I want to minimize the following function $$\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+2-\Gamma(1,\frac{eaf}{b(1-x)},\frac{eagf}{bx(1-y)})$$ where $a,b,c,d,e,f,g,H$ are constants and greater than $0$. $\Gamma(a,x_1,x_2)$ is the generalized incomplete Gamma function. The minimization is to be performed with respect to $x,y$ with $0\leq x\leq 1$ and $0\leq y\leq 1$.
I tried to plot $\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+1$ and $1-\Gamma(1,\frac{eaf}{b(1-x)},\frac{eagf}{bx(1-y)})$ for different values of $a,b,c,d,e,f,g,H$ and for those values it appears that both are convex. Since, the sum of two convex functions is also convex therefore I can conclude that the function is convex for those special values of $a,b,c,d,e,f,g,H$. However, I do not know how to show that the above function is convex or otherwise for general values of $a,b,c,d,e,f,g,H$.
I believe that the highest product of $x$ and $y$ minimizes $\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+1$ (because as $xy\to\infty$ the value approach 1). Can this information simplify this problem? I will be very thankful for any help in this regard.
The first summand $$f(x,y) = \dfrac{a}{bxy + cd} e^{\frac{a}{bxy+cd}} H$$ is not convex. Its Hessian at $(0,0)$ is $$ \left[ \begin {array}{cc} 0&-{\frac {Hab \left( a+{\it cd} \right) }{ {{\it cd}}^{3}}{{\rm e}^{{\frac {a}{{\it cd}}}}}}\\ -{\frac {Hab \left( a+{\it cd} \right) }{{{\it cd}}^{3}}{{\rm e}^{{ \frac {a}{{\it cd}}}}}}&0\end {array} \right] $$ which is not positive semidefinite (it has both positive and negative eigenvalues). Thus for sufficiently small positive $x, y$ there will still be a negative eigenvalue.