Let $h(x,y,z,t):ℝ³×(0,+∞)→ℝ$ be a given function depending on the variables $x,y,z$ and $t$ is a fixed parameter such that $h(x,y,z,t)>0$. Let $d(x,y,z):ℝ³→ℝ$ be a given function depending on the variables $x,y,z$ such that $d(x,y,z)≥0$.
My question is:
Can we find real functions $f(x,y,z,t),g(x,y,z,t):ℝ³×(0,+∞)→ℝ$ such that $f(x,y,z,t)>h(x,y,z,t)$ and $g(x,y,z,t)<0$ and the infinimum of the function $-(2g(x,y,z,t)/f(x,y,z,t))$ over the set $d(x,y,z)=0$ depends on only on the parameter $t$ and not on the variables $x,y,z$.