1 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
where
Working with the various differences, the assumptions of differentiability on
f
can be weakened.
$$ J_{1}(f) \geq 0, $$
(1)
$$\begin{aligned} J_{1}\bigl(f(\cdot)\bigr) =&\frac{1}{\mathbf{P}_{n}} \sum_{\rho =1}^{n}p_{\rho }f(2 \alpha -x_{\rho })-f\Biggl(\frac{1}{\mathbf{P}_{n}}\sum _{\rho =1}^{n}p_{ \rho }(2\alpha - x_{\rho })\Biggr)-\frac{1}{\mathbf{P} _{n}}\sum _{\rho =1} ^{n}p_{\rho }f(x_{\rho }) \\ &{}+ f\Biggl(\frac{1}{\mathbf{P}_{n}}\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr). \end{aligned}$$
(2)
Anzeige
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
and
\(p_{n}>0\). Then
where
$$ \max \{x_{1} \ldots x_{k}\} \leq \min \{y_{1} \ldots y_{k}\}, \qquad x_{1}+y_{1}= \cdots =x_{k}+y_{k} , $$
(3)
$$ J_{2}(f) \geq 0, $$
(4)
$$\begin{aligned} J_{2}\bigl(f(\cdot)\bigr) =&\frac{1}{\mathbf{P}_{k}} \sum_{\rho =1}^{k}p_{\rho }f(y _{\rho })-f\Biggl(\frac{1}{\mathbf{P}_{k}}\sum _{\rho =1}^{k}p_{\rho }y_{ \rho } \Biggr)- \frac{1}{\mathbf{P}_{k}}\sum_{\rho =1}^{k} p_{\rho }f(x_{ \rho }) \\ &{}+ f\Biggl(\frac{1}{\mathbf{P}_{k}}\sum_{\rho =1}^{k}p_{\rho }x_{ \rho } \Biggr). \end{aligned}$$
(5)
In [17], Pečarić weakened the 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.
Anzeige
In [15], Mercer presented 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
Then (1) holds.
$$ \sum_{\rho =1}^{n}p_{\rho } \Biggl(x_{\rho }-\sum_{\rho =1}^{n}p_{ \rho }x_{\rho } \Biggr)^{2}=\sum_{\rho =1}^{n}p_{\rho } \Biggl(y_{ \rho }- \sum_{\rho =1}^{n}p_{\rho }y_{\rho } \Biggr)^{2}. $$
(6)
On the other hand the error function \(e_{\mathcal{F}}(t)\) can be represented in terms of the Green function \(G_{\mathcal{F}, n}(t, s)\) of the boundary value problem
where
The following result holds in [1].
$$\begin{aligned}& z^{(n)}(t)=0, \\& z^{(i)}(a_{1}) = 0,\quad 0 \leq i \leq p, \\& z^{(i)}(a_{2}) = 0,\quad p+1 \leq i \leq n-1, \\& e_{F}(t)= \int ^{a_{2}}_{a_{1}}G_{F, n}(t, s)f^{(n)}(s)\,ds, \quad t \in [a, b], \end{aligned}$$
(7)
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
\(0 \leq p \leq n-2\),
where
\(G_{F, n}(t, s)\)
is the Green function, defined by (7).
$$\begin{aligned} f(t) =&P_{F}(t)+e_{F}(t) \\ =& \sum_{i=0}^{p}\frac{(t-a_{1})^{i}}{i!}f^{(i)}(a_{1}) \\ &{}+ \sum_{j=0}^{n-p-2} \Biggl(\sum _{i=0}^{j}\frac{(t-a_{1})^{p+1+i}(a _{1}-a_{2})^{j-i}}{(p+1+i)!(j-i)!} \Biggr)f^{(p+1+j)}(a_{2} ) \\ &{}+ \int ^{a_{2}}_{a_{1}}G_{F, n}(t, s)f^{(n)}(s)\,ds, \end{aligned}$$
(8)
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
where
For \(n=3\) and \(p=1\), (8) becomes
where
The presented work is organized as follows: in Sect. 2, Levinson’s inequality for the 3-convex function is generalized by using two Green 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 the f-divergence, the Rényi divergence, the Rényi entropy, the Shannon entropy and the Zipf–Mandelbrot law.
$$\begin{aligned} f(t) =& f(a_{1}) + (t-a_{1})f^{(1)}(a_{2})+ (t-a_{1}) (a_{1}-a_{2})f ^{(2)}(a_{2})+\frac{(t-a_{1})^{2}}{2}f^{(2)}(a_{2}) \\ &{}+ \int ^{a_{2}}_{a_{1}}G_{1}(t, s)f^{(3)}(s)\,ds, \end{aligned}$$
(9)
$$\begin{aligned} G_{1}(t, s) = \textstyle\begin{cases} (a_{1}-s)^{2}, & a_{1} \leq s \leq t; \\ - (t-a_{1})(a_{1}-s)+\frac{1}{2}(t-a_{1})^{2}, & t \leq s \leq a_{2}. \end{cases}\displaystyle \end{aligned}$$
(10)
$$\begin{aligned} f(t) = f(a_{1}) + (t-a_{1})f^{(1)}(a_{2})+ \frac{(t-a_{1})^{2}}{2}f ^{(2)}(a_{2})+ \int ^{a_{2}}_{a_{1}}G_{2}(t, s)f^{(3)}(s)\,ds, \end{aligned}$$
(11)
$$\begin{aligned} G_{2}(t, s) = \textstyle\begin{cases} \frac{1}{2}(a_{1}-s)^{2}+(t-a_{1})(a_{1}-s), & a_{1} \leq s \leq t; \\ - \frac{1}{2} (t-a_{1})^{2}, & t \leq s \leq a_{2}. \end{cases}\displaystyle \end{aligned}$$
(12)
2 Main results
First we give an identity involving Jensen’s difference of two different data points. Then we give equivalent form of identity by using Green 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
where
and
for
\(G_{k}(\cdot , s)\) (\(k=1, 2\)) defined in (10) and (12), respectively.
$$\begin{aligned} J\bigl(f(\cdot)\bigr) =&\frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y_{ \varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n}p_{\rho } x_{\rho }^{2}+\Biggl(\sum_{\rho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \\ &{}+ \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds, \end{aligned}$$
(13)
$$ J\bigl(f(\cdot)\bigr)=\sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })-f \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)-\sum_{\rho =1}^{n}p _{\rho }f(x_{\rho }) +f\Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) $$
(14)
$$ J\bigl(G_{k}(\cdot , s)\bigr)= \sum _{\varrho =1}^{m}q_{\varrho }G_{k}(y_{\varrho }, s)-G _{k}\Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }, s\Biggr)-\sum_{\rho =1} ^{n}p_{\rho }G_{k}(x_{\rho }, s) +G_{k}\Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho }, s\Biggr), $$
(15)
Proof
(i) For \(k=1\).
Using (9) in (14), we have
$$\begin{aligned}& J\bigl(f(\cdot)\bigr)=\sum_{\varrho =1}^{m}q_{\varrho } \bigg[f(\zeta _{1})+(y_{ \varrho }-\zeta _{1})f^{(1)}(\zeta _{2})+(y_{\varrho }- \zeta _{1}) (\zeta _{1}-\zeta _{2})f^{(2)} (\zeta _{2}) \\& \hphantom{J(f(\cdot))=}{}+\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\bigg] \\& \hphantom{J(f(\cdot))=}{}-\Bigg[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}) \\& \hphantom{J(f(\cdot))=}{}+\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\Bigg] \\& \hphantom{J(f(\cdot))=}{}-\sum_{\rho =1}^{n}p_{\rho } \bigg[f(\zeta _{1})+(x_{\rho }-\zeta _{1})f^{(1)}(\zeta _{2})+(x_{\rho }- \zeta _{1}) (\zeta _{1}-\zeta _{2})f ^{(2)} (\zeta _{2}) \\& \hphantom{J(f(\cdot))=}{}+\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\bigg] \\& \hphantom{J(f(\cdot))=}{}+\Bigg[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}) \\& \hphantom{J(f(\cdot))=}{}+\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\Bigg], \\& 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}) \\& \hphantom{J(f(\cdot))=}{}+\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 \\& \hphantom{J(f(\cdot))=}{}-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}) \\& \hphantom{J(f(\cdot))=}{}-\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} \\& \hphantom{J(f(\cdot))=}{}- \int _{\zeta _{1}}^{\zeta _{2}}G_{1}\Biggl(\sum _{\varrho =1} ^{m}q_{\varrho }y_{\varrho }, s\Biggr)f^{(3)}(s)\,ds \\& \hphantom{J(f(\cdot))=}{}-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}) \\& \hphantom{J(f(\cdot))=}{}-\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 \\& \hphantom{J(f(\cdot))=}{}+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}) \\& \hphantom{J(f(\cdot))=}{}+\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} \\& \hphantom{J(f(\cdot))=}{} + \int _{\zeta _{1}}^{\zeta _{2}}G_{1}\Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{ \rho }, s\Biggr)f^{(3)}(s)\,ds, \\& 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}) \\& \hphantom{J(f(\cdot))=}{}+\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 \\& \hphantom{J(f(\cdot))=}{}-\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}$$
After rearranging, we have (13).
(ii) For \(k=2\).
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
where
\(J(f(\cdot))\)
and
\(J(G(\cdot , s))\)
are defined in (14) and (15), respectively.
$$\begin{aligned} J\bigl(f(\cdot)\bigr) =& \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds, \quad 0\leq \zeta _{1}< \zeta _{2} \leq 2\alpha , \end{aligned}$$
(16)
Proof
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
then the following statements are equivalent:
$$ \Biggl[\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }^{2}- \Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1}^{n}p _{\rho }x_{\rho }^{2}+\Biggl(\sum _{\rho =1}^{n} p_{\rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \geq 0, $$
(17)
For
\(f \in C^{3}[\zeta _{1}, \zeta _{2}]\)
For all
\(s \in I\)
where
\(G_{k}(\cdot , s)\)
are defined by (10) and (12) for
\(k=1, 2\), respectively.
$$ \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })-f \Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho }\Biggr)\leq \sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })- f\Biggl( \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr). $$
(18)
$$\begin{aligned} \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s)-G_{k}\Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s\Biggr)\leq \sum _{\varrho =1}^{m}q_{\varrho }G_{k}(y _{\varrho }, s)- G_{k}\Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }, s\Biggr), \end{aligned}$$
(19)
Proof
(18) ⇒ (19): Let (18) is 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) holds. Then we can represent the function f in the form (9). Now by means of some simple calculations we can write
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. □
$$\begin{aligned}& \sum_{\varrho =1}^{m}q_{\varrho }f(y_{\varrho })-f \Biggl(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \Biggr)- \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })+ f\Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) \\& \quad = \frac{1}{2} \Biggl[\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } ^{2}-\Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }^{2}+ \Biggl(\sum _{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \\& \qquad {}+ \int _{\zeta _{1}}^{\zeta _{2}}\left ( \sum _{\varrho =1}^{m}q_{\varrho }G _{k}(y_{\varrho }, s)\right . \\& \qquad {}-\left .G_{k}\Biggl(\sum_{\varrho =1}^{m}q_{\varrho }(y_{\varrho }, s)\Biggr) - \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s)+G_{k}\Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s\Biggr)\right )f^{(3)}(s)\,ds. \end{aligned}$$
Remark 1
If the expression
and \(f^{(2)}(\zeta _{2})\) have different signs in (17) then inequalities (18) and (19) are reversed.
$$ \sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho }^{2}- \Biggl(\sum_{\varrho =1} ^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}-\sum_{\rho =1}^{n}p_{\rho }x_{ \rho }^{2}+ \Biggl(\sum_{\rho =1}^{n}p_{\rho }x_{\rho } \Biggr) ^{2} $$
Next we have results about generalization of Bullen’s type inequality (for real weights) given in [2] (see also [11, 16]).
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
and
Also let
\((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m})\in \mathbb{R}^{m}\)
such that
\(\sum_{\rho =1}^{n}p_{ \rho }=1\)
and
\(\sum_{\varrho =1}^{m}q_{\varrho }=1\)
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.
$$ \max \{x_{1}, \ldots , x_{n}\} \leq \min \{y_{1}, \ldots , y_{m}\} $$
(20)
$$ x_{1}+y_{1}= \cdots =x_{n}+y_{m}. $$
(21)
Proof
Remark 2
Next we have generalized form (for real weights) of Bullen’s 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
\(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), \((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\)
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
\(y_{1}, \ldots , y_{m} \in I\)
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 generalized form of Bullen’s inequality defined in [16, p. 32, Theorem 4].
In [15], Mercer made a notable work by replacing the 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 \(\varrho =1, \ldots , m\).
Corollary 4
Let
\(f: I= [\zeta _{1}, \zeta _{1}] \rightarrow \mathbb{R}\)
be
3-convex function and
\(f \in C^{3}[\zeta _{1}, \zeta _{2}]\), \(p_{\rho }\), \(q_{\rho }\)
are 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
If (17) holds, then (18) and (19) are equivalent.
$$ \sum_{\rho =1}^{n}p_{\rho } \Biggl(x_{\rho }-\sum_{\rho =1}^{n}p_{ \rho }x_{\rho } \Biggr)^{2}=\sum_{\rho =1}^{n}p_{\rho } \Biggl(y_{ \rho }-\sum_{\rho =1} ^{n}q_{\rho }y_{\rho } \Biggr)^{2}. $$
(22)
Next we have results that lean on the generalization of Levinson’s 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:
For all
\(s \in I\)
where
\(G_{k}(\cdot , s)\)
is defined in (10) and (12) for
\(k=1, 2\), respectively.
$$ \sum_{\rho =1}^{n}p_{\rho }f(x_{\rho })-f \Biggl(\sum_{\rho =1}^{n}p_{\rho }x _{\rho }\Biggr)\leq \sum_{\rho =1}^{n}p_{\rho }f(2 \alpha -x_{\rho })- f\Biggl( \sum_{\rho =1}^{n}p_{\rho }(2 \alpha -x_{\rho })\Biggr). $$
(23)
$$\begin{aligned} \sum_{\rho =1}^{n}p_{\rho }G_{k}(x_{\rho }, s) -G_{k}\Biggl(\sum_{\rho =1} ^{n}p_{\rho }x_{\rho }, s\Biggr) \leq& \sum _{\rho =1}^{n}p_{\rho }G_{k}(2 \alpha -x_{\rho }, s) \\ &{}-G_{k}\Biggl(\sum_{\rho =1}^{n}p_{\rho }(2 \alpha -x_{\rho }), s\Biggr), \end{aligned}$$
(24)
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 required result. □
3 New bounds for Levinson’s type functionals
Consider the Čebyšev functional for two Lebesgue integrable functions \(f_{1}, f_{2}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R,}\)
where the integrals are assumed to exist.
$$\begin{aligned} \varTheta (f_{1}, f_{2}) =& \frac{1}{\zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}} _{\zeta _{1}}f_{1}(x)f_{2}(x)\,dx \\ &{}- \frac{1}{\zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}}_{\zeta _{1}}f_{1}(x)\,dx\cdot \frac{1}{ \zeta _{2}-\zeta _{1}} \int ^{\zeta _{2}}_{\zeta _{1}}f_{2}(x)\,dx, \end{aligned}$$
(25)
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
\(\frac{1}{\sqrt{2}}\)
is the best possible.
$$\begin{aligned} \bigl\vert \varTheta (f_{1}, f_{2}) \bigr\vert \leq \frac{1}{\sqrt{2}}\bigl[\varTheta (f_{1}, f _{1})\bigr]^{\frac{1}{2}}\frac{1}{\sqrt{\zeta _{2}-\zeta _{1}}} \biggl( \int ^{ \zeta _{2}}_{\zeta _{1}} (t-\zeta _{1}) ( \zeta _{2}-t)\bigl[f'_{2}(t) \bigr]^{2}\,dt \biggr) ^{\frac{1}{2}}. \end{aligned}$$
(26)
Theorem G
([3])
Let
\(f_{1}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)
be an absolutely continuous with
\(f^{\prime }_{1} \in L_{\infty }[\zeta _{1}, \zeta _{2}]\)
and let
\(f_{2}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)
is monotonic non-decreasing on
\([\zeta _{1}, \zeta _{2}]\). Then
\(\frac{1}{2}\)
is the best possible.
$$\begin{aligned} \bigl\vert \varTheta (f_{1}, f_{2}) \bigr\vert \leq \frac{1}{2(\zeta _{2}-\zeta _{1})} \bigl\Vert f^{\prime } \bigr\Vert _{ \infty } \int ^{\zeta _{2}}_{\zeta _{1}} (t-\zeta _{1}) ( \zeta _{2}-t)\bigl[f'_{2}(t) \bigr]^{2}\,df _{2}(t). \end{aligned}$$
(27)
In the next result we construct the Čebyšev-type bound for our functional defined in (5).
Theorem 3
Let
\(f\in C^{3}[\zeta _{1}, \zeta _{2}]\)
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
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
for
\(G_{k}(\cdot , s)\) (\(k=1, 2\)) defined in (10) and (12), respectively.
$$\begin{aligned} J\bigl(f(\cdot)\bigr) =&\frac{1}{2} \Biggl[\sum _{\varrho =1}^{m}q_{\varrho }y_{ \varrho }^{2}- \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n}p_{\rho }x_{\rho }^{2} +\Biggl(\sum_{\varrho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr] f^{(2)}(\zeta _{2}) \\ &{}+\frac{f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2}-\zeta _{1})} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds + \mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f), \end{aligned}$$
(28)
$$\begin{aligned} \bigl\vert \mathcal{R}_{3}(\zeta _{1}, \zeta _{2}; f) \bigr\vert \leq& \frac{\zeta _{2}-\zeta _{1}}{\sqrt{2}} \bigl[\varTheta \bigl(J\bigl(G_{k}(\cdot , s) \bigr),J\bigl(G_{k}(\cdot , s)\bigr)\bigr) \bigr] ^{\frac{1}{2}}\times \\ &{} \frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s)\bigl[f^{(4)}(s)\bigr]^{2}\,ds \biggr)^{ \frac{1}{2}}, \end{aligned}$$
(29)
Proof
Setting \(f_{1} \mapsto J(G_{k}(\cdot , s))\) and \(f_{2} \mapsto f^{(3)}\) in Theorem F, we get
Multiplying \((\zeta _{2} - \zeta _{1})\) on both sides of the above inequality and using the estimation (29), we get
Using the identity (13), we get (28). □
$$\begin{aligned}& \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G _{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds - \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)\,ds\cdot \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f^{(3)}(s)\,ds \biggr\vert \\& \quad \leq \frac{1}{\sqrt{2}}\bigl[\varTheta \bigl(J\bigl(G_{k}( \cdot , s)\bigr), J\bigl(G_{k}(\cdot , s)\bigr)\bigr) \bigr]^{ \frac{1}{2}}\frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s)\bigl[f^{(4)}(s)\bigr]^{2}\,ds \biggr) ^{\frac{1}{2}}, \\& \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G _{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds - \frac{f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2} - \zeta _{1})^{2}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G _{k}(\cdot , s)\bigr)\,ds \biggr\vert \\& \quad \leq \frac{1}{\sqrt{2}}\bigl[\varTheta \bigl(J\bigl(G_{k}( \cdot , s)\bigr), J\bigl(G_{k}(\cdot , s)\bigr)\bigr) \bigr]^{ \frac{1}{2}}\frac{1}{\sqrt{\zeta _{2} - \zeta _{1}}} \biggl( \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s)\bigl[f^{(4)}(s)\bigr]^{2}\,ds \biggr) ^{\frac{1}{2}}. \end{aligned}$$
$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)f^{(3)}\,ds = \frac{f^{(2)}( \zeta _{2}) - f^{(2)}(\zeta _{1})}{(\zeta _{2} - \zeta _{1})} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)\,ds + \mathcal{R}_{3}(\zeta _{1}, \zeta _{1}; f). \end{aligned}$$
In the next result bounds of Grüss-type inequalities are estimated.
Theorem 4
Let
\(f\in C^{3}[\zeta _{1}, \zeta _{2}]\)
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
Since
using (13), (31) and (32), we have (28). □
$$\begin{aligned}& \biggl\vert \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G _{k}(\cdot , s)\bigr))f^{(3)}(s)\,ds - \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)\,ds\cdot \frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f^{(3)}(s)\,ds \biggr\vert \\& \quad \leq \frac{1}{2} \bigl\Vert J\bigl(G_{k}(\cdot , s)\bigr)^{\prime } \bigr\Vert _{\infty }\frac{1}{\zeta _{2} - \zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s)\bigl[f ^{(4)}(s)\bigr]^{2}\,ds. \end{aligned}$$
(31)
$$\begin{aligned}& \int _{\zeta _{1}}^{\zeta _{2}}(s - \zeta _{1}) ( \zeta _{2} - s)\bigl[f^{(4)}(s)\bigr]^{2}\,ds \\& \quad = \int _{\zeta _{1}}^{\zeta _{2}}[2s - \zeta _{1} - \zeta _{2}]f^{3}(s)\,ds \\& \quad = (\zeta _{2} - \zeta _{1})\bigl[f^{(2)}( \zeta _{2}) + f^{(2)}(\zeta _{1})\bigr] - 2 \bigl(f^{(2)}(\zeta _{2}) - f^{(2)}(\zeta _{1})\bigr), \end{aligned}$$
(32)
Ostrowski-type bounds for newly constructed functional defined in (5).
Theorem 5
Let
\(f\in C^{3}[\zeta _{1}, \zeta _{2}]\)
such that
\(f: I= [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)
and
\(f^{(2)}(\cdot)\)
is absolutely continuous. Also let
\((p_{1}, \ldots , p_{n}) \in \mathbb{R}^{n}\), \((q_{1}, \ldots , q_{m}) \in \mathbb{R}^{m}\)
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
\((r, s)\)
is a pair of conjugate exponents that is
\(1 \leq r, s, \leq \infty \), \(\frac{1}{r}+ \frac{1}{s}=1\). If
\(|f^{(3)}|^{r}: [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)
be 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 such a way
Employing the classical Hölder inequality for the R.H.S. of (34) yields (33). □
$$\begin{aligned}& \Biggl\vert J\bigl(f(\cdot)\bigr) -\frac{1}{2} \Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y _{\varrho }^{2}-\Biggl(\sum_{\varrho =1}^{m}q_{\varrho }y_{\varrho } \Biggr)^{2}- \sum_{\rho =1}^{n} p_{\rho } x_{\rho }^{2}+\Biggl(\sum _{\rho =1}^{m}p_{ \rho }x_{\rho } \Biggr)^{2} \Biggr)f^{(2)}(\zeta _{2}) \Biggr\vert \\& \quad \leq \int _{\zeta _{1}}^{\zeta _{2}}J\bigl(G_{k}(\cdot , s)\bigr)f^{(3)}(s)\,ds. \end{aligned}$$
(34)
4 Application to information theory
The idea of the Shannon entropy is the focal point of data hypothesis, now and then alluded to as the 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. The Shannon entropy permits one 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 become familiar with a measure of the difference between them. A variety of sorts of divergences exist, for instance the f-difference (particularly, the Kullback–Leibler divergence, the Hellinger distance and the total variation distance), the Rényi divergence, the 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. The Jensen inequality assumes a crucial role in a part of these inequalities. In any case, Jensen’s inequality deals with one sort of information focus and Levinson’s inequality manages two type information points.
The Zipf law is one of the central laws in data science, and it has been utilized in linguistics. 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 a steady \((C): C = r \times f\). Aside from the utilization of this law in data science and linguistics, the Zipf law is utilized in city population, sun powered flare power, site traffic, earthquack 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].
4.1 Csiszár divergence
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 the 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:
he stated that nonnegative probability distributions can also be used.
$$\begin{aligned} f(0) := \lim_{x \rightarrow 0^{+}}f(x); \qquad 0f \biggl( \frac{0}{0} \biggr):=0; \qquad 0f \biggl(\frac{a}{0} \biggr):= \lim_{x \rightarrow 0^{+}}xf \biggl(\frac{a}{0} \biggr), \quad a>0, \end{aligned}$$
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
Then
$$\begin{aligned} \frac{r_{s}}{k_{s}} \in I, \quad s= 1, \ldots , n. \end{aligned}$$
$$\begin{aligned} \hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{\mathbf{k}}) : = \sum _{s=1} ^{n}k_{s}f \biggl( \frac{r_{s}}{k_{s}} \biggr). \end{aligned}$$
Theorem 6
Let
\(\tilde{\mathbf{r}}= (r_{1}, \ldots , r_{n} )\), \(\tilde{\mathbf{k}}= (k_{1}, \ldots , k_{n} )\)
be in
\((0, \infty )^{n}\)
and
\(\tilde{\mathbf{w}}= (w_{1}, \ldots , w _{m} )\), \(\tilde{\mathbf{t}}= (t_{1}, \ldots , t_{m} )\)
are in
\((0, \infty )^{m}\)
such that
and
If
then the following are equivalent. where
$$\begin{aligned} \frac{r_{s}}{k_{s}} \in I, \quad s = 1, \ldots , n, \end{aligned}$$
$$\begin{aligned} \frac{w_{u}}{t_{u}} \in I, \quad u = 1, \ldots , m. \end{aligned}$$
$$\begin{aligned}& \Biggl[\frac{1}{\sum_{u=1}^{m}t_{u}}\sum_{u=1}^{m} \frac{(w_{u})^{2}}{t _{u}}- \Biggl(\sum_{u=1}^{m} \frac{w_{u}}{\sum_{u=1}^{m}t_{u}} \Biggr)^{2} -\frac{1}{\sum_{v=1}^{n}k_{v}}\sum _{v=1}^{n}\frac{(r_{v})^{2}}{k_{v}} \\& \quad {}+ \Biggl(\sum_{v=1}^{n} \frac{r_{v}}{\sum_{v=1}^{n}k_{v}} \Biggr)^{2} \Biggr]f^{(2)}(\zeta _{2})\geq 0, \end{aligned}$$
(35)
(i)
For every continuous 3-convex function
\(f: I \rightarrow \mathbb{R}\),
$$ J_{\hat{f}}(r, w, k, t)\geq 0. $$
(36)
(ii)
$$ J_{G_{k}}(r, w, k, t) \geq 0, $$
(37)
$$\begin{aligned} J_{\hat{f}}(r, w, k, t) =&\frac{1}{\sum_{u=1}^{m}t_{u}} \hat{I}_{f}( \tilde{\mathbf{w}}, \tilde{\mathbf{t}})- f \Biggl(\sum _{u=1}^{m}\frac{w _{u}}{\sum_{u=1}^{m}t_{u}} \Biggr) -\frac{1}{\sum_{v=1}^{n}k_{v}} \hat{I}_{f}(\tilde{\mathbf{r}}, \tilde{ \mathbf{k}}) \\ &{}+f \Biggl(\sum_{v=1}^{n} \frac{r_{v}}{\sum_{v=1}^{n}k_{v}} \Biggr). \end{aligned}$$
(38)
Proof
Using \(p_{v} = \frac{k_{v}}{\sum_{v=1}^{n}k_{v}}\), \(x_{v} = \frac{r _{v}}{k_{v}}\), \(q_{u} = \frac{t_{u}}{\sum_{u=1}^{m}t_{u}}\) and \(y_{u} = \frac{w_{u}}{t_{u}}\) in Theorem 2, we get the required results. □
4.2 Shannon entropy
Definition 3
(See [9])
The \(\mathcal{S}\)hannon entropy of a positive probability distribution \(\tilde{\mathbf{r}}=(r_{1}, \ldots , r_{n})\) is defined by
$$\begin{aligned} \mathcal{S} : = - \sum_{v=1}^{n}r_{v} \log (r_{v}). \end{aligned}$$
(39)
Corollary 6
Let
\(\tilde{\mathbf{r}}= (r_{1}, \ldots , r_{n} )\)
and
\(\tilde{\mathbf{w}}= (w_{1}, \ldots , w_{m} )\)
be probability distributions, \(\tilde{\mathbf{k}}= (k_{1}, \ldots , k_{n} )\)
be in
\((0, \infty )^{n}\)
and
\(\tilde{\mathbf{t}}= (t_{1}, \ldots , t_{m} )\)
be in
\((0, \infty )^{m}\). If
and
then
where
\(\mathcal{S}\)
is defined in (39) and
$$\begin{aligned}& \Biggl[\frac{1}{\sum_{u=1}^{m}t_{u}}\sum_{u=1}^{m} \frac{(w_{u})^{2}}{t _{u}}- \Biggl(\sum_{u=1}^{m} \frac{w_{u}}{\sum_{u=1}^{m}t_{u}} \Biggr)^{2} -\frac{1}{\sum_{v=1}^{n}k_{v}}\sum _{v=1}^{n}\frac{(r_{v})^{2}}{k_{v}} \\& \quad {}+ \Biggl(\sum_{v=1}^{n} \frac{r_{v}}{\sum_{v=1}^{n}k_{v}} \Biggr)^{2} \Biggr]\geq 0 \end{aligned}$$
(40)
$$ J_{G_{k}}(r, w, k, t)\leq 0, $$
(41)
$$ J_{s}(r, w, k, t) \leq 0, $$
(42)
$$\begin{aligned} J_{s}(r, w, k, t) =&\frac{1}{\sum_{u=1}^{m}t_{u}} \Biggl[\tilde{ \mathcal{S}}+\sum_{u=1}^{m}w_{u} \log (t_{u}) \Biggr]+ \Biggl[\sum_{u=1} ^{m}\frac{w_{u}}{\sum_{u=1}^{m} t_{u}}\log \Biggl(\sum _{u=1}^{m}\frac{w _{u}}{\sum_{u=1}^{m}t_{u}} \Biggr) \Biggr] \\ &{}-\frac{1}{\sum_{v=1}^{n}k_{v}} \Biggl[\mathcal{S}- \sum _{v=1}^{n}r_{v}\log (k_{v}) \Biggr] \\ &{} - \Biggl[\sum_{v=1}^{n} \frac{r_{v}}{ \sum_{v=1}^{n} k_{v}}\log \Biggl(\sum_{v=1}^{n} \frac{r_{v}}{\sum_{v=1} ^{n}k_{v}} \Biggr) \Biggr]; \end{aligned}$$
(43)
$$ \tilde{\mathcal{S}} : = - \sum_{u=1}^{m}w_{u} \log (w_{u}). $$
4.3 Rényi divergence and entropy
The Rényi divergence and the Rényi entropy are given in [19].
Definition 4
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 \(\delta \geq 0\), \(\delta \neq 1\).
(a)
The Rényi divergence of order δ is defined by
$$\begin{aligned} \mathcal{D}_{\delta }(\tilde{\mathbf{r}}, \tilde{ \mathbf{q}}) : = \frac{1}{\delta - 1} \log \Biggl(\sum _{i=1} ^{n}q_{i} \biggl( \frac{r_{i}}{q_{i}} \biggr)^{\delta } \Biggr). \end{aligned}$$
(44)
(b)
The Rényi entropy of order δ of \(\tilde{\mathbf{r}}\) is defined by
$$\begin{aligned} \mathcal{H}_{\delta }(\tilde{\mathbf{r}}): = \frac{1}{1 - \delta } \log \Biggl( \sum_{i=1}^{n} r_{i}^{\delta } \Biggr). \end{aligned}$$
(45)
These definitions also hold for non-negative probability distributions.
Theorem 7
Let
\(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n})\), \(\tilde{\mathbf{k}} = (k_{1}, \ldots , k_{n}) \in \mathbb{R}_{+}^{n}\), \(\tilde{\mathbf{w}} = (w_{1}, \ldots , w_{m})\), \(\tilde{\mathbf{t}} = (t_{1}, \ldots , t_{m}) \in \mathbb{R}_{+}^{m}\)
be such that
\(\sum_{1}^{n}r_{v}=1\), \(\sum_{1}^{n}k_{v}=1\), \(\sum_{1}^{m}w_{u}=1\)
and
\(\sum_{1}^{m}t_{u}=1\).
If either
\(1 < \delta \)
and the base of the log is greater than 1 or
\(\delta \in [0, 1)\)
and the base of the log is less than 1, and if
and
then
If either
\(1 < \delta \)
and the base of the log is greater than 1 or
\(\delta \in [0, 1)\)
and the base of the log is less than 1, then (47) and (48) hold in reverse direction.
$$\begin{aligned}& \Biggl[\sum_{u=1}^{m} \frac{(t_{u})^{2}}{w_{u}} \biggl(\frac{w_{u}}{t _{u}} \biggr)^{2\delta }- \Biggl(\sum_{u=1}^{m}t_{u} \biggl(\frac{w_{u}}{t _{u}} \biggr) ^{\delta } \Biggr)^{2} - \sum_{v=1}^{n}\frac{(k_{v})^{2}}{r _{v}} \biggl(\frac{r_{v}}{k_{v}} \biggr)^{2\delta } \\& \quad {}+ \Biggl(\sum_{v=1}^{n}k_{v} \biggl(\frac{r_{v}}{k_{v}} \biggr)^{\delta } \Biggr)^{2} \Biggr]\geq 0 \end{aligned}$$
(46)
$$\begin{aligned} \sum_{v=1}^{n}r_{v}G_{k} \biggl( \biggl(\frac{r_{v}}{k_{v}} \biggr)^{\delta -1}, s\biggr) -G_{k}\Biggl(\sum_{v=1}^{n}r_{v} \biggl(\frac{r_{v}}{k_{v}} \biggr)^{\delta -1}, s\Biggr) \geq& \sum _{u=1}^{m}w_{u}G_{k} \biggl( \biggl(\frac{w_{u}}{t_{u}} \biggr)^{ \delta -1}, s\biggr) \\ &{}-G_{k}\Biggl(\sum_{u=1}^{m}w_{u} \biggl(\frac{w_{u}}{t_{u}} \biggr)^{\delta -1}, s\Biggr), \end{aligned}$$
(47)
$$\begin{aligned} \sum_{v=1}^{n}r_{v} \log \biggl( \frac{r_{v}}{k_{v}} \biggr)-\mathcal{D}_{\delta }(\tilde{ \mathbf{r}}, \tilde{\mathbf{k}}) \geq \sum_{u=1}^{m}w_{u} \log \biggl( \frac{w_{u}}{t_{u}} \biggr)-\mathcal{D}_{\delta }(\tilde{ \mathbf{w}}, \tilde{\mathbf{t}}). \end{aligned}$$
(48)
Proof
The proof is only for the case when \(\delta \in [0, 1)\) and the base of the log is greater than 1 and similarly the remaining cases are simple to prove.
Choosing \(I = (0, \infty )\)
\(f=\log \) so \(f^{(2)}(x)\) is negative and \(f^{(3)}(x)\) is positive, therefore f is 3-convex. So using \(f=\log \) and the substitutions
and
in the reverse of inequality (18) (by Remark 1), we have
Dividing (49) with \((\delta -1)\) and using
we get (48). □
$$ p_{v} : = r_{v}, \qquad x_{v} : = \biggl( \frac{r_{v}}{k_{v}} \biggr)^{\delta - 1}, \quad v = 1, \ldots , n, $$
$$ q_{u} : = w_{u}, \qquad y_{u} : = \biggl( \frac{w_{u}}{t_{u}} \biggr)^{\delta - 1}, \quad u = 1, \ldots , m, $$
$$\begin{aligned} (\delta -1)\sum_{v=1}^{n}r_{v} \log \biggl( \frac{r_{v}}{k_{v}} \biggr)- \log \Biggl(\sum _{v=1}^{n}k_{v}\biggl( \frac{r_{v}}{k_{v}}\biggr)^{\delta } \Biggr) \geq & (\delta -1)\sum _{u=1}^{m}w_{u} \log \biggl( \frac{w_{u}}{t_{u}} \biggr) \\ &{}-\log \Biggl(\sum_{u=1}^{m}t_{u} \biggl(\frac{w_{u}}{t_{u}}\biggr)^{\delta } \Biggr). \end{aligned}$$
(49)
$$\begin{aligned}& \mathcal{D}_{\delta }(\tilde{\mathbf{r}}, \tilde{\mathbf{k}})= \frac{1}{\delta -1}\log \Biggl(\sum_{v=1}^{n}k_{v} \biggl(\frac{r _{v}}{k_{v}}\biggr)^{\delta } \Biggr), \\& \mathcal{D}_{\delta }(\tilde{\mathbf{w}}, \tilde{\mathbf{t}})= \frac{1}{\delta -1}\log \Biggl(\sum_{u=1}^{m}t_{u} \biggl(\frac{w _{u}}{t_{u}}\biggr)^{\delta } \Biggr), \end{aligned}$$
Corollary 7
Let
\(\tilde{\mathbf{r}} = (r_{1}, \ldots , r_{n}) \in \mathbb{R}_{+} ^{n}\), \(\tilde{\mathbf{w}} = (w_{1}, \ldots , w_{m}) \in \mathbb{R} _{+}^{m}\)
be such that
\(\sum_{1}^{n}r_{v}=1\)
and
\(\sum_{1}^{m}w_{u}=1\).
Also let
and
If
\(1 < \delta \)
and the base of the log is greater than 1, then
The reverse inequality holds in (51) and (52) if the base of the log is less than 1.
$$\begin{aligned}& \Bigg[\sum_{u=1}^{m} \frac{1}{m^{2}w_{u}} (mw_{u} )^{2\delta }- \Biggl(\sum _{u=1}^{m}\frac{1}{m} (m w_{u} )^{\delta } \Biggr)^{2}- \sum _{v=1}^{n}\frac{1}{n^{2}r_{v}} (nr_{v} )^{2\delta } \\& \quad+ \Biggl(\sum_{v=1}^{n} \frac{1}{n} (n r_{v} )^{\delta } \Biggr)^{2} \geq 0 \end{aligned}$$
(50)
$$\begin{aligned} \sum_{v=1}^{n}r_{v}G_{k} \bigl((nr_{v})^{\delta -1}, s \bigr) -G_{k} \Biggl( \sum_{v=1}^{n}r_{v}(nr_{v})^{\delta -1}, s \Biggr) \geq& \sum_{u=1} ^{m}w_{u}G_{k} \bigl((mw_{u})^{\delta -1}, s \bigr) \\ &{}-G_{k} \Biggl(\sum_{u=1}^{m}w_{u}(mw_{u})^{\delta -1}, s \Biggr). \end{aligned}$$
(51)
$$ \sum_{v=1}^{n}r_{v} \log (r_{v})+\mathcal{H}_{\delta }( \tilde{\mathbf{r}})\geq \sum_{u=1}^{m}w_{u}\log (w_{u})+\mathcal{H}_{\delta }(\tilde{\mathbf{w}}). $$
(52)
Proof
Suppose \(\tilde{\mathbf{k}}= (\frac{\textbf{1}}{\textbf{n}}, \ldots , \frac{\textbf{1}}{\textbf{n}} )\) and \(\tilde{\mathbf{t}}= (\frac{\textbf{1}}{\textbf{m}}, \ldots , \frac{\textbf{1}}{ \textbf{m}} )\). Then from (44), we have
and
This implies
and
It follows from Theorem 7, \(\tilde{\mathbf{k}}= \frac{ \textbf{1}}{\textbf{n}}\), \(\tilde{\mathbf{t}}= \frac{\textbf{1}}{ \textbf{m}}\), (53) and (54), that
After some simple calculations we get (52). □
$$\begin{aligned} \mathcal{D}_{\delta } (\tilde{\mathbf{r}}, \tilde{\mathbf{k}}) = \frac{1}{\delta - 1} \log \Biggl(\sum_{v=1}^{n}n ^{\delta - 1}r_{v}^{\delta } \Biggr) = \log (n) + \frac{1}{\delta - 1}\log \Biggl(\sum_{v=1}^{n}r_{v}^{\delta } \Biggr) \end{aligned}$$
$$\begin{aligned} \mathcal{D}_{\delta } (\tilde{\mathbf{w}}, \tilde{\mathbf{t}}) = \frac{1}{\delta - 1} \log \Biggl(\sum_{u=1}^{m}m ^{\delta - 1}w_{u}^{\delta } \Biggr) = \log (m) + \frac{1}{\delta - 1}\log \Biggl(\sum_{u=1}^{m}w_{u}^{\delta } \Biggr). \end{aligned}$$
$$\begin{aligned} \mathcal{H}_{\delta }(\tilde{\mathbf{r}}) = \log (n) - \mathcal{D}_{\delta } \biggl(\tilde{\mathbf{r}}, \frac{\textbf{1}}{ \textbf{n}} \biggr) \end{aligned}$$
(53)
$$\begin{aligned} \mathcal{H}_{\delta }(\tilde{\mathbf{w}}) = \log (m) - \mathcal{D}_{\delta } \biggl(\tilde{\mathbf{w}}, \frac{\textbf{1}}{ \textbf{m}} \biggr). \end{aligned}$$
(54)
$$ \sum_{v=1}^{n}r_{v} \log (nr_{v})-\log (n)+\mathcal{H}_{ \delta }(\tilde{ \mathbf{r}})\geq \sum_{u=1}^{m}w_{u} \log (m w_{u})- \log (m)+\mathcal{H}_{\delta }(\tilde{ \mathbf{w}}). $$
(55)
4.4 Zipf–Mandelbrot law
In [14] the authors gave some contribution in analyzing the Zipf–Mandelbrot law which is defined as follows.
Definition 5
Zipf–Mandelbrot law is a discrete probability distribution depending on the three parameters: \(\mathcal{N} \in \{1, 2, \ldots , \}\), \(\phi \in [0, \infty )\) and \(t > 0\), and is defined by
where
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
where
For \(\phi = 0\), the Zipf–Mandelbrot law becomes the Zipf law.
$$\begin{aligned} f(s; \mathcal{N}, \phi , t) : = \frac{1}{(s + \phi )^{t}\mathcal{H}_{\mathcal{N}, \phi , t}}, \quad s = 1, \ldots , \mathcal{N}, \end{aligned}$$
$$\begin{aligned} \mathcal{H}_{\mathcal{N}, \phi , t} = \sum_{\nu =1}^{ \mathcal{N}} \frac{1}{(\nu + \phi )^{t}}. \end{aligned}$$
$$\begin{aligned} f(s; \phi , t) = \frac{1}{(s + \phi )^{t}\mathcal{H}_{ \phi , t}}, \end{aligned}$$
$$\begin{aligned} \mathcal{H}_{\phi , t} = \sum_{\nu =1}^{\infty } \frac{1}{( \nu + \phi )^{t}}. \end{aligned}$$
Theorem 8
Let
\(\tilde{\mathbf{r}}\)
and
\(\tilde{\mathbf{w}}\)
be the Zipf–Mandelbrot laws.
If (50) and (51) hold for
\(r_{v}=\frac{1}{(v+k)^{v} {\mathcal{H}_{\mathcal{N}, k, v}}}\), \(w_{u}=\frac{1}{(u+w)^{u} {\mathcal{H}_{\mathcal{N}, w, u}}}\), and if the base of the log is greater than 1, then
The inequality is reversed in (51) and (56) if the base of the log is less than 1.
$$\begin{aligned}& \sum_{v=1}^{n}\frac{1}{(v+k)^{v}{\mathcal{H}_{ \mathcal{N}, k, v}}}\log \biggl(\frac{1}{(v+k)^{v}{\mathcal{H}_{\mathcal{N}, k, v}}} \biggr)+\frac{1}{1 - \delta }\log \Biggl( \frac{1}{\mathcal{H}_{\mathcal{N}, k, v}^{\delta }} \sum_{v=1}^{n} \frac{1}{(v + k)^{\delta v}} \Biggr) \\& \quad \geq \sum_{u=1}^{m} \frac{1}{(u+w)^{u}{\mathcal{H}_{ \mathcal{N}, w, u}}}\log \biggl(\frac{1}{(u+w)^{u}{\mathcal{H}_{\mathcal{N}, w, u}}} \biggr) \\& \qquad {} +\frac{1}{1 - \delta } \log \Biggl(\frac{1}{\mathcal{H}_{\mathcal{N}, w, u}^{ \delta }}\sum _{u=1}^{m}\frac{1}{(u + w)^{\delta u}} \Biggr). \end{aligned}$$
(56)
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).
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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