Here we maximize the Shannon entropy using the method of Lagrange multiplier under some equation constraints and get the Zipf–Mandelbrot law.
Under the assumption of Theorem
2.1(i), define the nonnegative functionals as follows:
$$\begin{aligned}& \varTheta_{3}(f) = \mathsf{A}_{m,r}^{[1]} - f \biggl( \frac{\sum_{s = 1}^{n} r _{s}}{\sum_{s = 1}^{n} q_{s}} \biggr) \sum_{s = 1}^{n} q_{s},\quad r = 1, \ldots,m, \\ \end{aligned}$$
(50)
$$\begin{aligned}& \varTheta_{4}(f) = \mathsf{A}_{m,r}^{[1]} - \mathsf{A}_{m,k}^{[1]},\quad 1 \le r < k \le m. \end{aligned}$$
(51)
Under the assumption of Theorem
2.1(ii), define the nonnegative functionals as follows:
$$\begin{aligned}& \varTheta_{5}(f) = \mathsf{A}_{m,r}^{[2]} - \Biggl( \sum _{s = 1}^{n} r_{s} \Biggr) f \biggl( \frac{\sum_{s = 1}^{n} r_{s}}{\sum_{s = 1}^{n} q_{s}} \biggr),\quad r = 1, \ldots,m, \end{aligned}$$
(52)
$$\begin{aligned}& \varTheta_{6}(f) = \mathsf{A}_{m,r}^{[2]} - \mathsf{A}_{m,k}^{[2]}, \quad 1 \le r < k \le m. \end{aligned}$$
(53)
Under the assumption of Corollary
3.1(i), define the following nonnegative functionals:
$$\begin{aligned}& \varTheta_{7}(f) = A_{m,r}^{[3]} + \sum _{i = 1}^{n} q_{i}\log (q_{i}), \quad r = 1, \ldots,n, \end{aligned}$$
(54)
$$\begin{aligned}& \varTheta_{8}(f) = A_{m,r}^{[3]} - A_{m,k}^{[3]},\quad 1 \le r < k \le m. \end{aligned}$$
(55)
Under the assumption of Corollary
3.1(ii), define the following nonnegative functionals given as
$$\begin{aligned}& \varTheta_{9}(f) = A_{m,r}^{[4]} - S,\quad r = 1, \ldots,m, \end{aligned}$$
(56)
$$\begin{aligned}& \varTheta_{10}(f) = A_{m,r}^{[4]} - A_{m,k}^{[4]},\quad 1 \le r < k \le m. \end{aligned}$$
(57)
Under the assumption of Corollary
3.2(i), let us define the nonnegative functionals as follows:
$$\begin{aligned}& \varTheta_{11}(f) = A_{m,r}^{[5]} - \sum _{s = 1}^{n} r_{s}\log \Biggl( \sum _{s = 1}^{n} \log \frac{r_{n}}{\sum_{s = 1}^{n} q_{s}} \Biggr),\quad r = 1, \ldots,m, \end{aligned}$$
(58)
$$\begin{aligned}& \varTheta_{12}(f) = A_{m,r}^{[5]} - A_{m,k}^{[5]}, \quad 1 \le r < k \le m. \end{aligned}$$
(59)
Under the assumption of Corollary
3.2(ii), define the nonnegative functionals as follows:
$$ \varTheta_{13}(f) = A_{m,r}^{[6]} - A_{m,k}^{[6]},\quad 1 \le r < k \le m. $$
(60)
Under the assumption of Theorem
4.1(i), consider the following functionals:
$$\begin{aligned}& \varTheta_{14}(f) = A_{m,r}^{[7]} - D_{\lambda } (\mathbf{r}, \mathbf{q}),\quad r = 1, \ldots,m, \end{aligned}$$
(61)
$$\begin{aligned}& \varTheta_{15}(f) = A_{m,r}^{[7]} - A_{m,k}^{[7]},\quad 1 \le r < k \le m. \end{aligned}$$
(62)
Under the assumption of Theorem
4.1(ii), consider the following functionals:
$$\begin{aligned}& \varTheta_{16}(f) = A_{m,r}^{[8]} - D_{1}(\mathbf{r},\mathbf{q}),\quad r = 1, \ldots,m, \\ \end{aligned}$$
(63)
$$\begin{aligned}& \varTheta_{17}(f) = A_{m,r}^{[8]} - A_{m,k}^{[8]}, \quad 1 \le r < k \le m. \end{aligned}$$
(64)
Under the assumption of Theorem
4.1(iii), consider the following functionals:
$$\begin{aligned}& \varTheta_{18}(f) = A_{m,r}^{[9]} - D_{\lambda } (\mathbf{r}, \mathbf{q}),\quad r = 1, \ldots,m, \end{aligned}$$
(65)
$$\begin{aligned}& \varTheta_{19}(f) = A_{m,r}^{[9]} - A_{m,k}^{[9]},\quad 1 \le r < k \le m. \end{aligned}$$
(66)
Under the assumption of Theorem
4.2, consider the following nonnegative functionals:
$$\begin{aligned}& \varTheta_{20}(f) = D_{\lambda } (\mathbf{r},\mathbf{q}) - A_{m,r} ^{[10]},\quad r = 1, \ldots,m, \end{aligned}$$
(67)
$$\begin{aligned}& \varTheta_{21}(f) = A_{m,k}^{[10]} - A_{m,r}^{[10]}, \quad 1 \le r < k \le m. \end{aligned}$$
(68)
$$\begin{aligned}& \varTheta_{22}(f) = A_{m,r}^{[11]} - D_{\lambda } (\mathbf{r}, \mathbf{q}),\quad r = 1, \ldots,m, \end{aligned}$$
(69)
$$\begin{aligned}& \varTheta_{23}(f) = A_{m,r}^{[11]} - A_{m,r}^{[11]},\quad 1 \le r < k \le m, \end{aligned}$$
(70)
$$\begin{aligned}& \varTheta_{24}(f) = A_{m,r}^{[11]} - A_{m,k}^{[10]},\quad r = 1, \ldots,m, k = 1, \ldots,m. \end{aligned}$$
(71)
Under the assumption of Corollary
4.3(i), consider the following nonnegative functionals:
$$\begin{aligned}& \varTheta_{25}(f) = H_{\lambda } (r) - A_{m,r}^{[12]}, \quad r = 1, \ldots,m, \end{aligned}$$
(72)
$$\begin{aligned}& \varTheta_{26}(f) = A_{m,k}^{[12]} - A_{m,r}^{[12]},\quad 1 \le r < k \le m. \end{aligned}$$
(73)
Under the assumption of Corollary
4.3(ii), consider the following functionals:
$$\begin{aligned}& \varTheta_{27}(f) = S - A_{m,r}^{[13]}, \quad r = 1, \ldots,m, \end{aligned}$$
(74)
$$\begin{aligned}& \varTheta_{28}(f) = A_{m,k}^{[13]} - A_{m,r}^{[13]},\quad 1 \le r < k \le m. \end{aligned}$$
(75)
Under the assumption of Corollary
4.3(iii), consider the following functionals:
$$\begin{aligned}& \varTheta_{29}(f) = H_{\lambda } (\mathbf{r}) - A_{m,r}^{[14]},\quad r = 1, \ldots,m, \end{aligned}$$
(76)
$$\begin{aligned}& \varTheta_{30}(f) = A_{m,k}^{[14]} - A_{m,r}^{[14]},\quad 1 \le r < k \le m. \end{aligned}$$
(77)
Under the assumption of Corollary
4.4, define the following functionals:
$$\begin{aligned}& \varTheta_{31} = A_{m,r}^{[15]} - H_{\lambda } (r),\quad r = 1, \ldots,m, \end{aligned}$$
(78)
$$\begin{aligned}& \varTheta_{32} = A_{m,r}^{[15]} - A_{m,k}^{[15]},\quad 1 \le r < k \le m, \end{aligned}$$
(79)
$$\begin{aligned}& \varTheta_{33} = H_{\lambda } (\mathbf{r}) - A_{m,r}^{[16]}, \quad r = 1, \ldots,m, \end{aligned}$$
(80)
$$\begin{aligned}& \varTheta_{34} = A_{m,k}^{[16]} - A_{m,r}^{[16]},\quad 1 \le r < k \le m, \end{aligned}$$
(81)
$$\begin{aligned}& \varTheta_{35} = A_{m,r}^{[15]} - A_{m,k}^{[16]},\quad r = 1, \ldots,m, k = 1, \ldots,m. \end{aligned}$$
(82)