Estimations of classes of integrals constructed with the help of the classical warping function

Let $G$ be a multiply connected plane domain. We denote by $\Gamma_0$ the outer boundary curve of $G$, and by $\Gamma_1,\ldots,\Gamma_n$ the internal boundary curves. The boundary-value problem that defines the warping function $u(x,G)$ of $G$ is
\begin{gather*} \Delta u =-2 \quad\text{in }G, \\ \begin{alignedat}{2} u&=0 &\qquad &\text{on } \Gamma_0, \\ u&=c_i &\qquad &\text{on } \Gamma_i,\ i=1,\dots,n, \end{alignedat} \end{gather*}
where the constants $c_i$ are determined by the conditions
$$ \oint_{\Gamma_i}\frac{\partial u}{\partial n}\, {\mathrm{d}} s=-2a_i,\qquad i=1,\dots,n, $$
$\partial/\partial n$ is the inward normal derivative, and $a_i$ is the area enclosed by $\Gamma_i$.
In the next two assertions we give estimates for a class of integrals of the warping function.
Let a function $F(t)$ have the representation
$$ F(t):=p\int\limits_0^t s^{p-1}f(s){\mathrm{d}} s, $$
where $p>0$, and $f(s)$ is another function, whose properties play an important role, as we see below.
Theorem 1. Let $G$ be a multiply connected domain and let $p>0$ such that $\mathbf{T}_{p}(G)<+\infty$. Then:
$1)$ If $f(s)$ is a non-decreasing function, then
$$ \int_G F(u(x,G))\,{\mathrm{dA}}\le \int_{R_p}F(u(x,R_p))\,{\mathrm{dA}}. $$

$2)$ if $f(s)$ is a non-increasing function, then an inverse inequality holds
$$ \int_G F(u(x,G))\,{\mathrm{dA}}\ge \int_{R_p} F(u(x,R_p))\,{\mathrm{dA}}. $$

Here $R_p$ is a concentric ring with the same joint area of the holes as on $G$, and the ring $R_p$ satisfy the equality $\mathbf{T}_p(R_p)=\mathbf{T}_p(G)$. Both equalities hold if and only if $G$ is a ring bounded by two concentric circles.
Using the functionals $\mathbf{T}_p(G)$ and ${\mathbf{u}(G)}$ we can get explicit bounds for integrals of the warping function.
Theorem 2. Under the assumptions of Theorem 1 the following estimates hold
$$ \int_G F(u(x,G))\,{\mathrm{dA}}\le \frac{\mathbf{T}_p(G)}{\mathbf{u}(G)^p}F({\mathbf{u}(G)})-\frac{2\pi {\mathbf{u}(G)}F({\mathbf{u}(G)})}{p+1}+2\pi\int\limits_{0}^{{\mathbf{u}(G)}}F(t)\,{\mathrm{d}} t, $$
where $f(s)$ is a non-decreasing function, and
$$ \int_G F(u(x,G))\,{\mathrm{dA}}\ge \frac{\mathbf{T}_p(G)}{{\mathbf{u}(G)}^p}F({\mathbf{u}(G)})-\frac{2\pi {\mathbf{u}(G)}F({\mathbf{u}(G)})}{p+1}+2\pi\int\limits_{0}^{{\mathbf{u}(G)}}F(t)\,{\mathrm{d}} t, $$
here $f(s)$ is a non-increasing function.
Equalities hold if and only if $G$ is a concentric ring.