When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$.

\begin{equation} f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( \gamma+b_1) } \end{equation}

and

\begin{equation} f_1(x)=\frac{[b_1 g'(x) +(g'(1) b_2x-g'(1) b_2) g(x)]\gamma+g'( 1) b_1b_2xg( x) }{b_1g'( x) \gamma+b_1 b_2 g'( x) } \end{equation}

where

1. $g(x)=\sum_{k=0}^{\infty} p(k)x^k$ is the generating function and $0 \leq p(k) \leq 1$ and $g(1)=p(0)+p(1)+p(2)+...=1$. $g'(x)$ is the derivative of $g(x)$. For example: $g(x)=0.3x+0.1x^2+0.6x^4$, and $g'(x)=0.3+0.2x+2.4x^3$

2. $0 \leq b_1 \leq 1$, $0 \leq b_2 \leq 1$, and $0 \leq \gamma \leq 1$

3. $n$ is a natural number and $n>2$

4. If it is helpful, $p(0)=0$, and $g(0)=0$ can also be used as a known condition.

Here is the programming code (in which $gd(x)$ means $g'(x)$) for this equation:

f(x):=(b_1*gd(x)*(((b_1*gd(x)+(gd(1)*b_2*x-gd(1)*b_2)*g(x))*gamma+gd(1)*b_1*b_2*x*g(x))/(b_1*gd(x)*gamma+b_1*b_2*gd(x)))^(n-2)+gd(1)*gamma)/(gd(1)*(gamma+b_1));

It is using the Maxima definition but can be used in Mathematica with minor modifications.

I have checked that many f(x) (given detailed $b_1,b_2,\gamma,g(x)$, and $n$ values) are convex but can not prove it.

It is of critical importance for me. Any help would be much appreciated!