, the span of moon pits, and so forth. In financial aspects this distribution is known as the Pareto law, which analyzes the distribution of the wealthiest individuals in the community [6, p. 125]. These two laws are equivalent in the mathematical sense, yet they are involved in different contexts [7, p. 294].

Csiszár divergence

In [4, 5] Csiszár gave the following definition:

Definition 1

Let f be a convex function from \(\mathbb{R}^{+}\) to \(\mathbb{R}^{+}\). Let \(\tilde{\mathbf{r}}, \tilde{\mathbf{k}} \in \mathbb{R}_{+}^{n}\) be such that \(\sum_{s=1}^{n}r_{s}=1\) and \(\sum_{s=1}^{n}q_{s}=1\). Then an f-divergence functional is defined by

$$\begin{aligned} I_{f}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}}) := \sum _{s=1}^{n}q_{s}f \biggl( \frac{r_{s}}{q_{s}} \biggr). \end{aligned}$$

By defining the following:

$$\begin{aligned} f(0) := \lim_{x \rightarrow 0^{+}}f(x); \qquad 0f \biggl( \frac{0}{0} \biggr):=0; \qquad 0f \biggl(\frac{a}{0} \biggr):= \lim_{x \rightarrow 0^{+}}xf \biggl(\frac{a}{0} \biggr), \quad a>0, \end{aligned}$$

he stated that nonnegative probability distributions can also be used.

Using the definition of f-divergence functional, Horv́ath et al. [9] gave the following functional:

Definition 2

Let I be an interval contained in \(\mathbb{R}\) and \(f: I \rightarrow \mathbb{R}\) be a function. Also let \(\tilde{\mathbf{r}}=(r_{1}, \ldots , r_{n})\in \mathbb{R}^{n}\) and \(\tilde{\mathbf{k}}=(k_{1}, \ldots , k_{n})\in (0, \infty )^{n}\) be such that

$$\begin{aligned} \frac{r_{s}}{k_{s}} \in I, \quad s= 1, \ldots , n. \end{aligned}$$

Then

$$\begin{aligned} \hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}}) : = \sum _{s=1} ^{n}k_{s}f \biggl( \frac{r_{s}}{k_{s}} \biggr). \end{aligned}$$

We apply a generalized form of Bullen’s inequality (18) (for positive weights) to \(\hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})\).

Let us denote the following set of assumptions by \(\mathcal{G}\):

Let \(f: I= [\alpha , \beta ] \rightarrow \mathbb{R}\) be a 3-convex function. Also let \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{+}\), \((q_{1}, \ldots , q_{m})\in \mathbb{R}^{+}\) be such that \(\sum_{s=1} ^{n}p_{s}=1\) and \(\sum_{s=1}^{m}q_{s}=1\) and \(x_{s}\), \(y_{s}\), \(\sum_{s=1} ^{n}p_{s}x_{s}\), \(\sum_{s=1}^{m}q_{s}y_{s} \in I\).

Theorem 6

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}}= (r_{1}, \ldots , r_{n} )\), \(\tilde{\mathbf{k}}= (k_{1}, \ldots , k_{n} )\) be in \((0, \infty )^{n}\), and \(\tilde{\mathbf{w}}= (w_{1}, \ldots , w _{m} )\), \(\tilde{\mathbf{t}}= (t_{1}, \ldots , t_{m} )\) be in \((0, \infty )^{m}\) such that

$$\begin{aligned} \frac{r_{s}}{k_{s}} \in I, \quad s = 1, \ldots , n, \end{aligned}$$

and

$$\begin{aligned} \frac{w_{u}}{t_{u}} \in I, \quad u = 1, \ldots , m. \end{aligned}$$

Then

$$ \mathrm{(i)} \quad \frac{1}{\sum_{s=1}^{n}k_{s}}\hat{I}_{f}( \tilde{\mathbf{r}}, \tilde{\mathbf{k}})-f \Biggl(\sum _{s=1}^{n} \frac{r_{s}}{\sum_{s=1}^{n}k_{s}} \Biggr) \leq \frac{1}{\sum_{u=1}^{m}t_{u}}\hat{I}_{f}( \tilde{\mathbf{w}}, \tilde{ \mathbf{t}})- f \Biggl(\sum_{u=1}^{m} \frac{w_{u}}{\sum_{u=1}^{m}t_{u}} \Biggr). $$
(35)

(ii) If \(x \rightarrow xf(x)\) (\(x \in [a, b]\)) is 3-convex, then

$$ \frac{1}{\sum_{s=1}^{n}k_{s}}\hat{I}_{idf} (\tilde{ \mathbf{r}}, \tilde{\mathbf{k}} )-f \Biggl(\sum_{s=1}^{n} \frac{r_{s}}{\sum_{s=1} ^{n}k_{s}} \Biggr)\leq \frac{1}{\sum_{u=1}^{m}t_{u}}\hat{I}_{idf}( \tilde{\mathbf{w}}, \tilde{\mathbf{t}})- f \Biggl(\sum _{u=1}^{m}\frac{w_{u}}{ \sum_{u=1}^{m}t_{u}} \Biggr), $$
(36)

where

$$ \hat{I}_{idf} (\tilde{\mathbf{r}}, \tilde{\mathbf{k}} )= \sum _{s=1}^{n}r_{s}f \biggl( \frac{r_{s}}{k_{s}} \biggr) $$

and

$$ \hat{I}_{idf} (\tilde{\mathbf{w}}, \tilde{\mathbf{t}} )= \sum _{u=1}^{m}w_{u}f \biggl( \frac{w_{u}}{t_{u}} \biggr). $$

Proof

(i) Taking \(p_{s} = \frac{k_{s}}{\sum_{s=1}^{n}k_{s}}\), \(x_{\rho } = \frac{r_{s}}{k_{s}}\), \(q_{s} = \frac{t_{u}}{\sum_{u=1}^{m}t_{u}}\), and \(y_{s} = \frac{w_{u}}{t_{u}}\) in inequality (18) (for positive weights), we have

$$ \sum_{s=1}^{n} \frac{k_{s}}{\sum_{s=1}^{n}k_{s}}f \biggl(\frac{r_{s}}{k_{s}} \biggr)-f \Biggl( \sum _{s=1}^{n}\frac{r_{s}}{\sum_{s=1}^{n}k_{s}} \Biggr) \leq \sum_{u=1}^{m}\frac{t _{u}}{\sum_{u=1}^{m}t_{u}}f \biggl( \frac{w_{u}}{t_{u}} \biggr)-f \Biggl(\sum_{u=1}^{m} \frac{w _{u}}{\sum_{u=1}^{m}t_{u}} \Biggr). $$
(37)

Multiplying (37) by the sum \(\sum_{s=1}^{n}k_{s}\), we get

$$\begin{aligned} \hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{ \mathbf{k}})-f \Biggl(\sum_{s=1}^{n} \frac{r _{s}}{\sum_{s=1}^{n}k_{s}} \Biggr)\sum_{s=1}^{n}k_{s} \leq & \sum_{u=1}^{m} \frac{t _{u}}{\sum_{u=1}^{m}t_{u}}f \biggl(\frac{w_{u}}{t_{u}} \biggr)\sum _{u=1}^{n}k_{u} \\ &{}-f \Biggl(\sum_{u=1}^{m} \frac{w_{u}}{\sum_{u=1}^{m}t_{u}} \Biggr)\sum_{s=1}^{n}k _{s}. \end{aligned}$$
(38)

Now again multiplying (38) by the sum \(\sum_{u=1}^{m}t_{u}\), we get

$$\begin{aligned}& \sum_{u=1}^{m}t_{u} \hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})-f \Biggl(\sum _{s=1}^{n}\frac{r_{s}}{\sum_{s=1}^{n}k_{s}} \Biggr) \sum_{s=1}^{n}k_{s}\sum _{u=1}^{m}t_{u} \\& \quad \leq \sum_{s=1}^{n}k_{s} \hat{I}_{f}(\tilde{\mathbf{w}}, \tilde{\mathbf{t}}) -f \Biggl(\sum _{u=1}^{m} \frac{w_{u}}{\sum_{u=1}^{m}t_{u}} \Biggr) \sum_{s=1}^{n}k_{s}\sum _{u=1}^{m}t _{u}. \end{aligned}$$

If we divide the above inequality with the product \(\sum_{s=1}^{n}k _{s} \sum_{u=1}^{m}t_{u}\), we get (35).

(ii) Using \(f:=id f\) (where “id” is the identity function) in (18)(for positive weights), we have

$$\begin{aligned} \sum_{s=1}^{n}p_{s}x_{s}f(x_{s})- \sum_{s=1}^{n}p_{s}x_{s}f \Biggl(\sum_{s=1} ^{n}p_{s}x_{s} \Biggr) \leq \sum_{u=1}^{m}q_{u}y_{u}f(y_{u})- \sum_{u=1}^{m}q _{u}y_{u}f \Biggl(\sum_{u=1}^{m}q_{u}y_{u} \Biggr). \end{aligned}$$

Using the same steps as in the proof of (i), we get (36). □

Shannon entropy

Definition 3

(see [9])

The \(\mathcal{S}\)hannon entropy of positive probability distribution \(\tilde{\mathbf{r}}=(r_{1}, \ldots , r_{n})\) is defined by

$$\begin{aligned} \boldsymbol{\mathcal{S}} : = - \sum _{s=1}^{n}r_{s}\log (r_{s}). \end{aligned}$$
(39)

Corollary 6

Assume \(\boldsymbol{\mathcal{G}}\).

If \(\tilde{\mathbf{k}}=(k_{1}, \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), \(\tilde{\mathbf{t}}=(t_{1}, \ldots , t_{m}) \in \mathbb{R}_{+}^{m}\) and if base of log is greater than 1, then

$$\begin{aligned}& \frac{1}{\sum_{s=1}^{n}k_{s}} \Biggl[\boldsymbol{\mathcal{S}}+\sum _{s=1} ^{n}r_{s}\log (k_{s}) \Biggr]+ \Biggl[\sum_{s=1}^{n} \frac{r_{s}}{\sum_{s=1}^{n} k_{s}}\log \Biggl(\sum_{s=1}^{n} \frac{r_{s}}{\sum_{s=1} ^{n}k_{s}} \Biggr) \Biggr] \\& \quad \leq \frac{1}{\sum_{u=1}^{m}t_{u}} \Biggl[\tilde{\boldsymbol{\mathcal{S}}}+ \sum _{u=1}^{m}w_{u}\log (t_{u}) \Biggr]+ \Biggl[\sum_{u=1}^{m} \frac{w _{u}}{\sum_{u=1}^{m} t_{u}}\log \Biggl(\sum_{u=1}^{m} \frac{w_{u}}{ \sum_{u=1}^{m}t_{u}} \Biggr) \Biggr], \end{aligned}$$
(40)

where \(\boldsymbol{\mathcal{S}}\) is defined in (39), and

$$\begin{aligned} \tilde{\boldsymbol{\mathcal{S}}} : = - \sum_{u=1}^{m}w_{u} \log (w_{u}). \end{aligned}$$

If base of log is less than 1, then inequality (40) is reversed.

Proof

The function \(f \mapsto -x\log (x)\) is 3-convex for base of log is greater than 1. So, using \(f:= -x\log (x)\) in Theorem 6(i), we get (40). □

Remark 5

If k and t are positive probability distributions, then (40) becomes

$$\begin{aligned} \Biggl[\boldsymbol{\mathcal{S}}+\sum _{s=1}^{n}r_{s}\log (k_{s}) \Biggr] + \Biggl[\sum_{s=1}^{n}r_{s} \log \Biggl(\sum_{s=1}^{n}r_{s} \Biggr) \Biggr] \leq& \Biggl[\tilde{\boldsymbol{\mathcal{S}}}+\sum _{s=1}^{m}w_{s}\log (t _{s}) \Biggr] \\ &{}+ \Biggl[\sum_{s=1}^{m}w_{s} \log \Biggl(\sum_{s=1}^{m}w_{s} \Biggr) \Biggr]. \end{aligned}$$
(41)

Definition 4

(see [9])

For \(\tilde{\mathbf{r}}\) and \(\tilde{\mathbf{q}}\), where \(\tilde{\mathbf{r}}, \tilde{\mathbf{q}} \in \mathbb{R}_{+}^{n}\) the Kullback–Leibler divergence is defined by

$$\begin{aligned} \boldsymbol{\mathcal{D}}(\tilde{\mathbf{r}}, \tilde{ \mathbf{q}}) : = \sum_{s=1}^{n}r_{s} \log \biggl( \frac{r_{s}}{q_{s}} \biggr). \end{aligned}$$
(42)

Corollary 7

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}} = (r_{1} , \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1} , \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), and \(\tilde{\mathbf{w}} : = (w_{1}, \ldots , w_{m})\), \(\tilde{\mathbf{t}} = (t_{1} , \ldots , t_{m}) \in \mathbb{R}_{+}^{m}\) be such that \(\sum_{s=1}^{n}r_{s}\), \(\sum_{s=1}^{n}k_{s}\), \(\sum_{s=1} ^{m}w_{s}\), and \(\sum_{s=1}^{m}t_{s}\) be equal to 1, then

$$\begin{aligned} \sum_{s=1}^{n} \biggl( \frac{r_{s}}{k_{s}} \biggr) \boldsymbol{\mathcal{D}}(\tilde{\mathbf{r}}, \tilde{ \mathbf{k}})- \sum_{s=1}^{m} \biggl( \frac{w_{s}}{t_{s}} \biggr)\boldsymbol{\mathcal{D}}( \tilde{\mathbf{w}}, \tilde{ \mathbf{t}})\geq 0, \end{aligned}$$
(43)

where base of log is greater than 1.

If base of log is less than 1, then the signs of inequality in (43) are reversed.

Proof

In Theorem 6(ii), replacing f by \(-x\log (x)\), we have

$$\begin{aligned}& \frac{\sum_{s=1}^{n} (\frac{r_{s}}{k_{s}} )}{\sum_{s=1} ^{n}k_{s}}\boldsymbol{\mathcal{D}}(\tilde{ \mathbf{r}}, \tilde{\mathbf{k}}) - \sum_{s=1}^{n} \frac{r_{s}}{ \sum_{s=1}^{n}k _{s}} \log \Biggl(\sum_{s=1}^{n} \frac{r_{s}}{\sum_{s=1}^{n}k_{s}} \Biggr) \\& \quad \geq \frac{\sum_{s=1}^{m} (\frac{w_{s}}{t_{s}} )}{\sum_{s=1} ^{m}t_{s}}\boldsymbol{ \mathcal{D}}( \tilde{\mathbf{w}}, \tilde{\mathbf{t}}) - \sum_{s=1}^{m} \frac{w_{s}}{ \sum_{s=1}^{n}t_{s}}\log \Biggl(\sum_{s=1}^{m} \frac{w_{s}}{\sum_{s=1}^{m}t_{s}} \Biggr). \end{aligned}$$
(44)

Now simply taking \({\sum_{s=1}^{n}r_{s}}\), \({\sum_{s=1}^{n}k_{s}}\), \({\sum_{s=1}^{m}w_{s}}\), and \({\sum_{s=1}^{m}t_{s}}\) are equal to 1 and after rearranging, we get (43). □

Rényi divergence and entropy

The Rényi divergence and Rényi entropy are given in [19].

Definition 5

Let \(\tilde{\mathbf{r}}, \tilde{\mathbf{q}} \in \mathbb{R}_{+}^{n}\) be such that \(\sum_{1}^{n}r_{i}=1\) and \(\sum_{1}^{n}q_{i}=1\), and let \(\delta \geq 0\), \(\delta \neq 1\).

  1. (a)

    The Rényi divergence of order δ is defined by

    $$\begin{aligned} \boldsymbol{\mathcal{D}}_{\delta }(\tilde{\mathbf{r}}, \tilde{\mathbf{q}}) : = \frac{1}{\delta - 1} \log \Biggl(\sum _{i=1} ^{n}q_{i} \biggl( \frac{r_{i}}{q_{i}} \biggr)^{\delta } \Biggr). \end{aligned}$$
    (45)
  2. (b)

    The Rényi entropy of order δ of \(\tilde{\mathbf{r}}\) is defined by

    $$\begin{aligned} \boldsymbol{\mathcal{H}}_{\delta }(\tilde{\mathbf{r}}): = \frac{1}{1 - \delta } \log \Biggl( \sum_{i=1}^{n} r_{i}^{\delta } \Biggr). \end{aligned}$$
    (46)

These definitions also hold for nonnegative probability distributions. If \(\delta \rightarrow 1\) in (45), we have (42), and if \(\delta \rightarrow 1\) in (46), then we have (39).

Now we obtain inequalities for the Rényi divergence.

Theorem 7

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1}, \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), \(\tilde{\mathbf{w}} = (w_{1}, \ldots , w_{m})\), and \(\tilde{\mathbf{t}} = (t_{1}, \ldots , t_{m})\in \mathbb{R}_{+}^{m}\).

  1. (i)

    If base of log is greater than 1 and \(0 \leq \delta \leq \theta \) are such that \(\delta , \theta \neq 1\), then

    $$ \boldsymbol{\mathcal{D}}_{\theta }(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})-\boldsymbol{\mathcal{D}}_{\delta }( \tilde{\mathbf{r}}, \tilde{\mathbf{k}})\leq \boldsymbol{\mathcal{D}} _{\theta }(\tilde{ \mathbf{w}}, \tilde{\mathbf{t}})-\boldsymbol{\mathcal{D}}_{\delta }(\tilde{ \mathbf{w}}, \tilde{\mathbf{t}}). $$
    (47)

    If base of log is less than 1, then inequality (47) holds in reverse.

  2. (ii)

    If \(\theta >1\) and if base of log is greater than 1, then

    $$ \boldsymbol{\mathcal{D}}_{\theta }(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})-\boldsymbol{\mathcal{D}}_{1}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})\leq \boldsymbol{\mathcal{D}}_{\theta }( \tilde{ \mathbf{w}}, \tilde{\mathbf{t}})-\boldsymbol{\mathcal{D}}_{1}( \tilde{\mathbf{w}}, \tilde{\mathbf{t}}). $$
    (48)
  3. (iii)

    If \(\delta \in [0, 1)\) and if base of log is greater than 1, then

    $$ \boldsymbol{\mathcal{D}}_{1}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})-\boldsymbol{\mathcal{D}}_{\delta }(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})\leq \boldsymbol{\mathcal{D}}_{1}(\tilde{ \mathbf{w}}, \tilde{\mathbf{t}})-\boldsymbol{\mathcal{D}}_{\delta }(\tilde{ \mathbf{w}}, \tilde{\mathbf{t}}). $$
    (49)

Proof

With the mapping f defined by \(f: (0, \infty ) \rightarrow \mathbb{R}\) by \(f(t):= t^{\frac{\theta - 1}{\delta -1}}\) and using

$$ p_{s} : = r_{s}, \qquad x_{s} : = \biggl( \frac{r_{s}}{k_{s}} \biggr)^{\delta - 1}, \quad s = 1, \ldots , n, $$

and

$$ q_{u} : = w_{u}, \qquad y_{u} : = \biggl( \frac{w_{u}}{t_{u}} \biggr)^{\delta - 1}, \quad u = 1, \ldots , m, $$

in (18) (for positive weights) and after simplifications, we have

$$ \sum_{s=1}^{n}k_{s} \biggl(\frac{r_{s}}{k_{s}} \biggr)^{\theta }- \Biggl(\sum _{s=1}^{n}k _{s} \biggl( \frac{r_{s}}{k_{s}} \biggr)^{\delta } \Biggr)^{\frac{\theta -1}{\delta -1}} \leq \sum _{u=1}^{m}t_{u} \biggl( \frac{w_{u}}{t_{u}} \biggr)^{\theta }- \Biggl(\sum _{u=1} ^{m}t_{u} \biggl( \frac{w_{u}}{t_{u}} \biggr)^{\delta } \Biggr)^{ \frac{\theta -1}{\delta -1}} $$
(50)

if either \(0 \leq \delta < 1 < \gamma \) or \(1 < \delta \leq \theta \), and inequality (50) holds in reverse if \(0 \leq \delta \leq \gamma < 1\). Raising the power \(\frac{1}{\theta - 1}\) in (50),

$$ \begin{aligned}[b] &\Biggl(\sum_{s=1}^{n}k_{s} \biggl(\frac{r_{s}}{k_{s}} \biggr)^{\theta } \Biggr)^{\frac{1}{\theta -1}}- \Biggl(\sum_{s=1}^{n}k_{s} \biggl(\frac{r_{s}}{k_{s}} \biggr)^{\delta } \Biggr)^{\frac{1}{ \delta -1}} \\ &\quad \leq \Biggl(\sum_{u=1}^{m}t_{u} \biggl(\frac{w_{u}}{t_{u}} \biggr)^{\theta } \Biggr)^{\frac{1}{ \theta -1}}- \Biggl(\sum_{u=1}^{m}t_{u} \biggl(\frac{w_{u}}{t_{u}} \biggr)^{\delta } \Biggr)^{\frac{1}{ \delta -1}}. \end{aligned} $$
(51)

For base of log is greater than 1, the log function is increasing, therefore on taking log in (51), we get (47). If base of log is less than 1, inequality in (47) is reversed. If \(\delta = 1= \theta \), and by taking the limit, we have (48) and (49) respectively. □

Theorem 8

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1}, \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), \(\tilde{\mathbf{w}} = (w_{1}, \ldots , w_{m})\), and \(\tilde{\mathbf{t}} = (t_{1}, \ldots , t_{m}) \in \mathbb{R}_{+}^{m}\).

If either \(1 < \delta \) and base of log is greater than 1 or \(\delta \in [0, 1)\) and base of log is less than 1, then

$$\begin{aligned}& \frac{1}{\sum_{s=1}^{n}k_{s} (\frac{r_{s}}{k_{s}} )^{ \delta }} \sum_{s=1}^{n}k_{s} \biggl(\frac{r_{s}}{k_{s}} \biggr)^{ \delta } \log \biggl( \frac{r_{s}}{k_{s}} \biggr)-\boldsymbol{\mathcal{D}}_{\delta }(\tilde{ \mathbf{r}}, \tilde{\mathbf{k}}) \\& \quad \leq \frac{1}{\sum_{s=1}^{n}k_{s} (\frac{r_{s}}{k_{s}} ) ^{\delta }} \sum_{s=1}^{m}t_{s} \biggl(\frac{w_{s}}{t_{s}} \biggr) ^{\delta } \log \biggl( \frac{w_{s}}{t_{s}} \biggr)-\frac{\sum_{s=1} ^{m}t_{s} (\frac{w_{s}}{t_{s}} )^{\delta }}{\sum_{s=1} ^{n}k_{s} (\frac{r_{s}}{k_{s}} )^{\delta }} \boldsymbol{ \mathcal{D}}_{\delta }(\tilde{\mathbf{w}}, \tilde{\mathbf{t}}). \end{aligned}$$
(52)

If either \(1 < \delta \) and base of log is greater than 1 or \(\delta \in [0, 1)\) and base of log is less than 1, inequality in (52) is reversed.

Proof

The proof is only for the case when \(\delta \in [0, 1)\) and base of log is greater than 1, and similarly the remaining cases are simple to prove.

The function \(x \mapsto xf(x)\) \((x > 0)\) is 3-convex for base of log is less than 1. Also \(0>\frac{1}{1 - \delta }\) and choosing \(I = (0, \infty )\),

$$ p_{s} : = r_{s}, \qquad x_{s} : = \biggl( \frac{r_{s}}{k_{s}} \biggr)^{\delta - 1}, \quad s = 1, \ldots , n, $$

and

$$ q_{u} : = w_{u}, \qquad y_{u} : = \biggl( \frac{w_{u}}{t_{u}} \biggr)^{\delta - 1}, \quad u = 1, \ldots , m, $$

in (18) (for positive weights) and after simplifications, we have (52). □

Corollary 8

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1}, \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), \(\tilde{\mathbf{w}} = (w_{1}, \ldots , w_{m})\), and \(\tilde{\mathbf{t}} = (t_{1}, \ldots , t_{m}) \in \mathbb{R}_{+}^{m}\) be such that \({\sum_{s=1}^{n}r_{s}}\), \({\sum_{s=1}^{n}k_{s}}\), \({\sum_{u=1}^{m}w_{u}}\), and \({\sum_{u=1}^{m}t_{u}}\) are equal to 1.

  1. (i)

    If base of log is greater than 1 and \(0 \leq \delta \leq \theta \) such that \(\delta , \theta \neq 1\), then

    $$ \boldsymbol{\mathcal{H}}_{\theta }(\tilde{\mathbf{r}})- \boldsymbol{\mathcal{H}}_{\delta }(\tilde{\mathbf{r}})\geq \boldsymbol{ \mathcal{H}}_{\theta }(\tilde{\mathbf{w}})-\boldsymbol{ \mathcal{H}}_{\delta }(\tilde{\mathbf{w}}). $$
    (53)

    The reverse inequality holds in (53) if base of log is less than 1.

  2. (ii)

    If \(1 < \theta \) and base of log is greater than 1, then

    $$ \boldsymbol{\mathcal{H}}_{\theta }(\tilde{\mathbf{r}})- \boldsymbol{\mathcal{S}} \geq \boldsymbol{\mathcal{H}}_{\theta }( \tilde{ \mathbf{w}})-\tilde{\boldsymbol{\mathcal{S}}}. $$
    (54)

    The reverse inequality holds in (54) if base of log is greater than 1.

  3. (iii)

    If \(0 \leq \delta < 1\) and base of log is greater than 1, then

    $$ \boldsymbol{\mathcal{S}}-\boldsymbol{\mathcal{H}}_{\delta }( \tilde{\mathbf{r}})\geq \tilde{\boldsymbol{\mathcal{S}}}-\boldsymbol{ \mathcal{H}}_{\delta }(\tilde{\mathbf{w}}). $$
    (55)

    If base of log is less than 1, the inequality in (55) is reversed.

Proof

(i) Suppose \(\tilde{\mathbf{k}}, \tilde{\mathbf{t}}= \frac{ \textbf{1}}{\textbf{n}}\). Then from (45) we have

$$\begin{aligned} \boldsymbol{\mathcal{D}}_{\delta } (\tilde{\mathbf{r}}, \tilde{ \mathbf{q}}) = \frac{1}{\delta - 1} \log \Biggl(\sum _{s=1}^{n}n ^{\delta - 1}r_{s}^{\delta } \Biggr) = \log (n) + \frac{1}{\delta - 1}\log \Biggl(\sum _{s=1}^{n}r_{s}^{\delta } \Biggr) \end{aligned}$$

and

$$\begin{aligned} \boldsymbol{\mathcal{D}}_{\delta } (\tilde{\mathbf{w}}, \tilde{ \mathbf{t}}) = \frac{1}{\delta - 1} \log \Biggl(\sum _{s=1}^{n}n ^{\delta - 1}w_{s}^{\delta } \Biggr) = \log (n) + \frac{1}{\delta - 1}\log \Biggl(\sum _{s=1}^{n}w_{s}^{\delta } \Biggr). \end{aligned}$$

We have

$$\begin{aligned} \boldsymbol{\mathcal{H}}_{\delta }(\tilde{\mathbf{r}}) = \log (n) - \boldsymbol{\mathcal{D}}_{\delta } \biggl(\tilde{\mathbf{r}}, \frac{\textbf{1}}{ \textbf{n}} \biggr) \end{aligned}$$
(56)

and

$$\begin{aligned} \boldsymbol{\mathcal{H}}_{\delta }(\tilde{\mathbf{w}}) = \log (n) - \boldsymbol{\mathcal{D}}_{\delta } \biggl(\tilde{\mathbf{w}}, \frac{\textbf{1}}{ \textbf{n}} \biggr). \end{aligned}$$
(57)

We get (53) after using Theorem 7(i), (56) and (57).

Statements (ii) and (iii) are similarly proved. □

Corollary 9

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1}, \ldots , k_{n})\), \(\tilde{\mathbf{w}} = (w _{1}, \ldots , w_{m})\), and \(\tilde{\mathbf{t}} = (t_{1}, \ldots , t _{n})\) be positive probability distributions.

If either \(\delta \in [0, 1)\) and base if log is greater than 1, or \(\delta >1\) and base if log is less than 1, then

$$ -\frac{1}{\sum_{s=1}^{n}r_{s}^{\delta }} \sum_{s=1}^{n}r_{s}^{\delta } \log (r_{s})-\boldsymbol{\mathcal{H}}_{\delta }(r) \geq \frac{1}{\sum_{s=1}^{m}w_{s}^{\delta }}{\sum_{s=1}^{m}w_{s}^{\delta }} \log (w_{s}) -\frac{\sum_{s=1}^{m}w_{s}^{\delta }}{\sum_{s=1}^{n}r_{s}^{\delta }}\boldsymbol{\mathcal{H}}_{\delta }(w). $$
(58)

The inequality in (58) is reversed if either \(\delta \in [0, 1)\) and base if log is less than 1, or \(\delta > 1\) and the base of log is greater than 1.

Proof

Proof is similar to Corollary 8 □

Zipf–Mandelbrot law

In [14] the authors gave some contribution in analyzing the Zipf–Mandelbrot law which is defined as follows:

Definition 6

The Zipf–Mandelbrot law is a discrete probability distribution depending on three parameters: \(\mathcal{N} \in \{1, 2, \ldots , \}\), \(\phi \in [0, \infty )\), and \(t > 0\), and is defined by

$$\begin{aligned} f(s; \mathcal{N}, \phi , t) : = \frac{1}{(s + \phi )^{t}\boldsymbol{\mathcal{H}}_{\mathcal{N}, \phi , t}}, \quad s = 1, \ldots , \mathcal{N}, \end{aligned}$$

where

$$\begin{aligned} \boldsymbol{\mathcal{H}}_{\mathcal{N}, \phi , t} = \sum_{\nu =1}^{\mathcal{N}} \frac{1}{(\nu + \phi )^{t}}. \end{aligned}$$

For all values of \(\mathcal{N}\), if the total mass of the law is taken, then for \(0 \leq \phi \), \(1< t\), \(s \in \mathcal{N}\), the density function of the Zipf–Mandelbrot law becomes

$$\begin{aligned} f(s; \phi , t) = \frac{1}{(s + \phi )^{t}\boldsymbol{\mathcal{H}}_{ \phi , t}}, \end{aligned}$$

where

$$\begin{aligned} \boldsymbol{\mathcal{H}}_{\phi , t} = \sum_{\nu =1}^{\infty } \frac{1}{( \nu + \phi )^{t}}. \end{aligned}$$

For \(\phi = 0\), the Zipf–Mandelbrot law becomes Zipf’s law.

Conclusion 1

Assume \(\boldsymbol{\mathcal{G}}\).

Let \(\tilde{\mathbf{r}}\) and \(\tilde{\mathbf{w}}\) be the Zipf–Mandelbrot laws. By Corollary 8(iii). If \(\delta \in [0, 1)\) and base of log is greater than 1, then

$$\begin{aligned} \boldsymbol{\mathcal{S}} =&-\sum_{s=1}^{n} \frac{1}{(s+k)^{s}{\boldsymbol{\mathcal{H}}_{\mathcal{N}, k, v}}}\log \biggl(\frac{1}{(s+k)^{s}{\boldsymbol{\mathcal{H}}_{\mathcal{N}, k, v}}} \biggr)- \frac{1}{1 - \delta } \log \Biggl(\frac{1}{\boldsymbol{\mathcal{H}}_{\mathcal{N}, k, v}^{\delta }}\sum _{s=1}^{n}\frac{1}{(s + k)^{\delta s}} \Biggr) \\ \geq &\tilde{\boldsymbol{\mathcal{S}}}\\ =& -\sum_{s=1}^{m} \frac{1}{(s+w)^{s} {\boldsymbol{\mathcal{H}}_{\mathcal{N}, w, v}}}\log \biggl(\frac{1}{(s+w)^{s} {\boldsymbol{\mathcal{H}}_{\mathcal{N}, w, v}}} \biggr)- \frac{1}{1 - \delta } \log \Biggl(\frac{1}{\boldsymbol{\mathcal{H}}_{\mathcal{N}, w, v}^{ \delta }}\sum _{s=1}^{m}\frac{1}{(s + w)^{\delta s}} \Biggr). \end{aligned}$$

The inequality is reversed if base of log is less than 1.

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Acknowledgements

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. The research of the 4th author is supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008).

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Correspondence to Muhammad Adeel.

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Adeel, M., Khan, K.A., Pečarić, Ð. et al. Generalization of the Levinson inequality with applications to information theory. J Inequal Appl 2019, 212 (2019) doi:10.1186/s13660-019-2166-8

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Keywords

  • Levinson’s inequality
  • Information theory