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given is the function [tex] h(y_s) = \ln (1-y_s) - \ln y_s - \gamma + \frac{\gamma}{\theta + \beta (1-y_s)} [/tex]

my job is now to show that [itex] h'(y_s) < 0, \forall y_s \in ]0,1[ [/itex] when

[tex] \frac{\gamma \beta}{\theta (\beta + \theta)} < 4 [/tex]

I guess that all constants can be assumed to be real and positive.

My first tought was to introduce the new variable [itex] z=1-y_s [/itex] so that after differentiation i would get

[tex] - 1/z + 1/(z-1) + \frac{\gamma \beta}{(\theta + \beta z)^2} < 0[/tex]

but i cant derive the above expression from this inequality. Any help would be greatly appreciated.

Thank you.