1 Introduction and preliminary results

The most commonly used words, population ranks of cities in various countries, corporation sizes, income rankings can be described in terms of Zipf’s law. The f-divergence measures the difference between two probability distributions by making an average value, which is weighted by a specified function. There are other probability distributions like Csiszar f-divergence [10, 11], a special case of which is Kullback–Leibler-divergence which is used to find the appropriate distance between the probability distributions (see [18, 19]). The notion of distance is stronger than divergence, because it gives the properties of symmetry and triangle inequalities. Probability theory has application in many fields and the divergence between probability distributions has many applications in these fields.

Many natural phenomena such as distributions of wealth and income in a society, Facebook likes, football goals, and city sizes follow power-law distributions (Zipf’s Law). Auerbach [2] was the first to explore the idea that the distribution of city size can be well approximated with the help of Pareto distribution (power-law distribution). This idea was well refined by many researchers, but Zipf [27] worked significantly in this field. The distribution of city sizes is investigated by many scholars of the urban economics, like Rosen and Resnick [24], Black and Henderson [3], Ioannides and Overman [17], Soo [25], Anderson and Ge [1], and Bosker et al. [4]. Zipf’s law states that: “The rank of cities with a certain number of inhabitants varies proportional to the city sizes with some negative exponent, say that is close to unit". In other words, Zipf’s Law states that the product of city sizes and their ranks appears roughly constant. This indicates that the population of the second largest city is one half of the population of the largest city and the third largest city equal to one-third of the population of the largest city and the population of n-th city is \(\frac{1}{n}\) of the largest city population. This rule is called rank, size rule, and also named as Zipf’s Law. Hence, Zip’s law shows that the city size distribution follows the Pareto distribution and, in addition, that the estimated value of the shape parameter is equal to unity.

Horváth et al. [16] introduced and obtained some estimates for new functionals based on the f-divergence functionals. and obtained some estimates for the new functionals. They obtained f-divergence and Rényi divergence by applying a cyclic refinement of Jensen’s inequality. They also constructed new inequalities for Rényi and Shannon entropies and used Zipf–Madelbrot law to illustrate the results.

The inequalities involving higher order convexity are used by many physicists in higher dimension problems since the founding of higher order convexity by Popoviciu (see [22, p. 15]). It is quite interesting that some results are true for convex functions, but in higher order convexity, they are not valid anymore.

In [22, p. 16], the following criteria are given to check the m-convexity of the function.

If \(f^{(m)}\) exists, then f is m-convex if and only if \(f^{(m)} \ge 0\).

In recent years, many researchers have generalized the inequalities for m-convex functions. For instance, Butt et al. generalized the Popoviciu’s inequality for m-convex function using Taylor’s formula, Lidstone polynomial, Montgomery identity, Fink’s identity, Abel–Goncharov interpolation, and Hermite interpolating polynomial (see [5,6,7,8,9]).

For many years, Jensen’s inequality has been of great interest. It was refined by defining some new functions (see [14, 15]). Horváth and Pečarić ([12, 15], see also [13, p. 26]) gave a refinement of Jensen’s inequality for convex function. They defined some essential notions to prove the refinement given as follows:

Let X be a set, and: \(P(X):=\) Power set of X, |X|:= Number of elements of X, \({\mathbb {N}}\):= Set of natural numbers with 0. Consider \(q \ge 1\) and \(r \ge 2\) be fixed integers. Define the functions:

$$\begin{aligned}&F_{r, s}:\{1, \ldots , q\}^{r}\rightarrow \{1, \ldots , q\}^{r-1} \,\ \,\ \,\ 1 \le s \le r,\\&\qquad F_{r}: \{1, \ldots , q \}^{r} \rightarrow P\left( \{1, \ldots , q \}^{r-1}\right) , \end{aligned}$$

and

$$\begin{aligned} T_{r}: P\left( \{1, \ldots , q\}^{r} \right) \rightarrow P\left( \{1, \ldots , q\}^{r-1}\right) , \end{aligned}$$

by

$$\begin{aligned} F_{r, s}(i_1, \ldots , i_{r}):= & {} (i_1, i_2, \ldots , i_{s-1}, i_{s+1}, \ldots , i_r) \,\ \,\ \,\ 1\le s \le r,\\ F_{r}(i_1, \ldots , i_r):= & {} \bigcup \limits _{s=1}^{r}\{F_{r, s}(i_1, \ldots , i_r)\}, \end{aligned}$$

and

$$\begin{aligned} T_{r}(I)=\left\{ \begin{array}{ll} \phi , &{} {I = \phi ;} \\ \bigcup \limits _{(i_1, \ldots , i_r)\in I}F_{r}(i_1, \ldots , i_r), &{} {I \ne \phi .} \end{array} \right. \end{aligned}$$

Next, the function

$$\begin{aligned} \alpha _{r, i}: \{1, \ldots , q\}^{r} \rightarrow {\mathbb {N}} \,\ \,\ \,\ 1\le i \le q \end{aligned}$$

is defined by:

$$\begin{aligned} \alpha _{r, i}(i_1, \ldots , i_r) \,\ \text {is the number of occurrences of } i \text { in the sequence}\,\ (i_1, \ldots , i_r). \end{aligned}$$

For each \(I \in P(\{1, \ldots , q\}^r)\), let

$$\begin{aligned} \alpha _{I, i}:=\sum \limits _{(i_1, \ldots , i_r)\in I}\alpha _{r, i}(i_1, \ldots , i_r) \,\ \,\ \,\ 1\le i \le q. \end{aligned}$$

\(\left( H_1\right) \) Let n, m be fixed positive integers, such that \(n\ge 1\), \(m\ge 2\), and let \(I_m\) be a subset of \(\{1, \ldots , n \}^m\), such that:

$$\begin{aligned} \alpha _{I_m, i} \ge 1 \,\ \,\ \,\ 1 \le i \le n. \end{aligned}$$

Introduce the sets \(I_{l}\subset \{1, \ldots , n\}^{l} (m-1 \ge l \ge 1)\) inductively by:

$$\begin{aligned} I_{l-1}:= T_{l}(I_l) \,\ \,\ \,\ m \ge l \ge 2. \end{aligned}$$

Obviously, the sets \(I_1= \{1, \ldots , n\}\), by \((H_1)\), and this insures that \(\alpha _{I_1, i}=1(1 \le i \le n)\). From \((H_1)\), we have \(\alpha _{I_l, i} \ge 1(m-1 \ge l \ge 1, 1 \le i \le n)\).

For \(m \ge l \ge 2\), and for any \((j_1, \ldots , j_{l-1})\in I_{l-1}\), let:

$$\begin{aligned} {\mathscr {H}}_{I_l}(j_1, \ldots , j_{l-1}):=\{((i_1, \ldots , i_l), k) \times \{1, \ldots , l\}|F_{l, k}(i_1, \ldots , i_l)=(j_1, \ldots , j_{l-1})\}. \end{aligned}$$

With the help of these sets, they define the functions \(\eta _{I_m, l}: I_l \rightarrow {\mathbb {N}}(m \ge l \ge 1)\) inductively by:

$$\begin{aligned}&\eta _{I_m, m}(i_1, \ldots , i_m):=1\,\ \,\ \,\ (i_1, \ldots , i_m)\in I_m;\\&\quad \eta _{I_m, l-1}(j_1, \ldots , j_{l-1}):=\sum \limits _{((i_1, \ldots , i_l),k)\in {\mathscr {H}}_{I_l}(j_1, \ldots , j_{l-1})}\eta _{I_m, l}(i_1, \ldots ,i_l). \end{aligned}$$

They define some special expressions for \(1 \le l \le m\), as follows:

$$\begin{aligned} {\mathscr {A}}_{m,l}= & {} {\mathscr {A}}_{m,l}(I_m, x_1, \ldots , x_n, p_1, \ldots , p_n ; f):= \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \\&\times \left( \sum \limits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}\right) f\left( \frac{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}x_{i_j}}{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}}\right) \end{aligned}$$

and prove the following theorem.

Theorem 1.1

Assume \((\mathrm{H}_1)\), and let \(f: I \rightarrow {\mathbb {R}}\) be a convex function where \(I \subset {\mathbb {R}}\) is an interval. If \(x_1, \ldots , x_n \in I\), and \(p_1, \ldots , p_n\) are positive real numbers, such that \(\sum \nolimits _{i=1}^{n}p_i=1\), then

$$\begin{aligned} f\left( \sum \limits _{s=1}^{n}p_{s}x_s\right) \le {\mathscr {A}}_{m, m} \le {\mathscr {A}}_{m, m-1} \le \cdots \le {\mathscr {A}}_{m, 2} \le {\mathscr {A}}_{m, 1} = \sum \limits _{s=1}^{n}p_{s}f\left( x_s\right) . \end{aligned}$$
(1)

We define the following functionals by taking the differences of refinement of Jensen’s inequality given in (1):

$$\begin{aligned} \Theta _{1}(f)= & {} {\mathscr {A}}_{m, r} - f\left( \sum \limits _{s=1}^{n}p_s x_s\right) , \,\ \,\ \,\ r=1, \ldots , m, \end{aligned}$$
(2)
$$\begin{aligned} \Theta _{2}(f)= & {} {\mathscr {A}}_{m, r} - {\mathscr {A}}_{m, k}, \,\ \,\ \,\ 1\le r <k \le m. \end{aligned}$$
(3)

Under the assumptions of Theorem 1.1, we have:

$$\begin{aligned} \Theta _{i}(f) \ge 0, \,\ \,\ \,\ i=1,2. \end{aligned}$$
(4)

Inequalities (4) are reversed if f is concave on I.

1.1 Lidstone polynomial

We generalize the refinement of Jensen’s inequality for higher order convex function using Lidstone interpolating polynomial. In [26], Widder gives the following result.

Lemma A

If \(g \in C^{\infty }([0, 1])\), then:

$$\begin{aligned} g(u)= \sum _{l=0}^{m-1}\left[ g^{(2l)}(0){\mathfrak {F}}_{l}(1 - u) + g^{(2l)}(0){\mathfrak {F}}_{l}(t) \right] + \int _{0}^{1}G_{m}(u,s)g^{(2m)}(s)\mathrm{d}s, \end{aligned}$$

where \({\mathfrak {F}}_{l}\) is a polynomial of degree \(2l+1\) defined by the relation:

$$\begin{aligned} {\mathfrak {F}}_{0}(u)=u, \,\ {\mathfrak {F}}_{m}''(u)={\mathfrak {F}}_{m-1}(u), \,\ {\mathfrak {F}}_{m}(0)={\mathfrak {F}}_{m}(1)=0, \,\ m\ge 1, \end{aligned}$$
(5)

and

$$\begin{aligned} G_{1}(u, s)= G(u, s)=\left\{ \begin{array}{ll} (u-1)s, &{} \quad {\alpha _1 \le s \le u \le \alpha _2;} \\ (s-1)u, &{} \quad {\alpha _1 \le u \le s \le \alpha _2,} \end{array} \right. \end{aligned}$$

is a homogeneous Green’s function of the differential operator \(\frac{\mathrm{d}^2}{\mathrm{d}^2s}\) on [0, 1], and with the successive iterates of G(u, s):

$$\begin{aligned} G_{m}(u,s)=\int _{0}^{1}G_{1}(u,p)G_{m-1}(p,s)\mathrm{d}p, \,\ \,\ m \ge 2. \end{aligned}$$

The Lidstone polynomial can be expressed in terms of \(G_{m}(u, s)\) as:

$$\begin{aligned} {\mathfrak {F}}_{m}(u)=\int _{0}^{1}G_{m}(u,s)s\mathrm{d}s. \end{aligned}$$

Lidstone series representation of \(g \in C^{2m}[\alpha _1, \alpha _2]\) is given by:

$$\begin{aligned} g(u)= & {} \sum _{l=0}^{m-1}(\alpha _2 - \alpha _1)^{2l}g^{(2l)}(\alpha _1){\mathfrak {F}}_{l}\left( \frac{\alpha _2 - u}{\alpha _2 - \alpha _1}\right) + \sum _{l=0}^{m-1}(\alpha _2 - \alpha _1)^{2l}g^{(2l)}(\alpha _2){\mathfrak {F}}_{l}\left( \frac{u - \alpha _1}{\alpha _2 - \alpha _1}\right) \nonumber \\&+ \, (\alpha _2 - \alpha _1)^{2l-1}\int _{\alpha _1}^{\alpha _2}G_{m}\left( \frac{u - \alpha _1}{\alpha _2 - \alpha _1}, \frac{t - \alpha _1}{\alpha _2 - \alpha _1}\right) g^{(2l)}(t)\mathrm{d}t. \end{aligned}$$
(6)

2 Inequalities for Csiszár divergence

In [10, 11], Csiszár introduced the following notion.

Definition 2.1

Let \(f : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}\) be a convex function, let \({\mathbf {r}}=\left( r_1, \ldots , r_n\right) \), and \({\mathbf {q}}=\left( q_1, \ldots , q_n\right) \) be positive probability distributions. Then, f-divergence functional is defined by:

$$\begin{aligned} I_{f}({\mathbf {r}}, {\mathbf {q}}) := \sum _{i=1}^{n}q_{i}f\left( \frac{r_i}{q_i}\right) . \end{aligned}$$
(7)

And he stated that by defining:

$$\begin{aligned} f(0) := \lim \limits _{x \rightarrow 0^{+}}f(x); \,\ \,\ \,\ 0f\left( \frac{0}{0}\right) :=0; \,\ \,\ \,\ 0f\left( \frac{a}{0}\right) := \lim \limits _{x \rightarrow 0^{+}}xf\left( \frac{a}{0}\right) , \,\ \,\ a>0, \end{aligned}$$
(8)

we can also use the non-negative probability distributions, as well.

Horv́ath et al. [16] gave the following functional based on the previous definition.

Definition 2.2

Let \(I \subset {\mathbb {R}}\) be an interval and let \(f: I \rightarrow {\mathbb {R}}\) be a function. Let \({\mathbf {r}}=(r_1, \ldots , r_n)\in {\mathbb {R}}^n\) and \({\mathbf {q}}=(q_1, \ldots , q_n)\in (0, \infty )^{n}\), such that:

$$\begin{aligned} \frac{r_s}{q_s} \in I, \,\ \,\ \,\ s= 1, \ldots , n. \end{aligned}$$

Then, they define the sum \({\hat{I}}_{f}({\mathbf {r}}, {\mathbf {q}})\) as:

$$\begin{aligned} {\hat{I}}_{f}({\mathbf {r}}, {\mathbf {q}}) : = \sum _{s=1}^{n}q_{s}f\left( \frac{r_s}{q_s}\right) . \end{aligned}$$
(9)

We apply Theorem 1.1 to \({\hat{I}}_{f}({\mathbf {r}}, {\mathbf {q}})\).

Theorem 2.3

Assume \((H_1)\), let \(I \subset {\mathbb {R}}\) be an interval and let \({\mathbf {r}}=\left( r_1, \ldots , r_n\right) \) and \({\mathbf {q}}=\left( q_1, \ldots , q_n\right) \) be in \((0, \infty )^{n}\), such that

$$\begin{aligned} \frac{r_s}{q_s} \in I, \,\ \,\ \,\ s = 1, \ldots , n. \end{aligned}$$

(i) If \(f: I \rightarrow {\mathbb {R}}\) is a convex function, then:

$$\begin{aligned} {\hat{I}}_{f}({\mathbf {r}}, {\mathbf {q}})= & {} \sum _{s=1}^{n}q_{s}f\left( \frac{r_s}{q_s}\right) =A_{m,1}^{[1]}\ge A_{m,2}^{[1]} \ge \ldots \ge A_{m,m-1}^{[1]} \ge A_{m,m}^{[1]} \ge f\left( \frac{\sum _{s=1}^{n}r_s}{\sum _{s=1}^{n}q_s} \right) \sum _{s=1}^{n}q_s, \end{aligned}$$
(10)

where

$$\begin{aligned} A_{m,l}^{[1]}= & {} \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_m, i_j}}\right) f\left( \frac{\sum _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_m, i_j}}}\right) . \end{aligned}$$
(11)

If f is a concave function, then inequality signs in (10) are reversed.

(ii) If \(f: I \rightarrow {\mathbb {R}}\) is a function, such that \(x \rightarrow xf(x) (x \in I)\) is convex, then:

$$\begin{aligned}&\left( \sum _{s=1}^{n}r_s\right) f\left( \sum _{s=1}^{n}\frac{r_s}{\sum _{s=1}^{n}q_s}\right) \le A_{m, m}^{[2]} \le A_{m,m-1}^{[2]} \le \cdots \le A_{m,2}^{[2]} \le A_{m,1}^{[2]} = \sum _{s=1}^{n}r_sf\left( \frac{r_s}{q_S}\right) = {\hat{I}}_{id f}({\mathbf {r}}, {\mathbf {q}}),\nonumber \\ \end{aligned}$$
(12)

where

$$\begin{aligned} A_{m, l}^{[2]}= & {} \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{{q_{i_j}}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) f\left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) . \end{aligned}$$

Proof

(i) Consider \(p_{s} = \frac{q_{s}}{\sum _{s=1}^{n}q_s}\) and \(x_{s} = \frac{r_s}{q_s}\) in Theorem 1.1, we have:

$$\begin{aligned}&f\left( \sum _{s=1}^{n}\frac{q_s}{\sum _{s=1}^{n}q_s}\frac{r_s}{q_s}\right) \le \cdots \le \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\nonumber \\&\quad \times \, \left( \sum \limits _{j=1}^{l}\frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_{s}}}{\alpha _{I_m, i_j}}\right) f\left( \frac{\sum \nolimits _{j=1}^{l}\frac{ \frac{q_{i_j}}{\sum _{i=1}^{n}q_{i}} }{\alpha _{I_m, i_j}}\frac{r_{i_j}}{q_{i_j}}}{\sum \nolimits _{j=1}^{l}\frac{ \frac{q_{i_j}}{\sum _{i=1}^{n}q_{i}} }{\alpha _{I_m, i_j}}}\right) \le \ldots \le \sum _{s=1}^{n}\frac{q_s}{\sum _{i=1}^{n}q_{s}}f\left( \frac{r_s}{q_s} \right) . \end{aligned}$$
(13)

And taking the sum \(\sum _{s=1}^{n}q_{i}\), we have (10).

(ii) Using \(f:=id f\) (where “id” is the identity function) in Theorem 1.1, we have:

$$\begin{aligned}&\sum _{s=1}^{n}p_{s}x_{s}f\left( \sum _{s=1}^{n}p_s x_s\right) \le \ldots \le \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\nonumber \\&\quad \times \left( \sum \limits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}x_{i_j}}{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}}\right) f\left( \frac{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}x_{i_j}}{\sum \nolimits _{j=1}^{l}\frac{p_{i_j}}{\alpha _{I_m, i_j}}}\right) \le \cdots \le \sum _{s=1}^{n}p_{s}x_{s}f(x_s). \end{aligned}$$
(14)

Now, using \(p_s = \frac{q_s}{\sum _{s=1}^{n}q_s}\) and \(x_s = \frac{r_s}{q_s}, \,\ s = 1, \ldots , n\), we get:

$$\begin{aligned}&\sum _{s=1}^{n}\frac{q_s}{\sum _{s=1}^{n}q_s}\frac{r_s}{q_s}f\left( \sum _{s=1}^{n}\frac{q_s}{\sum _{s=1}^{n}q_s}\frac{r_s}{q_s}\right) \le \cdots \le \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\nonumber \\&\quad \times \left( \sum \limits _{j=1}^{l}\frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_s}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum _{j=1}^{l} \frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_s}}{\alpha _{I_{m}, i_j}}\frac{r_{i_j}}{q_{i_j}}}{\sum _{j=1}^{l} \frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_s}}{\alpha _{I_{m}, i_j}}} \right) f\left( \frac{\sum _{j=1}^{l} \frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_s}}{\alpha _{I_{m}, i_j}}\frac{r_{i_j}}{q_{i_j}}}{\sum _{j=1}^{l} \frac{\frac{q_{i_j}}{\sum _{s=1}^{n}q_s}}{\alpha _{I_{m}, i_j}}} \right) \le \cdots \le \sum _{s=1}^{n}\frac{q_s}{\sum _{s=1}^{n}q_s} \frac{r_s}{q_s}f\left( \frac{r_s}{q_S}\right) .\nonumber \\ \end{aligned}$$
(15)

By taking sum \(\sum _{s=1}^{n}q_s\) on both sides, we get (12). \(\square \)

3 Inequalities for Shannon entropy

Definition 3.1

(See [16]) The Shannon entropy of positive probability distribution \({\mathbf {r}}=(r_1, \ldots , r_n)\) is defined by:

$$\begin{aligned} S : = - \sum _{s=1}^{n}r_{s}\log (r_s). \end{aligned}$$
(16)

Corollary 3.2

Assume \((\mathrm{H}_1)\).

  1. (i)

    If \({\mathbf {q}}=(q_1, \ldots , q_n) \in (0, \infty )^{n}\), and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned} S \le A_{m,m}^{[3]} \le A_{m,m-1}^{[3]} \le \cdots \le A_{m,2}^{[3]} \le A_{m,1}^{[3]} = \log \left( \frac{n}{\sum _{s=1}^{n}q_s}\right) \sum _{s=1}^{n}q_s, \end{aligned}$$
    (17)

    where

    $$\begin{aligned}&A_{m,l}^{[3]} = - \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\left( \sum \limits _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \sum \limits _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_m, i_j}}\right) . \end{aligned}$$
    (18)

    If the base of \(\log \) is between 0 and 1, then inequality signs in (17) are reversed.

  2. (ii)

    If \({\mathbf {q}}= (q_1, \ldots , q_n)\) is a positive probability distribution and the base of \(\log \) is greater than 1, then we have the estimates for the Shannon entropy of \({\mathbf {q}}\):

    $$\begin{aligned}&S \le A_{m,m}^{[4]} \le A_{m,m-1}^{[4]} \le \cdots \le A_{m,2}^{[4]} \le A_{m,1}^{[4]} = \log (n), \end{aligned}$$
    (19)

    where

    $$\begin{aligned} A_{m,l}^{[4]} = - \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \sum \limits _{j=1}^{l} \frac{q_{i_j}}{\alpha _{I_m, i_j}}\right) . \end{aligned}$$

Proof

(i) Using \(f:= \log \) and \({\mathbf {r}} = (1, \ldots , 1)\) in Theorem 2.3 (i), we get (17).

(ii) It is a special case of (i). \(\square \)

Definition 3.3

(See [16]) The Kullback–Leibler divergence between the positive probability distribution \({\mathbf {r}}=(r_1, \ldots , r_n)\) and \({\mathbf {q}}= (q_1, \ldots , q_n)\) is defined by:

$$\begin{aligned} D({\mathbf {r}}, {\mathbf {q}}) : = \sum _{s=1}^{n}r_{i} \log \left( \frac{r_i}{q_i}\right) . \end{aligned}$$
(20)

Corollary 3.4

Assume \((\mathrm{H}_1)\).

  1. (i)

    Let \({\mathbf {r}} = (r_1 , \ldots , r_n) \in (0, \infty )^{n}\) and \({\mathbf {q}} : = (q_1, \ldots , q_n) \in (0, \infty )^{n}\). If the base of \(\log \) is greater than 1, then:

    $$\begin{aligned}&\sum _{s=1}^{n}r_s \log \left( \sum _{s=1}^{n}\frac{r_s}{\sum _{s=1}^{n}q_s}\right) \le A_{m, m}^{[5]} \le A_{m, m-1}^{[5]} \le \cdots \le A_{m, 2}^{[5]} \le A_{m, 1}^{[5]} = \sum _{s=1}^{n}r_s \log \left( \frac{r_s}{q_s}\right) = D({\mathbf {r}}, {\mathbf {q}}),\nonumber \\ \end{aligned}$$
    (21)

    where

    $$\begin{aligned}&A_{m, l}^{[5]} = \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{{q_{i_j}}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) \log \left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) . \end{aligned}$$

    If the base of \(\log \) is between 0 and 1, then inequality in (21) is reversed.

  2. (ii)

    If \(\mathbf{r }\) and \(\mathbf{q }\) are positive probability distributions, and the base of \(\log \) is greater than 1, then we have:

    $$\begin{aligned}&D(\mathbf{r }, \mathbf{q }) = A_{m, 1}^{[6]} \ge A_{m, 2}^{[6]} \ge \ldots \ge A_{m, m-1}^{[6]} \ge A_{m, m}^{[6]} \ge 0, \end{aligned}$$
    (22)

    where

    $$\begin{aligned} A_{m, l}^{[6]}= & {} \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{{q_{i_j}}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) \log \left( \frac{\sum _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{q_{i_j}}{\alpha _{I_{m}, i_j}}} \right) . \end{aligned}$$

    If the base of \(\log \) is between 0 and 1, then inequality signs in (22) are reversed.

Proof

(i) On taking \(f: = \log \) in Theorem 2.3 (ii), we get (21).

(ii) It is a special case of (i). \(\square \)

4 Inequalities for Rényi divergence and entropy

The Rényi divergence and entropy come from [23].

Definition 4.1

Let \(\mathbf{r } := (r_1, \ldots , r_n)\) and \(\mathbf{q } : = (q_1, \ldots , q_n)\) be positive probability distributions, and let \(\lambda \ge 0\), \(\lambda \ne 1\).

(a):

The Rényi divergence of order \(\lambda \) is defined by:

$$\begin{aligned} D_{\lambda }(\mathbf{r }, \mathbf{q }) : = \frac{1}{\lambda - 1} \log \left( \sum _{i=1}^{n}q_{i}\left( \frac{r_i}{q_i}\right) ^{\lambda } \right) . \end{aligned}$$
(23)
(b):

The Rényi entropy of order \(\lambda \) of \(\mathbf{r }\) is defined by:

$$\begin{aligned} H_{\lambda }(\mathbf{r }) : = \frac{1}{1 - \lambda } \log \left( \sum _{i=1}^{n} r_{i}^{\lambda }\right) . \end{aligned}$$
(24)

The Rényi divergence and the Rényi entropy can also be extended to non-negative probability distributions. If \(\lambda \rightarrow 1\) in (23), we have the Kullback–Leibler divergence, and if \(\lambda \rightarrow 1\) in (24), then we have the Shannon entropy. In the next two results, inequalities can be found for the Rényi divergence.

Theorem 4.2

Assume \((\mathrm{H}_{1})\), let \(\mathbf{r } = (r_1, \ldots , r_n)\) be \(\mathbf{q } = (q_1, \ldots , q_n)\) be probability distributions.

  1. (i)

    If \( 0 \le \lambda \le \mu \), such that \( \lambda , \mu \ne 1\), and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned}&D_{\lambda } (\mathbf{r }, \mathbf{q }) \le A_{m, m}^{[7]} \le A_{m, m-1}^{[7]} \le \cdots \le A_{m, 2}^{[7]} \le A_{m, 1}^{[7]} = D_{\mu } (\mathbf{r }, \mathbf{q }), \end{aligned}$$
    (25)

    where

    $$\begin{aligned} A_{m, l}^{[7]}= & {} \frac{1}{\mu -1}\log \left( \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \right. \\&\times \left. \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) ^{\frac{\mu - 1}{\lambda - 1}}\right) . \end{aligned}$$

    The reverse inequalities hold in (25) if the base of \(\log \) is between 0 and 1.

  2. (ii)

    If \(1 < \mu \) and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned}&D_{1} (\mathbf{r }, \mathbf{q }) = D (\mathbf{r }, \mathbf{q }) = \sum _{s=1}^{n}r_s\log \left( \frac{r_s}{q_s}\right) \le A_{m, m}^{[8]} \le A_{m, m-1}^{[8]} \le \cdots \le A_{m, 2}^{[8]} \le A_{m, 1}^{[8]} = D_{\mu } (\mathbf{r }, \mathbf{q }),\nonumber \\ \end{aligned}$$
    (26)

    where

    $$\begin{aligned} A_{m, l}^{[8]}= & {} \le \frac{1}{\mu -1}\log \left( \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \exp \right. \\&\times \left. \left( \frac{(\mu -1)\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}} \log \left( \frac{r_{i_j}}{q_{i_j}}\right) }{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}} \right) \right) ; \end{aligned}$$

    here the base of \(\exp \) is same as the base of \(\log \), and the reverse inequalities hold if the base of \(\log \) is between 0 and 1.

  3. (iii)

    If \(0 \le \lambda < 1\), and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned}&D_{\lambda } (\mathbf{r }, \mathbf{q }) \le A_{m, m}^{[9]} \le A_{m, m-1}^{[9]} \le \cdots \le A_{m, 2}^{[9]} \le A_{m, 1}^{[9]} = D_{1} (\mathbf{r }, \mathbf{q }), \end{aligned}$$
    (27)

    where

    $$\begin{aligned}&A_{m, l}^{[9]} = \frac{1}{\lambda -1}\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) .\nonumber \\ \end{aligned}$$
    (28)

Proof

By applying Theorem 1.1 with \(I=(0, \infty )\), \(f: (0, \infty ) \rightarrow {\mathbb {R}}\), \(f(t):= t^{\frac{\mu - 1}{\lambda -1}}\):

$$\begin{aligned} p_s : = r_s, \,\ \,\ \,\ x_s : = \left( \frac{r_s}{q_s} \right) ^{\lambda - 1}, \,\ s = 1, \ldots , n, \end{aligned}$$

we have:

$$\begin{aligned}&\left( \sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s} \right) ^{\lambda }\right) ^{\frac{\mu - 1}{\lambda - 1}} = \left( \sum _{s=1}^{n}r_s \left( \frac{r_s}{q_s}\right) ^{\lambda - 1} \right) ^{\frac{\mu - 1}{\lambda - 1}} \nonumber \\&\quad \ldots \le \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) ^{\frac{\mu - 1}{\lambda - 1}} \nonumber \\&\quad \le \ldots \le \sum _{s=1}^{n}r_s \left( \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}\right) ^{\frac{\mu - 1}{\lambda - 1}}, \end{aligned}$$
(29)

if either \(0 \le \lambda< 1 < \beta \) or \(1 < \lambda \le \mu \), and the reverse inequality in (29) holds if \(0 \le \lambda \le \beta < 1\). By raising to power \(\frac{1}{\mu - 1}\), we have from all:

$$\begin{aligned}&\left( \sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s}\right) ^{\lambda }\right) ^{\frac{1}{\lambda - 1}} \nonumber \\&\quad \le \ldots \le \left( \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) ^{\frac{\mu - 1}{\lambda - 1}}\right) ^{\frac{1}{\mu - 1}} \nonumber \\&\quad \le \cdots \le \left( \sum _{s=1}^{n}r_s \left( \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}\right) ^{\frac{\mu - 1}{\lambda - 1}}\right) ^{\frac{1}{\mu - 1}} = \left( \sum _{s=1}^{n}q_{s}\left( \frac{r_s}{q_s} \right) ^{\mu } \right) ^{\frac{1}{\mu - 1}}. \end{aligned}$$
(30)

Since \(\log \) is increasing if the base of \(\log \) is greater than 1, it now follows (25). If the base of log is between 0 and 1, then \(\log \) is decreasing and, therefore, inequality in (25) is reversed. If \(\lambda = 1\) and \(\beta = 1\), we have (ii) and (iii), respectively, by taking limit. \(\square \)

Theorem 4.3

Assume \((\mathrm{H}_{1})\); let \(\mathbf{r } = (r_1, \ldots , r_n)\) and \(\mathbf{q } = (q_1, \ldots , q_n)\) be probability distributions. If either \(0 \le \lambda < 1\) and the base of \(\log \) is greater than 1, or \(1 < \lambda \) and the base of \(\log \) is between 0 and 1, then:

$$\begin{aligned} \frac{1}{\sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s}\right) ^{\lambda }} \sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s}\right) ^{\lambda } \log \left( \frac{r_s}{q_s}\right)= & {} A_{m, 1}^{[10]} \le A_{m, 2}^{[10]} \le \cdots \le A_{m, m-1}^{[10]} \le A_{m, m}^{[10]} \le D_{\lambda } (\mathbf{r }, \mathbf{q }) \le A_{m, m}^{[11]} \nonumber \\\le & {} A_{m, m}^{[11]} \le \cdots \le A_{m, 2}^{[11]} \le A_{m, 1}^{[11]} = D_{1} (\mathbf{r }, \mathbf{q }), \end{aligned}$$
(31)

where

$$\begin{aligned} A_{m, m}^{[10]}= & {} \frac{1}{(\lambda - 1)\sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }} \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\\&\times \left( {\sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \end{aligned}$$

and

$$\begin{aligned} A_{m, m}^{[11]}&= \frac{1}{\lambda - 1}\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) . \end{aligned}$$

The inequalities in (31) are reversed if either \(0 \le \lambda < 1\) and the base of \(\log \) is between 0 and 1, or \(1 < \lambda \) and the base of \(\log \) is greater than 1.

Proof

We prove only the case when \(0 \le \lambda < 1\) and the base of \(\log \) is greater than 1 and the other cases can be proved similarly. Since \(\frac{1}{\lambda - 1} < 0\) and the function \(\log \) is concave and then choose \(I = (0, \infty )\), \(f : = \log \), \(p_{s} = r_{s}\), \(x_{s}: = \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}\) in Theorem 1.1, we have:

$$\begin{aligned}&D_{\lambda } (\mathbf{r }, \mathbf{q }) = \frac{1}{\lambda -1}\log \left( \sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) =\frac{1}{\lambda -1}\log \left( \sum _{s=1}^{n}r_{s} \left( \frac{r_{s}}{q_s}\right) ^{\lambda - 1}\right) \nonumber \\&\quad \le \cdots \le \frac{1}{\lambda - 1}\frac{(m-1)!}{(l-1)!} \sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l} (i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \nonumber \\&\quad \le \cdots \le \frac{1}{\lambda - 1}\sum _{s=1}^{n}r_{s}\log \left( \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}\right) = \sum _{s=1}^{n}r_{s}\log \left( \frac{r_s}{q_s}\right) = D_{1} (\mathbf{r }, \mathbf{q }), \end{aligned}$$
(32)

and this gives the upper bound for \(D_{\lambda } (\mathbf{r }, \mathbf{q })\).

Since the base of \(\log \) is greater than 1, the function \(x \mapsto xf(x)\) \((x > 0)\) is convex; therefore, \(\frac{1}{1 - \lambda } < 0\), and Theorem 1.1 gives:

$$\begin{aligned} D_{\lambda } (\mathbf{r }, \mathbf{q })= & {} \frac{1}{\lambda -1}\log \left( \sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) = \frac{1}{\lambda -1\left( \sum _{s=1}^{n}q_{s} \left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) } \left( \sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) \log \left( \sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) \nonumber \\\ge & {} \ldots \ge \frac{1}{\lambda - 1\left( \sum _{s=1}^{n}q_{s}\left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) } \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \nonumber \\&\times \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \nonumber \\= & {} \frac{1}{\lambda - 1\left( \sum _{s=1}^{n}q_{s} \left( \frac{r_{s}}{q_s}\right) ^{\lambda }\right) }\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \nonumber \\&\times \left( {\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\left( \frac{r_{i_j}}{q_{i_j}}\right) ^{\lambda - 1}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \nonumber \\\ge & {} \ldots \ge \nonumber \\&\times \frac{1}{\lambda - 1} \sum _{s=1}^{n}r_s \left( \frac{r_s}{q_s}\right) ^{\lambda - 1} \log \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}\frac{1}{\sum _{s=1}^{n}r_s \left( \frac{r_s}{q_s}\right) ^{\lambda - 1}} \nonumber \\= & {} \frac{1}{\sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s}\right) ^{\lambda }} \sum _{s=1}^{n}q_s \left( \frac{r_s}{q_s}\right) ^{\lambda } \log \left( \frac{r_s}{q_s}\right) , \end{aligned}$$
(33)

which give the lower bound of \(D_{\lambda } (\mathbf{r }, \mathbf{q })\). \(\square \)

Using the previous results, some inequalities of Rényi entropy are obtained. Let \(\frac{\mathbf{1 }}{\mathbf{n }} = (\frac{1}{n}, \ldots , \frac{1}{n})\) be a discrete probability distribution.

Corollary 4.4

Assume \((\mathrm{H}_1)\); let \(\mathbf{r }= (r_1, \ldots , r_n)\) and \(\mathbf{q }= (q_1, \ldots , q_n)\) be positive probability distributions.

  1. (i)

    If \(0 \le \lambda \le \mu \), \(\lambda , \mu \ne 1\), and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned}&H_{\lambda }(\mathbf{r }) = \log (n) - D_{\lambda }\left( \mathbf{r }, \frac{1}{\mathbf{n }}\right) \ge A_{m, m}^{[12]} \ge A_{m, m}^{[12]} \ge \cdots A_{m, 2}^{[12]} \ge A_{m, 1}^{[12]} = H_{\mu }(\mathbf{r }), \end{aligned}$$
    (34)

    where

    $$\begin{aligned} A_{m, l}^{[12]}= & {} \frac{1}{1 - \mu }\log \left( \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \nonumber \right. \\&\times \left. \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) ^{\frac{\mu - 1}{\lambda - 1}} \right) . \end{aligned}$$

    The reverse inequalities hold in (34) if the base of \(\log \) is between 0 and 1.

  2. (ii)

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

    $$\begin{aligned} S= -\sum _{s=1}^{n}p_i\log (p_i) \ge A_{m, m}^{[13]} \ge A_{m, m-1}^{[13]} \ge \ldots \ge A_{m, 2}^{[13]} \ge A_{m, 1}^{[13]} = H_{\mu }(\mathbf{r }), \end{aligned}$$
    (35)

    where

    $$\begin{aligned} A_{m, l}^{[13]}= & {} \log (n) + \frac{1}{1 -\mu }\log \left( \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \exp \right. \\&\times \left. \left( \frac{(\mu -1)\sum \nolimits _{j=1}^{l} \frac{r_{i_j}}{\alpha _{I_m, i_j}}\log \left( nr_{i_j}\right) }{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \right) ; \end{aligned}$$

    the base of \(\exp \) is the same as the base of \(\log \). The inequalities in (35) are reversed if the base of \(\log \) is between 0 and 1.

  3. (iii)

    If \(0 \le \lambda < 1\), and the base of \(\log \) is greater than 1, then:

    $$\begin{aligned} H_{\lambda }(\mathbf{r }) \ge A_{m, m}^{[14]} \ge A_{m, m-1}^{[14]} \ge \cdots \ge A_{m, 2}^{[14]} \le A_{m, 1}^{[14]} = S, \end{aligned}$$
    (36)

    where

    $$\begin{aligned} A_{m, m}^{[14]} = \frac{1}{1 - \lambda } \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) . \end{aligned}$$
    (37)

    The inequalities in (36) are reversed if the base of \(\log \) is between 0 and 1.

Proof

(i) Suppose \(\mathbf{q }= \frac{\mathbf{1 }}{\mathbf{n }}\); then from (23), we have:

$$\begin{aligned} D_{\lambda } (\mathbf{r }, \mathbf{q }) = \frac{1}{\lambda - 1} \log \left( \sum _{s=1}^{n}n^{\lambda - 1}r_{s}^{\lambda } \right) = \log (n) + \frac{1}{\lambda - 1}\log \left( \sum _{s=1}^{n}r_{s}^{\lambda } \right) ; \end{aligned}$$
(38)

therefore, we have:

$$\begin{aligned} H_{\lambda }(\mathbf{r }) = \log (n) - D_{\lambda } (\mathbf{r }, \frac{\mathbf{1 }}{\mathbf{n }}). \end{aligned}$$
(39)

Now, using Theorem 4.2 (i) and (39), we get:

$$\begin{aligned} H_{\lambda }(\mathbf{r })= & {} \log (n) - D_{\lambda } \left( \mathbf{r }, \frac{\mathbf{1 }}{\mathbf{n }}\right) \ge \cdots \ge \log (n) - \frac{1}{\mu -1}\log \left( n^{\mu - 1}\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \right. \nonumber \\&\left. \times \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) ^{\frac{\mu - 1}{\lambda - 1}}\right) \ge \cdots \ge \log (n) - D_{\mu } (\mathbf{r }, \mathbf{q }) = H_{\mu }(\mathbf{r }); \end{aligned}$$
(40)

(ii) and (iii) can be proved similarly. \(\square \)

Corollary 4.5

Assume \((H_1)\), and let \(\mathbf{r }= (r_1, \ldots , r_n)\) and \(\mathbf{q }= (q_1, \ldots , q_n)\) be positive probability distributions.

If either \(0 \le \lambda < 1\) and the base of \(\log \) is greater than 1, or \(1 < \lambda \) and the base of \(\log \) is between 0 and 1, then:

$$\begin{aligned}&- \frac{1}{\sum _{s=1}^{n}r_s^{\lambda } } \sum _{s=1}^{n}r_s^{\lambda } \log (r_s) = A_{m, 1}^{[15]} \ge A_{m, 2}^{[15]} \ge \ldots \ge A_{m, m-1}^{[15]} \ge A_{m, m}^{[15]} \ge H_{\lambda }(\mathbf{r }) \ge A_{m, m}^{[16]} \ge A_{m, m-1}^{[16]} \nonumber \\&\quad \ge \ldots A_{m, 2}^{[16]} \ge A_{m, 1}^{[16]} = H\left( \mathbf{r } \right) , \end{aligned}$$
(41)

where

$$\begin{aligned} A_{m, l}^{[15]}= & {} \frac{1}{(\lambda - 1)\sum _{s=1}^{n}r_s^{\lambda }}\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( {\sum \limits _{j=1}^{l}\frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}\right) \log \left( n^{\lambda - 1}\frac{\sum \nolimits _{j=1}^{l} \frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) \\&\times \, { \hbox {and}} \\ A_{m, 1}^{[16]}= & {} \frac{1}{1 - \lambda }\frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}\right) \log \left( \frac{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}^{\lambda }}{\alpha _{I_m, i_j}}}{\sum \nolimits _{j=1}^{l}\frac{r_{i_j}}{\alpha _{I_m, i_j}}}\right) . \end{aligned}$$

The inequalities in (41) are reversed if either \(0 \le \lambda < 1\) and the base of \(\log \) is between 0 and 1, or \(1 < \lambda \) and the base of \(\log \) is greater than 1.

Proof

The proof is similar to Corollary 4.4 using Theorem 4.3. \(\square \)

5 Inequalities using Zipf–Mandelbrot law

The Zipf–Mandelbrot law is defined as follows (see [20]).

Definition 5.1

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

$$\begin{aligned} f(s; N, q, t) : = \frac{1}{(s + q)^{t}H_{N, q, t}}, \,\ \,\ \,\ s = 1, \ldots , N, \end{aligned}$$
(42)

where

$$\begin{aligned} H_{N, q, t} = \sum _{j=1}^{N}\frac{1}{(j + q)^{t}}. \end{aligned}$$
(43)

If the total mass of the law is taken over all \({\mathbb {N}}\), then for \(q \ge 0\), \(t > 1\), \(s \in {\mathbb {N}}\), density function of Zipf–Mandelbrot law becomes:

$$\begin{aligned} f(s; q, t) = \frac{1}{(s + q)^{t}H_{q, t}}, \end{aligned}$$
(44)

where

$$\begin{aligned} H_{q, t} = \sum _{j=1}^{\infty } \frac{1}{(j + q)^{t}}. \end{aligned}$$
(45)

For \(q = 0\), the Zipf–Mandelbrot law. By Corollary 4.4 (iii), we get:

Conclusion 5.2

Assume \((H_1)\); let \(\mathbf{r }\) be a Zipf–Mandelbrot law, by Corollary 4.4 (iii), we get. If \(0 \le \lambda < 1\), and the base of \(\log \) is greater than 1, then:

$$\begin{aligned} H_{\lambda }(\mathbf{r })= & {} \frac{1}{1 - \lambda }\log \left( \frac{1}{H_{N, q, t}^{\lambda }}\sum _{s=1}^{n}\frac{1}{(s + q)^{\lambda s}} \right) \nonumber \\\ge & {} \ldots \ge \frac{1}{1 - \lambda } \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l) \in I_l}\eta _{I_m, l}(i_1, \ldots , i_l) \left( \sum \limits _{j=1}^{l}\frac{1}{\alpha _{I_m, i_j}(i_j + q)H_{N. q, t}} \right) \log \nonumber \\&\times \left( \frac{1}{H_{N, q, t}^{\lambda - 1}}\frac{\sum \nolimits _{j=1}^{l}\frac{1}{\alpha _{I_m, i_j}(i_j - q)^{\lambda s}}}{\sum \nolimits _{j=1}^{l}\frac{1}{\alpha _{I_m, i_j}(i_j - q)^{s}}}\right) \nonumber \\\ge & {} \ldots \ge \frac{t}{H_{N, q, t}}\sum _{s=1}^{N} \frac{\log (s + q)}{(s + q)^{t}} + \log (H_{N, q, t}) = S. \end{aligned}$$
(46)

The inequalities in (46) are reversed if the base of \(\log \) is between 0 and 1.

Conclusion 5.3

Assume \((H_1)\); let \(\mathbf{r }_{1}\) and \(\mathbf{r }_2\) be the Zipf–Mandelbort law with parameters \(N \in \{1, 2, \ldots \}\), \(q_1, q_2 \in [0, \infty )\) and \(s_1, s_2 > 0\), respectively. Then, from Corollary 3.4 (ii), we have: if the base of \(\log \) is greater than 1, then:

$$\begin{aligned} {\bar{D}}(\mathbf{r }_{1}, \mathbf{r }_{2})= & {} \sum _{s=1}^{n}\frac{1}{(s + q_1)^{t_1} H_{N, q_1, t_1}} \log \left( \frac{(s + q_2)^{t_2}H_{N, q_2, t_2}}{(s + q_1)^{t_1}H_{N, q_2, t_1}} \right) \ge \ldots \ge \frac{(m-1)!}{(l-1)!}\sum \limits _{(i_1, \ldots , i_l)\in I_l}\eta _{I_m, l}(i_1, \ldots , i_l)\nonumber \\&\times \left( \sum \limits _{j=1}^{l}\frac{{\frac{1}{(i_j + q_2)^{t_2}H_{N, q_2, t_2}}}}{\alpha _{I_m, i_j}}\right) \left( \frac{\sum _{j=1}^{l} \frac{\frac{1}{(i_j + q_1)^{t_1}H_{N, q_1, t_1}}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{\frac{1}{(i_j + q_2)^{t_2}H_{N, q_2, t_2}}}{\alpha _{I_{m}, i_j}}} \right) \log \left( \frac{\sum _{j=1}^{l} \frac{\frac{1}{(i_j + q_1)^{t_1}H_{N, q_1, t_1}}}{\alpha _{I_{m}, i_j}}}{\sum _{j=1}^{l}\frac{\frac{1}{(i_j + q_2)^{t_2}H_{N, q_2, t_2}}}{\alpha _{I_{m}, i_j}}}\right) \ge \ldots \ge 0.\nonumber \\ \end{aligned}$$
(47)

The inequalities in (47) are reversed if base of \(\log \) is between 0 and 1.

6 Shannon entropy, Zipf–Mandelbrot law, and hybrid Zipf–Mandelbrot law

Here, we maximize the Shannon entropy using the method of Lagrange multiplier under some equations constraints and get the Zipf–Mandelbrot law.

Theorem 6.1

If \(J = \{1, 2, \ldots , N \}\), for a given \(q \ge 0\), a probability distribution that maximizes the Shannon entropy under the constraints:

$$\begin{aligned} \sum \limits _{s\in J}r_s = 1, \,\ \,\ \,\ \sum \limits _{s\in J} r_s \left( \ln (s + q) \right) := \Psi , \end{aligned}$$

is Zipf–Mandelbrot law.

Proof

If \(J = \{1, 2, \ldots , N \}\). We set the Lagrange multipliers \(\lambda \) and t and consider the expression:

$$\begin{aligned} {\widetilde{S}} = - \sum _{s=1}^{N}r_s \ln r_s - \lambda \left( \sum _{s=1}^{N}r_s - 1\right) - t\left( \sum _{s=1}^{N}r_{s}\ln (s + q) - \Psi \right) . \end{aligned}$$

Just for the sake of convenience, replace \(\lambda \) by \(\ln \lambda -1\), and thus, the last expression gives:

$$\begin{aligned} {\widetilde{S}} = - \sum _{s=1}^{N}r_s \ln r_s - \left( \ln \lambda - 1\right) \left( \sum _{s=1}^{N}r_s - 1\right) - t\left( \sum _{s=1}^{N}r_{s}\ln (s + q) - \Psi \right) . \end{aligned}$$

From \({\widetilde{S}}_{r_s} = 0\), for \(s =1, 2, \ldots , N\), we get:

$$\begin{aligned} r_s = \frac{1}{\lambda \left( s + q\right) ^t}, \end{aligned}$$

and on using the constraint \(\sum _{s = 1}^Nr_s = 1 \), we have:

$$\begin{aligned} \lambda = \sum _{s=1}^{N} \left( \frac{1}{(s+1)^t} \right) , \end{aligned}$$

where \(t > 0\), concluding that:

$$\begin{aligned} r_s = \frac{1}{(s + q)^t H_{N, q, t}}, \,\ \,\ \,\ s = 1, 2, \ldots , N. \end{aligned}$$

\(\square \)

Remark 6.2

Observe that the Zipf–Mandelbrot law and Shannon entropy can be bounded from above (see [21]):

$$\begin{aligned} S = - \sum _{s=1}^{N} f\left( s, N, q, t\right) \ln f(s, N, q, t) \le - \sum _{s=1}^{N}f(s, N, q, t) \ln q_s, \end{aligned}$$

where \(\left( q_1, \ldots , q_N\right) \) is a positive N-tuple, such that \(\sum _{s=1}^{N}q_s = 1\).

Theorem 6.3

If \(J = \{1, \ldots , N\}\), then probability distribution that maximizes Shannon entropy under constraints

$$\begin{aligned} \sum \limits _{s \in J}r_s = 1, \,\ \,\ \,\ \sum \limits _{s \in J} r_s \ln (s + q) := \Psi , \,\ \,\ \,\ \sum \limits _{s \in J}sr_s := \eta \end{aligned}$$

is hybrid Zipf–Mandelbrot law given as:

$$\begin{aligned} r_s = \frac{w^s}{\left( s+q\right) ^k \Phi ^{*}(k, q, w)}, \,\ \,\ \,\ s \in J, \end{aligned}$$

where

$$\begin{aligned} \Phi _{J}(k, q, w) = \sum \limits _{ s \in J} \frac{w^s}{(s + q)^k}. \end{aligned}$$

Proof

First, consider \(J = \{1, \ldots , N\}\); we set the Lagrange multiplier and consider the expression:

$$\begin{aligned} {\tilde{S}} = - \sum _{s=1}^{N}r_s \ln r_s + \ln w \left( \sum _{s=1}^{N}sr_s - \eta \right) - \left( \ln \lambda - 1\right) \left( \sum _{s=1}^{N}r_s - 1 \right) - k \left( \sum _{s=1}^{N}r_s \ln (s + q) - \Psi \right) . \end{aligned}$$

On setting \({\tilde{S}}_{r_s} = 0\), for \(s= 1, \ldots , N\), we get:

$$\begin{aligned} - \ln r_s + s \ln w - \ln \lambda - k \ln (s+q) = 0; \end{aligned}$$

after solving for \(r_s\), we get:

$$\begin{aligned} \lambda = \sum _{s=1}^{N}\frac{w^s}{\left( s+q\right) ^k}; \end{aligned}$$

and we recognize this as the partial sum of Lerch’s transcendent that we will denote with:

$$\begin{aligned} \Phi _{N}^{*}\left( k, q, w\right) = \sum _{s=1}^{N}\frac{w^s}{(s+q)^k} \end{aligned}$$

with \(w \ge 0, k > 0\). \(\square \)

Remark 6.4

Observe that for Zipf–Mandelbrot law, Shannon entropy can be bounded from above (see [21]):

$$\begin{aligned} S = -\sum _{s=1}^{N} f_{h}\left( s, N, q, k\right) \ln f_{h}\left( s, N, q, k\right) \le - \sum _{s=1}^{N}f_{h}\left( s, N, q, k\right) \ln q_s, \end{aligned}$$

where \(\left( q_1, \ldots , q_N\right) \) is any positive N-tuple, such that \(\sum _{s=1}^{N}q_s = 1\).

Under the assumption of Theorem 2.3 (i), define the non-negative functionals as follows:

$$\begin{aligned} \Theta _{3}(f)= & {} {\mathscr {A}}_{m, r}^{[1]} - f\left( \frac{\sum _{s=1}^{n}r_s}{\sum _{s=1}^{n}q_s}\right) \sum _{s=1}^{n}q_{s}, \,\ \,\ \,\ r=1, \ldots , m, \end{aligned}$$
(48)
$$\begin{aligned} \Theta _{4}(f)= & {} {\mathscr {A}}_{m, r}^{[1]} - {\mathscr {A}}_{m, k}^{[1]}, \,\ \,\ \,\ 1\le r <k \le m. \end{aligned}$$
(49)

Under the assumption of Theorem 2.3 (ii), define the non-negative functionals as follows:

$$\begin{aligned} \Theta _{5}(f)= & {} {\mathscr {A}}_{m, r}^{[2]} - \left( \sum _{s=1}^{n}r_s\right) f\left( \frac{\sum _{s=1}^{n}r_s}{\sum _{s=1}^{n}q_s}\right) , \,\ \,\ \,\ r=1, \ldots , m, \end{aligned}$$
(50)
$$\begin{aligned} \Theta _{6}(f)= & {} {\mathscr {A}}_{m, r}^{[2]} - {\mathscr {A}}_{m, k}^{[2]}, \,\ \,\ \,\ 1\le r <k \le m. \end{aligned}$$
(51)

Under the assumption of Corollary 3.2 (i), define the following non-negative functionals:

$$\begin{aligned} \Theta _{7}(f)= & {} A_{m,r}^{[3]}+\sum _{i=1}^{n}q_{i}\log (q_{i}), \,\ r=1,\ldots , n \end{aligned}$$
(52)
$$\begin{aligned} \Theta _{8}(f)= & {} A_{m,r}^{[3]}-A_{m,k}^{[3]},\,\ 1 \le r <k \le m. \end{aligned}$$
(53)

Under the assumption of Corollary 3.2 (ii), define the following non-negative functionals are given as:

$$\begin{aligned} \Theta _{9}(f)= & {} A_{m,r}^{[4]}- S, \,\ r=1,\ldots ,m \end{aligned}$$
(54)
$$\begin{aligned} \Theta _{10}(f)= & {} A_{m,r}^{[4]}-A_{m,k}^{[4]},\,\ 1 \le r <k \le m. \end{aligned}$$
(55)

Under the assumption of Corollary 3.4 (i), let us define the non-negative functionals as follows:

$$\begin{aligned} \Theta _{11}(f)= & {} A_{m,r}^{[5]}-\sum _{s=1}^{n}r_{s}\log \left( \sum _{s=1}^{n} \log \frac{r_{n}}{\sum _{s=1}^{n}q_{s}}\right) , \,\ r=1,\ldots ,m \end{aligned}$$
(56)
$$\begin{aligned} \Theta _{12}(f)= & {} A_{m,r}^{[5]}-A_{m,k}^{[5]},\,\ 1 \le r <k \le m. \end{aligned}$$
(57)

Under the assumption of Corollary 3.4 (ii), define the non-negative functionals as follows:

$$\begin{aligned} \Theta _{13}(f) = A_{m,r}^{[6]}-A_{m,k}^{[6]},\,\ 1 \le r <k \le m. \end{aligned}$$
(58)

Under the assumption of Theorem 4.2 (i), consider the following functionals:

$$\begin{aligned} \Theta _{14}(f)= & {} A_{m,r}^{[7]}- D_{\lambda }({\mathbf {r}},{\mathbf {q}}), \,\ r=1, \ldots ,m \end{aligned}$$
(59)
$$\begin{aligned} \Theta _{15}(f)= & {} A_{m,r}^{[7]}-A_{m,k}^{[7]},\,\ 1 \le r <k \le m. \end{aligned}$$
(60)

Under the assumption of Theorem 4.2 (ii), consider the following functionals:

$$\begin{aligned} \Theta _{16}(f) = A_{m,r}^{[8]}-D_{1}({\mathbf {r}},{\mathbf {q}}), \,\ r=1, \ldots ,m \end{aligned}$$
(61)
$$\begin{aligned} \Theta _{17}(f)=A_{m,r}^{[8]}-A_{m,k}^{[8]},\,\ 1 \le r <k \le m. \end{aligned}$$
(62)

Under the assumption of Theorem 4.2 (iii), consider the following functionals:

$$\begin{aligned} \Theta _{18}(f) = A_{m,r}^{[9]}-D_{\lambda }({\mathbf {r}},{\mathbf {q}}), \,\ r=1, \ldots ,m \end{aligned}$$
(63)
$$\begin{aligned} \Theta _{19}(f) = A_{m,r}^{[9]}-A_{m,k}^{[9]},\,\ 1 \le r <k \le m. \end{aligned}$$
(64)

Under the assumption of Theorem 4.3, consider the following non-negative functionals:

$$\begin{aligned} \Theta _{20}(f)= & {} D_{\lambda } (\mathbf{r }, \mathbf{q }) - A_{m,r}^{[10]}, \,\ \,\ \,\ r = 1, \ldots , m \end{aligned}$$
(65)
$$\begin{aligned} \Theta _{21}(f)= & {} A_{m,k}^{[10]}-A_{m,r}^{[10]},\,\ 1 \le r <k \le m. \end{aligned}$$
(66)
$$\begin{aligned} \Theta _{22}(f)= & {} A_{m,r}^{[11]} - D_{\lambda } (\mathbf{r }, \mathbf{q }), \,\ r = 1, \ldots , m \end{aligned}$$
(67)
$$\begin{aligned} \Theta _{23}(f)= & {} A_{m,r}^{[11]} - A_{m,r}^{[11]} ,\,\ 1 \le r <k \le m. \end{aligned}$$
(68)
$$\begin{aligned} \Theta _{24}(f)= & {} A_{m,r}^{[11]} - A_{m,k}^{[10]} ,\,\ r = 1, \ldots , m, \,\ k = 1, \ldots , m. \end{aligned}$$
(69)

Under the assumption of Corollary 4.4 (i), consider the following non-negative functionals:

$$\begin{aligned} \Theta _{25}(f)= & {} H_{\lambda }(\mathbf{r }) - A_{m,r}^{[12]}, \,\ r=1, \ldots ,m \end{aligned}$$
(70)
$$\begin{aligned} \Theta _{26}(f)= & {} A_{m,k}^{[12]}-A_{m,r}^{[12]}, \,\ 1 \le r <k \le m. \end{aligned}$$
(71)

Under the assumption of Corollary 4.4 (ii), consider the following functionals:

$$\begin{aligned} \Theta _{27}(f)= & {} S - A_{m,r}^{[13]}, \,\ r=1, \ldots ,m \end{aligned}$$
(72)
$$\begin{aligned} \Theta _{28}(f)= & {} A_{m,k}^{[13]}-A_{m,r}^{[13]}, \,\ 1 \le r <k \le m. \end{aligned}$$
(73)

Under the assumption of Corollary 4.4 (iii), consider the following functionals:

$$\begin{aligned} \Theta _{29}(f)= & {} H_{\lambda }(\mathbf{r }) - A_{m,r}^{[14]}, \,\ r=1, \ldots ,m \end{aligned}$$
(74)
$$\begin{aligned} \Theta _{30}(f)= & {} A_{m,k}^{[14]}-A_{m,r}^{[14]}, \,\ 1 \le r <k \le m. \end{aligned}$$
(75)

Under the assumption of Corollary 4.5, defined the following functionals.

$$\begin{aligned} \Theta _{31}= & {} A_{m,r}^{[15]} - H_{\lambda }(\mathbf{r }), \,\ r = 1, \ldots , m \end{aligned}$$
(76)
$$\begin{aligned} \Theta _{32}= & {} A_{m,r}^{[15]} - A_{m,k}^{[15]}, \,\ 1 \le r <k \le m. \end{aligned}$$
(77)
$$\begin{aligned} \Theta _{33}= & {} H_{\lambda }(\mathbf{r }) - A_{m,r}^{[16]}, \,\ r =1, \ldots , m \end{aligned}$$
(78)
$$\begin{aligned} \Theta _{34}= & {} A_{m,k}^{[16]} - A_{m,r}^{[16]}, \,\ 1 \le r <k \le m. \end{aligned}$$
(79)
$$\begin{aligned} \Theta _{35}= & {} A_{m,r}^{[15]} - A_{m,k}^{[16]} , \,\ r = 1, \ldots , m, \,\ k = 1, \ldots , m. \end{aligned}$$
(80)

7 Generalization of refinement of Jensen-, Rényi-, and Shannon-type inequalities via Lidstone polynomial

We construct some new identities with the help of generalized Lidstone polynomial (6).

Theorem 7.1

Assume \((H_1)\); let \(f: [\alpha _1, \alpha _2] \rightarrow {\mathbb {R}}\) be a function, where \([\alpha _1, \alpha _2] \subset {\mathbb {R}}\), such that be an interval, such that \(f\in C^{2m}[\alpha _1, \alpha _2]\) for \(m \ge 1\). Also let \(x_1, \ldots , x_n \in [\alpha _1, \alpha _2]\) and \(p_1, \ldots , p_n\) be positive real numbers, such that \(\sum \nolimits _{i=1}^{n}p_i=1\), and \({\mathfrak {F}}_m(t)\) are the same as defined in (5), and then:

$$\begin{aligned} \Theta _{i}(f)= & {} \sum _{k=1}^{m-1}(\alpha _2 - \alpha _1)^{2k}f^{(2k)}(\alpha _1)\Theta _{i}\left( {\mathfrak {F}}_l\left( \frac{\alpha _2 -x}{\alpha _2 - \alpha _1}\right) \right) \nonumber \\&+ \sum _{k=1}^{m-1}(\alpha _2 - \alpha _1)^{2k}f^{(2k)}(\alpha _2) \Theta _{i}\left( {\mathfrak {F}}_l\left( \frac{x - \alpha _1}{\alpha _2 - \alpha _1}\right) \right) \nonumber \\&+ (\alpha _2 -\alpha _1)^{2k-1}\int _{\alpha _1}^{\alpha _2} \Theta _{i}\left( G_{m}\left( \frac{x - \alpha _1}{\alpha _2 - \alpha _1}, \frac{t - \alpha _1}{\alpha _2 - \alpha _1}\right) \right) f^{(2m)}(t)\mathrm{d}t, \,\ \,\ i = 1, 2, \ldots , 35. \end{aligned}$$
(81)

Proof

Using (6) in place of f in \(\Theta _{i}(f),\) \(i = 1, 2, \ldots , 35\), we get (81). \(\square \)

Theorem 7.2

Assume \((H_1)\); let \(f: [\alpha _1, \alpha _2] \rightarrow {\mathbb {R}}\) be a function, where \([\alpha _1, \alpha _2] \subset {\mathbb {R}}\), such that be an interval, such that \(f \in C^{2m}[\alpha _1, \alpha _2]\) for \( m \ge 1\). Also let \(x_1, \ldots , x_n \in [\alpha _1, \alpha _2]\) and \(p_1, \ldots , p_n\) be positive real numbers, such that \(\sum \nolimits _{i=1}^{n}p_i=1\), and \({\mathfrak {F}}_m(t)\) are the same as defined in (5); let for \(m \ge 1\):

$$\begin{aligned} \Theta _{i}\left( G_{m}\left( \frac{x - \alpha _1}{\alpha _2 - \alpha _1}, \frac{t - \alpha _1}{\alpha _2 - \alpha _1}\right) \right) \ge 0, \,\ for all \,\ t \in [\alpha _1, \alpha _2]. \end{aligned}$$
(82)

If f is 2m-convex function, then we have:

$$\begin{aligned} \Theta _{i}(f)\ge & {} \sum _{k=1}^{m-1}(\alpha _2 - \alpha _1)^{2k}f^{(2k)}(\alpha _1)\Theta _{i}\left( {\mathfrak {F}}_{l}\left( \frac{\alpha _2 - x}{\alpha _2 - \alpha _1}\right) \right) \nonumber \\&+ \sum _{k=1}^{m-1}(\alpha _2 - \alpha _1)^{2k}f^{(2k)}(\alpha _2)\Theta _{i}\left( {\mathfrak {F}}_{l}\left( \frac{x - \alpha _1}{\alpha _2 - \alpha _1}\right) \right) , \,\ \,\ i = 1, 2, \ldots , 35. \end{aligned}$$
(83)

Proof

Since f is 2m-convex; therefore, \(f^{(2m)} \ge 0\) for all \(x \in [\alpha _1, \alpha _2]\), then using (82) in (81), we get the required result. \(\square \)

Theorem 7.3

Assume \((H_1)\); let \(f: [\alpha _1, \alpha _2] \rightarrow {\mathbb {R}}\) be a function, where \([\alpha _1, \alpha _2] \subset {\mathbb {R}}\) be an interval. Also let \(x_1, \ldots , x_n \in [\alpha _1, \alpha _2]\) and \(p_1, \ldots , p_n\) be positive real numbers, such that \(\sum \nolimits _{i=1}^{n}p_i=1\), and also suppose that \(f:[\alpha _1, \alpha _2] \rightarrow {\mathbb {R}}\) is 2m-convex. Then, the following results are valid.

  1. (i)

    If m is odd integer, then for every 2m-convex function, (83) holds. (ii) Suppose that (83) holds. If the function

    $$\begin{aligned} \lambda (u) = \sum _{l=0}^{m-1}(\alpha _2 - \alpha _1)^{2l}g^{(2l)}(\alpha _1){\mathfrak {F}}_{l}\left( \frac{\alpha _2 - u}{\alpha _2 - \alpha _1}\right) + \sum _{l=0}^{m-1}(\alpha _2 - \alpha _1)^{2l}g^{(2l)}(\alpha _2){\mathfrak {F}}_{l}\left( \frac{u - \alpha _1}{\alpha _2 - \alpha _1}\right) \end{aligned}$$

    is convex, then the right-hand side of (83) is non-negative and we have:

    $$\begin{aligned} \Theta _{i}(f) \ge 0, \,\ \,\ i = 1, 2, \ldots , 35. \end{aligned}$$
    (84)

Proof

(i) Note that \(G_{1}(u, s) \le 0\) for \(1 \le u, s, \le 1\) and also note that \(G_{m}(u, s) \le 0\) for odd integer m and \(G_{m}(u, s) \ge 0\) for even integer m. As \(G_1\) is convex function and \(G_{m-1}\) is positive for odd integer m, therefore:

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}^{2}u}\left( G_{m}(u,s)\right) = \int _{0}^{1}\frac{\mathrm{d}^2}{\mathrm{d}^2u}G_{1}(u,p)G_{m-1}(p,s)\mathrm{d}p \ge 0, \,\ m\ge 2. \end{aligned}$$

This shows that \(G_m\) is convex in the first variable u if m is convex. Similarly, \(G_m\) is concave in the first variable if m is even. Hence, if m is odd, then:

$$\begin{aligned} \Theta _{i}\left( G_{m}\left( \frac{x - \alpha _1}{\alpha _2 - \alpha _1}, \frac{t - \alpha _1}{\alpha _2 - \alpha _1}\right) \right) \ge 0; \end{aligned}$$

therefore, (84) is valid.

(ii) Using the linearity of \(\Theta _{i}(f)\), we can write the right-hand side of (83) in the form \(\Theta _{i}(\lambda )\). As \(\lambda \) is supposed to be convex, therefore, the right-hand side of (83) is non-negative, and so \(\Theta _{i}(f) \ge 0\). \(\square \)

Remark A

We can investigate the bounds for the identities related to the generalization of refinement of Jensen inequality using inequalities for the C̆ebys̆ev functional and some results relating to the Gr̈uss and Ostrowski-type inequalities can be constructed as given in Section 3 of [5]. Also we can construct the non-negative functionals from inequality (83) and give related mean value theorems and we can construct the new families of m-exponentially convex functions and Cauchy means related to these functionals as given in Section 4 of [5].