Take
$$ \int_{I}\big|f{'''}(x)\big|^{2}w(x)\,dx= \int_{I}f{'''}(x) \bigl(f{'''}(x)w(x) \bigr)\,dx. $$
Using the integration by parts formula and making use of condition (
1.1), we get
$$ \begin{aligned}[b] \int_{I}\big|f{'''}(x)\big|^{2}w(x)\,dx={}&{-} \int_{I} f{''}(x)f^{(\mathit{iv})}(x)w(x)\, dx \\ &- \int_{I} f{''}(x)f{'''}(x)w{'}(x) \,dx.\end{aligned} $$
(3.8)
Now take the first integral of (
3.8) on the right-hand side. Using the integration by parts formula and condition (
1.1), we have
$$ \begin{aligned}[b] ={}& \int_{I} f{'}(x)f^{(v)}(x)w(x)\,dx+ \int_{I} f{'}(x)f^{(\mathit{iv})}(x)w{'}(x) \,dx \\ &- \int_{I} f{''}(x)f{'''}(x)w{'}(x) \,dx.\end{aligned} $$
(3.9)
Now take the first and the second integrals on the right-hand side of the latter expression. Using the integration by parts formula and making use of condition (
1.1), we get
$$ \begin{aligned}[b] ={}&{-} \int_{I} f(x)f^{(\mathit{vi})}(x)w(x)\,dx- \int_{I} f(x)f^{(v)}(x)w{'}(x)\, dx- \int_{I} f(x)f^{(v)}(x)w{'}(x)\,dx \\ &- \int_{I} f(x)f^{(\mathit{iv})}(x)w{''}(x)\,dx- \int_{I} f{''}(x)f{'''}(x)w{'}(x) \,dx.\end{aligned} $$
(3.10)
Proceeding in the similar way and using condition (
1.1) and the definition of weight function, we obtain
$$ \begin{aligned}[b] ={}&{-} \int_{I} f(x)f^{(\mathit{vi})}(x)w(x)\,dx+ \int_{I} f(x)f^{(\mathit{iv})}(x)w{''}(x)\, dx \\ &+\frac{7}{2} \int_{I} \bigl(f{''}(x) \bigr)^{2}w{''}(x) \,dx- \int_{I} \bigl(f{'}(x) \bigr)^{2}w^{(\mathit{iv})}(x) \,dx.\end{aligned} $$
(3.11)
Now we take
$$\int_{I} \bigl(f{''}(x) \bigr)^{2}w{''}(x) \,dx. $$
Using Theorem 2.1 from [
16], we have
$$ \begin{aligned}[b] \int_{I} \bigl(f{''}(x) \bigr)^{2}w{''}(x) \,dx={}& \int_{I} f(x)f^{(\mathit{iv})}(x)w{''}(x)\,dx - 2 \int_{I} f(x)f{''}(x)w^{(\mathit{iv})}(x)\, dx \\ &+\frac{1}{2} \int_{I} f^{2}(x)w^{(\mathit{vi})}(x)\,dx.\end{aligned} $$
(3.12)
Now, using (2.6) of [
15], we have
$$ \begin{aligned}[b] \int_{I} \bigl(f{'}(x) \bigr)^{2} w^{(\mathit{iv})}(x)\,dx ={}&\frac{1}{2} \int_{I} f^{2}(x) w^{(\mathit{vi})}(x)\,dx \\ &- \int_{I} f(x)f{''}(x)w^{(\mathit{iv})}(x)\,dx.\end{aligned} $$
(3.13)
Substituting (
3.12) and (
3.13) in (
3.11), we have
$$\begin{aligned} ={}&{-} \int_{I} f(x)f^{(\mathit{vi})}(x)w(x)\,dx + \frac{9}{2} \int_{I} f(x)f^{(\mathit{iv})}(x)w{''}(x)\,dx \\ &-6 \int_{I} f(x)f{''}(x)w^{(\mathit{iv})}(x)\,dx+ \frac{5}{4} \int_{I} f^{2}(x)w^{(\mathit{vi})}(x)\,dx \\\leq{}&\sup_{x\in I}\big|f(x)\big| \int_{I} \big|f^{(\mathit{vi})}(x)\big|\big|w(x)\big|\,dx + \frac{9}{2} \sup_{x\in I}\big|f(x)\big| \int_{I} \big|f^{(\mathit{iv})}(x)\big|\big|w{''}(x)\big|\,dx \\&+6 \sup_{x\in I}\big|f(x)\big| \int_{I} \big|f{''}(x)\big|\big|w^{(\mathit{iv})}(x)\big|\,dx + \frac {5}{4} \int_{I} f^{2}(x)\big|w^{(\mathit{vi})}\big|(x)\,dx. \end{aligned}$$
(3.14)
Here,
$$\begin{gathered} \big|f{''}(x)\big|\leq F{''}(x), \\\big|f^{(\mathit{iv})}(x)\big|\leq F^{(\mathit{iv})}(x), \\\big|f^{(\mathit{vi})}(x)\big|\leq F^{(\mathit{vi})}(x), \\w{''}(x)\leq0, \\w^{(\mathit{iv})}(x)\geq0.\end{gathered} $$
Using the above conditions, we obtain
$$\begin{aligned} \leq{}&\sup_{x\in I}\big|f(x)\big| \int_{I} F^{(\mathit{vi})}(x)w(x)\,dx - \frac{9}{2} \sup _{x\in I} \int_{I} F^{(\mathit{iv})}(x)w{''}(x)\,dx \\&+6 \sup_{x\in I}\big|f(x)\big| \int_{I} F{''}(x)w^{(\mathit{iv})}(x)\,dx + \frac {5}{4} \int_{I} f^{2}(x)\big|w^{(\mathit{vi})}\big|(x)\,dx.\end{aligned} $$
Using the integration by parts formula, we obtain
$$\begin{aligned} \leq{}&\sup_{x\in I}\big|f(x)\big| \int_{I} F(x)w^{(\mathit{vi})}(x)\,dx - \frac{9}{2} \sup _{x\in I} \int_{I} F(x)w^{(\mathit{vi})}(x)\,dx \\&+6 \sup_{x\in I}\big|f(x)\big| \int_{I} F(x)w^{(\mathit{vi})}(x)\,dx + \frac{5}{4} \int _{I} f^{2}(x)\big|w^{(\mathit{vi})}\big|(x)\,dx.\end{aligned} $$