REVISED
From Theorem 2.1 in your citation $[1]$ we have: for $p_k=h_k=x_{k}-x_{*}$ and $x_k$ is a sequence of points in $\textbf{R}^N$ such that $\lim_{k\rightarrow \infty}x_k=x_{*}$ , $\nabla f(x_{*})=0$, $\nabla^2 f(x_{*})\neq 0$. Hence for all $x$ in an open ball $B$ with center $x_{*}$ we have $||\nabla^2f(x)||>0$.
$$
\nabla f(x_k)=\nabla f(x_{*}+h_k)=\nabla f (x_*)+\int^{1}_{0}\nabla^2 f(x_{*}+th_k)h_kdt\Rightarrow
$$
$$
\nabla f(x_k)=\int^{1}_{0}\nabla^2 f(x_{*}+th_k)h_kdt.\tag 1
$$
Since $L_H\geq ||\nabla^2f(x)||\geq L'>0$ : $(2)$ in $B$, we get
$$
||g_k||=||\int^{1}_{0}\nabla^2 f(x_{*}+th_k)h_kdt||\leq \int^{1}_{0}||\nabla^2f(x_*+th_k)||\cdot ||h_k||dt=
$$
$$
\leq L_H\int^{1}_{0}dt(||h_k||)=L_H ||h_k||.
$$
Also it holds from the definition of norm of matrices $A,B$, where $A\in M_{n\times n}$ and $B\in M_{n\times 1}$ that $||A\cdot B||\leq ||A||\cdot ||B||$ hence we can define $\Gamma(A,B)$ to be such that $\Gamma(A,B)=\frac{A\cdot B}{||A||\cdot ||B||}$. (Note that $\Gamma(A,B)$ is a vector and if $A\in M_{n\times n}$, $B\in M_{n\times 1}$, then $\Gamma=\Gamma(A,B)\in M_{n\times 1}$ and $0\leq ||\Gamma(A,B)||\leq 1$ see also Matrix Norm provited that $B$ is small enough and positive). One can see that one such norm is
$$
||A||_2:=\left(\sum^{n}_{i,j=1}|a_{i,j}|^2\right)^{1/2}\textrm{, }A\in M_{n\times n}
$$
and
$$
||B||_2:=\left(\sum^{n}_{i=1}|b_i|^2\right)^{1/2}\textrm{, }B\in M_{n\times 1}.\tag 2
$$
Hence there holds if $A,C\in M_{n\times n}$ and $B\in M_{n\times 1}$
$$
i)\textrm{ }||A||_2\geq 0\textrm{ and }||A||_2=0\Leftrightarrow A=0
$$
$$
ii)\textrm{ }||cA_2||_2=|c|\times ||A||_2
$$
$$
iii)\textrm{ }||A+C||_2\leq ||A||_2+||C||_2
$$
$$
iv)\textrm{ }||A\cdot C||_2\leq ||A||_2\cdot ||C||_2
$$
$$
v)\textrm{ }||A\cdot B||_2\leq ||A||_2||B||_2.
$$
Also if $A=(a_{i,j})_{n\times n}$ and $0<||A||_2\leq 1$, then exists $i_0,j_0$ such that $a_{i_0,j_0}\neq 0$ and for all $i,j\Rightarrow |a_{i,j}|\leq 1$ and if $B=(b_{i,j})_{n\times 1}$ and $0<||B||_2\leq 1$, then exists $i_0$ such that $b_{i_01}$ and for all $i$, $0<|b_{i,1}|\leq 1$. In our case it is
$$
A=\nabla^2f(x_{*}+th_k)\in M_{n\times n}\textrm{ and }\Gamma_k(t),g_k,h_k\in M_{n\times 1}.
$$
Also
$$
\Gamma_k(t)=\frac{\nabla^2f(x_{*}+th_k)\cdot h_k}{||\nabla^2f(x_{*}+th_k)||_2\cdot ||h_k||_2}.
$$
Hence
$$
g_k=\int^{1}_{0}\nabla^2f(x_{*}+th_{k})\cdot h_kdt=\int^{1}_{0}||\nabla^2 f(x_{*}+th_k)||_2\cdot ||h_k||_2\Gamma_k(t)dt=
$$
$$
=||h_k||_2\int^{1}_{0}||\nabla^2 f(x_{*}+th_k)||_2\Gamma_k(t)dt.
$$
Here $\Gamma_k(t)=\sum^{n}_{j=1}\Gamma^{(j)}_{k}(t)\left(\overline{e}_{j}\right)^T$, where $\overline{e}_1=\{1,0,0,\ldots,0\}$, $\overline{e}_{2}=\{0,1,0,\ldots,0\}$,$\ldots$, $\overline{e}_{N}=\{0,0,0,\ldots,1\}$ is the usual base of $\textbf{R}^{n}$ and $\Gamma^{(j)}_{k}(t)$ are continuous functions of a single variable in $\textbf{R}$ such that $0\leq ||\Gamma_k(t)||_2\leq 1\Rightarrow 0\leq |\Gamma^{(j)}_k(t)|\leq 1$, (since we can choose such norm). Hence in our case it is
$$
g_k=||h_k||_2\cdot \int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\left(\sum^{n}_{j=1}\Gamma^{(j)}_{k}(t)(\overline{e}_j)^T\right)dt=
$$
$$
=||h_k||_2\cdot \sum^{n}_{j=1}\left(\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\cdot \Gamma^{(j)}_{k}(t)dt\right)(\overline{e}_j)^T.
$$
Hence
$$
||g_k||_2=||h_k||_2\cdot ||\sum^{n}_{j=1}\left(\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\cdot \Gamma^{(j)}_{k}(t)dt\right)(\overline{e}_j)^T||_2.
$$
But
$$
0<||\sum^{n}_{j=1}\int^{1}_{0}||\nabla^2 f(x_{*}+th_k)||_2\cdot \Gamma^{(j)}_k(t)dt(\overline{e}_j)^T||_2=
$$
$$
=\left(\sum^{n}_{j=1}\left|\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt\right|^2\right)^{1/2}\leq C.
$$
Otherwise for all $j$ it would hold
$$
\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt=0\Rightarrow ||g_k||_2=0\Rightarrow g_k=0,
$$
for a certain $k=k_0\in\textbf{N}$ hence we have two points $x_{k_0},x_{*}\in B$, such that $\nabla f(x_{*})=\nabla f(x_{k_0})=0\Rightarrow \exists \xi\in B: \nabla^2f(\xi)=0$, which is imposible (see Mean value theorem). If we have two distinct points $P_1,P_2$ near each other with zero gradient, then exists point $P_0$ near $P_1,P_2$ such that Hessian is zero at $P_0$.
Thus from not vanish of
$$
||\sum^{n}_{j=1}\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt(\overline{e}_j)^T||_2=
$$
$$
=\left(\sum^{n}_{j=1}\left|\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt\right|^2\right)^{1/2}
$$
we have that exists at least one $j=j_0$ such that
$$
0<m^{(j_0)}_k\leq \left|\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j_0)}_{k}(t)dt\right|.
$$
Hence from the continuity of all $\Gamma^{(j)}_{k}(t)$, we can write for some costants $m,M$ such that $0<m<M<\infty$ that
$$
0<m ||h_k||_2\leq ||g_k||_2\leq ||h_k||_2 M.
$$
Hence
$$
\frac{m ||h_{k+1}||_2}{M^2 ||h_k||_2^2}\leq\frac{||g_{k+1}||_2}{||g_k||_2^2}\leq \frac{M||h_{k+1}||_2}{m^2||h_k||_2^2}
$$
Hence we have using $||g_{k+1}||_2\leq c ||g_k||_2^2$ that
$$
||h_{k+1}||_2\leq c \frac{M^2}{m}||h_k||_2^2=c' ||h_k||_2^2\textrm{, when }k>>1.
$$
Note
In this way $||.||_2$ it is a traditional norm plus the condition $||A\cdot x^{T}||\leq ||A||_2\cdot ||x^{T}||_2$. For $A$ given and $x^T$ small enough (where $x=\{x^{(1)},x^{(2)},\ldots,x^{(n)}\}$).
We have
$$
S=||\sum^{n}_{j=1}\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt(\overline{e}_j)^T||_2=
$$
$$
=||\sum^{n}_{j=1}\left(\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(j)}_k(t)dt\right)(\overline{e}_j)^T||_2=
$$
$$
||\left(
\begin{array}{cc}
\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(1)}_k(t)dt\\
\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(2)}_k(t)dt\\
\ldots
\\
\int^{1}_{0}||\nabla^2f(x_{*}+th_k)||_2\Gamma^{(n)}_k(t)dt
\end{array}
\right)||_2
$$
Hence from relation $(2)$, we get
$$
S=\left(\sum^{n}_{j=1}\left|\int^{1}_{0}||\nabla^2f(x_{*}+h_kt)||_2\Gamma^{(j)}_{k}dt\right|^2\right)^{1/2}
$$
