\\[1.75 ex]
\!\!&\!\! \!\!&\!\!+\left(\nfrac{1552}{9}-\nfrac{16128}{\pi^4}-\nfrac{16}{9}\pi^2\right) x^6+\left(\nfrac{8064}{\pi^3}+\nfrac{104}{135}\pi^3-\nfrac{3632}{45}\pi\right) x^5                         \\[1.75 ex]
\!\!&\!\! \!\!&\!\!+\left(16\!-\!\nfrac{28}{135}\pi^4\!-\!\nfrac{2688}{\pi^2}\!+\!\nfrac{1124}{45}\pi^2\right)x^4\!+\!\Big {(}\nfrac{2}{63}\pi^5\!+\!\nfrac{576}{\pi}\!-\!\nfrac{208}{45}\pi^3      \\[1.75 ex]
\!\!&\!\! \!\!&\!\!-16\pi \Big {)} x^3+\left(-72-\nfrac{2}{945}\pi^6+\nfrac{16}{45}\pi^4+4\pi^2 \right) x^2 \\[1.5 ex]
\!\!&\!\!=\!\!&\!\!-\nfrac{2}{945} \nfrac{x^2}{\pi^8}\hat{Q}_8(x).
\end{array}$$ Then, we have to determine sign of the polynomial $$\label{Ineq_76}
\begin{array}{rcl}
\hat{Q}_8(x)\!\!&\!\!=\!\!&\!\!(128\pi^6-10752\pi^4+967680) x^8+(-576\pi^7+48384\pi^5                              \\[1.5 ex]
\!\!&\!\! \!\!&\!\!-4354560\pi) x^7+(1120\pi^8-96768\pi^6+8709120\pi^2) x^6                                        \\[1.5 ex]
\!\!&\!\! \!\!&\!\!+(-1232\pi^9+112896\pi^7-10160640\pi^3) x^5+(840\pi^{10}                                        \\[1.5 ex]
\!\!&\!\! \!\!&\!\!-81480\pi^8+7620480\pi^4) x^4+(-364\pi^{11}+38136\pi^9                                          \\[1.5 ex]
\!\!&\!\! \!\!&\!\!-3810240\pi^5)x^3\!+\!(98\pi^{12}\!-\!11802\pi^{10}\!-\!7560\pi^8\!+\!1270080\pi^6)x^2          \\[1.5 ex]
\!\!&\!\! \!\!&\!\!+(\!-\!15\pi^{13}\!+\!2184\pi^{11}\!+\!7560\pi^9\!-\!272160\pi^7)x\!+\!\pi^{14}\!-\!168\pi^{12} \\[1.5 ex]
\!\!&\!\! \!\!&\!\!-1890\pi^{10}+34020\pi^8,
\end{array}$$ for $ x\!\in\!(0,c_2)$.

The fourth derivate of the polynomial $\hat{Q}_8(x)$ is the polynomial of $4^{th}$ degree $$\label{Ineq_77}
\begin{array}{rcl}
\hat{Q}_8^{(iv)}(x)\!\!&\!\!=\!\!&\!\!1680(128\pi^6\!-\!10750\pi^4\!+\!967680)x^4\!+\!840(\!-\!576\pi^7\!+\!48384\pi^5 \\[1.25 ex]
\!\!&\!\! \!\!&\!\!-4354560\pi) x^3+360(1120\pi^8-96768\pi^6+8709120\pi^2) x^2                                         \\[1.25 ex]
\!\!&\!\! \!\!&\!\!+120(-1232\pi^9+112896\pi^7-10160640\pi^3) x +20160\pi^{10}                                         \\[1.25 ex]
\!\!&\!\! \!\!&\!\!-1955520\pi^8+182891520\pi^4.
\end{array}$$ Using MATLAB software we can determine the real numerical factorization of the polynomial $$\label{Ineq_78}
\hat{Q}_8^{(iv)}(x)= \alpha(x^2+p_1 x+q_1)(x^2+p_2 x+q_2),$$ with values $\alpha= 7.29\ldots 10^7$, $p_1=-0.798\ldots $, $ q_1=1.417\ldots $, $p_2=-6.27\ldots $, $ q_2=11.111\ldots\;$. Also, holds that inequalies $p_1^2-4q_1<0$ and $p_2^2-4q_2<0$ are true. The polynomial $\hat{Q}_8^{(iv)}(x)$ has no simple real roots but has two pairs of complex conjugate. Roots and constants $p_1$, $ q_1$, $p_2$, $ q_2$ can be represented in symbolic form. The polynomial $\hat{Q}_8^{(iv)}(x)$ has no simple real roots for $x\!\in\!\left(0,\nfrac{\pi}{2}\right)$ and $\hat{Q}_8^{(iv)}(0)>0$. That means that $\hat{Q}_8^{(iv)}(x)>0$ for $ x\!\in\!(0, c_2) \subset \left(0,\nfrac{\pi}{2}\right)$ and the function $\hat{Q}_8^{'''}(x)$ 