I have in a resarche article "Distributions of Transmit Power and SINR in Device-to-Device Networks" an equation that I can't prouve it (Eq 7). I applied Jensen's inequality, but I can't go further. Following the equation:
$$ E\{\,exp[-s.\sum_{t\epsilon\phi,R_t\gt R_0}\,P_t.H_t.R_t^{-a}]\,\} \ge exp\,\{-πλ (s.E[P_0])^{2/a}.G_a[\frac{R_0²}{(s.E[P_0])^{2/a}}]\,\} $$
When: $G_a(d)= \binom{1/sinc(2/a) \quad ,\,d=0}{\frac{2d}{(a-2)(1+d^{a/2})}.\, _2F_1(1\,;\,1\,;\,2-\frac2a\,;\,\frac{1}{1+d^{a/2}})\quad ,\,d\gt0} $
and: $\,_2F_1(a;b;c;z)\,$ is the hypergeometric function. E is the expectation value. "a" is the path loss exponent. $H_t ∼Exp(μ)$. Rt is the distance between mobiles follow homogeneous PPP Φ with (constant) intensity λ. $P_t$ transmit powers.
I know that it's not easy and it's not enough readable, but I hope that There is some one of you who see this equation in other article with its demonstration ?
If I forgot to mention some thing, please ask me to do.
Many thanks in advance.