As done in the above answers and comments, the positivity of the function
\begin{equation*}
g(u)= \frac{1}{e^u-1}-\frac{1}{e^{u/p}-1} - \frac{1}{e^{u/q}-1}
\end{equation*}
is equivalent to the inequality
\begin{equation}
\frac{1}{e^u-1} \geq \frac{1}{e^{u/p}-1} + \frac{1}{e^{u/q}-1}\tag{1}\label{1}
\end{equation}
for $u>0$ and $p,q>0$ such that $p+q=1$.

The inequality \eqref{1} was recovered in [1, Lemma 1] by Alzer in the form of
\begin{equation}
\frac{1}{y-1}-\frac{1}{y^{1/\lambda}-1}-\frac{1}{y^{1/(1-\lambda)}-1}>0, \quad y>1, \quad \lambda\in(0,1).\tag{2}\label{2}
\end{equation}
In [2, Lemma 1.4], Ouimet generalized the inequality \eqref{2} to
\begin{equation}
\frac1{y-1}-\sum_{k=1}^n\frac{1}{y^{1/\lambda_k}-1}>0,\tag{3}\label{3}
\end{equation}
where $y>1$ and $\lambda_1,\lambda_2,\dotsc,\lambda_n\in(0,1)$ such that $\sum_{k=1}^{n}\lambda_k=1$.

In the paper [3], the inequality
\begin{equation}
\Biggl(\sum_{k=1}^n\lambda_k\Biggr) H\biggl(\frac{x}{\sum_{k=1}^n\lambda_k}\biggr) \ge\sum_{k=1}^{n}\lambda_k H\biggl(\frac{x}{\lambda_k}\biggr)\tag{4}\label{4}
\end{equation}
for $\lambda_k>0$ and $x>0$ was proved to be true, where
$$
H(t)=\frac{t}{e^t-1}, \quad t\in\mathbb{R}
$$
is the generating function of the Bernoulli numbers.

In the preprint [4], the inequality
\begin{equation}
\Biggl(\sum_{k=1}^n\lambda_k\Biggr)^3 H\biggl(\frac{x}{\sum_{k=1}^n\lambda_k}\biggr)
\ge\sum_{k=1}^{n}\lambda_k^3 H\biggl(\frac{x}{\lambda_k}\biggr)\tag{5}\label{5}
\end{equation}
was proved to be true for $\lambda_k>0$ and $x>0$.

In the preprint [5], Ouimet proved the inequality
\begin{equation}
\sum_{i=1}^m\frac1{y^{1/\nu_i}-1}+\sum_{j=1}^n\frac1{y^{1/\tau_j}-1}
>\sum_{i=1}^m\sum_{j=1}^n\frac1{y^{1/\lambda_{ij}}-1}\tag{6}\label{6}
\end{equation}
for $y>1$ and $0<\lambda_{ij}\le1$, where $\nu_i=\sum_{j=1}^n\lambda_{ij}$ and $\tau_j=\sum_{i=1}^m\lambda_{ij}$ satisfying $\sum_{i=1}^m\nu_i=\sum_{j=1}^n\tau_j=1$.

In the paper [6, Theorem 3.1], Ouimet's inequality \eqref{6} was refined as
\begin{equation}
\sum_{i=1}^m\frac1{e^{x/\nu_i}-1}+\sum_{j=1}^n\frac1{e^{x/\tau_j}-1}
\ge2\sum_{i=1}^m\sum_{j=1}^n\frac1{e^{x/\lambda_{ij}}-1}\tag{7}\label{7}
\end{equation}
for $x>0$ and $\lambda_{ij}>0$, where $\nu_i=\sum_{j=1}^n\lambda_{ij}$ and $\tau_j=\sum_{i=1}^m\lambda_{ij}$.

The inequality \eqref{7} established in [6, Theorem 3.1] extends, generalizes, and refines all of the above inequalities other than \eqref{5}.

In the paper [7], the inequalities \eqref{5} and \eqref{7} were generalized as follows.

- For $\alpha\ge1$, $x>0$, and $\lambda_{ij}>0$ for $1\le i\le m$ and $1\le j\le n$, denote $\nu_i=\sum_{j=1}^n\lambda_{ij}$ and $\tau_j=\sum_{i=1}^m\lambda_{ij}$. Then
\begin{equation}\label{alpha>=1-inequal}\tag{8}
\sum_{i=1}^m\frac{\nu_i^{\alpha-1}}{e^{x/\nu_i}-1}+ \sum_{j=1}^n\frac{\tau_j^{\alpha-1}}{e^{x/\tau_j}-1}
\ge2\sum_{i=1}^m\sum_{j=1}^n\frac{\lambda_{ij}^{\alpha-1}}{e^{x/\lambda_{ij}}-1}.
\end{equation}
- Let $\alpha\ge1$, $x>0$, and $\lambda_{ijk}>0$ for $1\le i\le\ell$, $1\le j\le m$, and $1\le k\le n$. Then
\begin{multline}\label{2:1Sum-Ineq}
\sum_{k=1}^n\sum_{j=1}^m\frac{\bigl(\sum_{i=1}^\ell\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{i=1}^\ell\lambda_{ijk}}-1}
+\sum_{i=1}^\ell\sum_{k=1}^n\frac{\bigl(\sum_{j=1}^m\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{j=1}^m\lambda_{ijk}}-1}\\
+\sum_{j=1}^m\sum_{i=1}^\ell\frac{\bigl(\sum_{k=1}^n\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{k=1}^n\lambda_{ijk}}-1}
\ge3\sum_{k=1}^n\sum_{j=1}^m\sum_{i=1}^\ell\frac{\lambda_{ijk}^{\alpha-1}}{e^{x/\lambda_{ijk}}-1}
\end{multline}
and
\begin{multline}\label{1:2Sum-Ineq}
\sum_{k=1}^n\frac{\bigl(\sum_{j=1}^m\sum_{i=1}^\ell\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{j=1}^m\sum_{i=1}^\ell\lambda_{ijk}}-1}
+\sum_{i=1}^\ell\frac{\bigl(\sum_{k=1}^n\sum_{j=1}^m\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{k=1}^n\sum_{j=1}^m\lambda_{ijk}}-1}\\
+\sum_{j=1}^{m} \frac{\bigl(\sum_{i=1}^\ell\sum_{k=1}^n\lambda_{ijk}\bigr)^{\alpha-1}}{e^{x/\sum_{i=1}^\ell\sum_{k=1}^n\lambda_{ijk}}-1}
\ge3\sum_{k=1}^n\sum_{j=1}^m\sum_{i=1}^\ell\frac{\lambda_{ijk}^{\alpha-1}}{e^{x/\lambda_{ijk}}-1}.
\end{multline}

The inequality \eqref{alpha>=1-inequal} was also employed in the paper [8].

In the preprint [9], whose revised version has been accepted by Acta Mathematica Scientia, the inequality
\begin{equation}
\Biggl(\sum_{k=1}^{n}w_k\lambda_k\Biggr)^{\alpha+1}H\Biggl(\frac{x}{\sum_{k=1}^{n}w_k\lambda_k}\Biggr)
\le\sum_{k=1}^{n}w_k\lambda_k^{\alpha+1}H\biggr(\frac{x}{\lambda_k}\biggr)
\end{equation}
was proved, where $\alpha\ge0$, $n\ge2$, $\lambda_k\in(0,\infty)$ and $w_k\in(0,1)$ for $1\le k\le n$, and $\sum_{k=1}^{n}w_k=1$.

Finally, a closely-related or similar inequality was proved in the preprint [10, Lemma 2.1].

References

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*Complete monotonicity for a new ratio of finitely many gamma functions*, Acta Mathematica Scientia Series B English Edition **42B** (2022), no. 2, 511--520; available online at https://doi.org/10.1007/s10473-022-0206-9.
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*Logarithmically complete monotonicity of a matrix-parametrized analogue of the multinomial distribution*, arXiv (2021), available online at https://arxiv.org/abs/2105.01494.