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I want to prove that for $a<b$ \begin{align*} \left\vert erf\left(\sqrt{\pi\gamma+\mathrm{i} a}\right) \right\vert^{2}-\left\vert erf\left(\sqrt{\pi\gamma+\mathrm{i} b}\right) \right\vert^{2}>0, \end{align*} where $a,b,\gamma\in\mathbb{R}^{+}$ and \begin{align*} erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^2}~dt \end{align*} is error function. I tried by series expansion but there is no conclusion at the end.

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False. For $a=1, b=2, \gamma=1$ we get $$ \left( \left| {\rm erf} \left(\sqrt {\pi +i}\right) \right| \right) ^{2}- \left( \left| {\rm erf} \left(\sqrt {\pi +2\,i}\right) \right| \right) ^{2}=- 0.0244743830 $$

Perhaps you didn't write what you meant?

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