If I know $$\frac{\alpha}{\alpha+\beta}<\frac{\lambda}{\lambda+\gamma}$$ can I know the sign of $$\frac{\alpha+1}{\alpha+1+\beta}<\frac{\lambda+1}{\lambda+1+\gamma} $$ And the sign of $$(\frac{\alpha+1}{\alpha+1+\beta}-\frac{\alpha}{\alpha+\beta})-(\frac{\lambda+1}{\lambda+1+\gamma}- \frac{\lambda}{\lambda+\gamma})$$
The background of this question is that I want to know if the speed of learning for beta distribution is affected by the prior. If I have a smaller prior, do I learn faster ?
Assuming everything is positive, the first inequality is equivalent to $\alpha\gamma<\lambda\beta$ and the second to $(\alpha+1)\gamma<(\lambda+1)\beta$, or to $\alpha\gamma+\gamma<\lambda\beta+\beta$.
So the question is: If
$$\alpha\gamma<\lambda\beta,\tag 1$$
then does it follow that $\alpha\gamma+\gamma<\lambda\beta+\beta$, or
$$(\alpha+1)\beta<(\lambda+1)\gamma\text{ ?}\tag 2$$
So I would try to figure out what the spaces of solutions of $(1)$ and $(2)$ within the space $\alpha,\beta,\gamma,\lambda>0$ look like.