Let the density of the generalzied gamma:

\( f(x)=\frac{|a|}{\Gamma(g)}g^gl^{ag}x^{ag-1}e^{-g(lx)^a}\)

with survival function:

\( S(x)= I(g(lt)^a,g), \ \ if \ \ a<0 \),

\( S(x)= 1-I(g(lt)^a,g), \ \ if \ \ a>0 \),

where the I(.,.) is the incomplete gamma function defined as:

\( I(x,a)= \frac{1}{\Gamma(a)}\int_{0}^{x}e^{-t}t^{a-1}dt \),

The hazard rate h(x)=f(x)/S(x) is one of the following:

1. increasing

2. decreasing

3. U shaped

4. bath shaped

5. constant. (that is easy, it the exponential distribution case for a=g=1)

I want to know for what values of the parameters the above forms of the hazard rate take place? For example for a>1 and a>g/2 then the h(x) is increasing (I just made up the constraint.)

So I evaluated the derivative of the h(x)=hder using matlab.

First I try to find when hder>0. Matlab seems to not support solving inequalities, maple does support but I had no luck because the inequality in nonlinear.

Are you aware of any software that could do the job? And maybe provide the one or two lines of code required?

Can mathematica do this? I have no access but soon I will. I searced the net and saw InequalityGraphics package and looks promising.

Is anyone aware of this here? Can someone try and tell me??

Thanx in advance for any answers!!

If you have any questions please feel free to tell me!