Consider two real-valued function $\phi $ and $\Delta$, defined as \begin{equation*} \phi (m,\theta ,j,a)=\frac{m-1}{\left( j+(m-1)\theta \right) ^{a}}-\frac{m+1 }{\left( j+m\theta \right) ^{a}}\text{,} \end{equation*} \begin{equation*} \Delta(m,\theta,a)=\left( \sum \limits_{j=1}^{\theta }\phi (m,\theta ,j,a)\right) -\left( \sum \limits_{j=1}^{\theta +1}\phi (m,\theta +1,j,a)\right) \text{.} \end{equation*}
where $m \in \{2,3,\dots\}, \theta \in \{1,2,\dots\}, j \in \{1,2,\dots\}$, and $a$ is nonnegative real number.
My conjecture: For any given $m$ and $\theta $, there exist $\underline{a}=\underline{a}(m,\theta )$ and $\overline{a}=\overline{a} (m,\theta )$ (with $1<\underline{a}<\overline{a}$) such that \begin{equation*} \Delta(m,\theta,a)<0\text{ if and only if }a\in (\underline{a},\overline{a})\text{.} \end{equation*}
So far, I have tried many approaches ranging from Taylor's Expansion to Jensen's inequality. None of them worked.
Any comments will be highly appreciated.