Is this expression positive, negative or both under different regions of positive real numbers?

Please show if this expression is positive, negative or both under different
regions of positive real numbers?

$$
g\left(y\right)=\int_{0}^{Q}2\left(Q-x\right)\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2\left\{ \int_{0}^{Q}F\left(x,y\right)\,dx\right\} \int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

Given,

$$
x,y,Q\geq0;\;F\left(x,y\right)>0;\;F\left(\infty,y\right)=1;\;\frac{\partial F\left(x,y\right)}{\partial y}\leq0;\;\frac{\partial F\left(x,y\right)}{\partial x}\geq0
$$

Please let me know if the question is not well formed or if anything
is not clear.

Steps Tried,

$$
g\left(y\right)\leq\int_{0}^{Q}2\left(Q-x\right)\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2F\left(Q,y\right)\,\left\{ \int_{0}^{Q}dx\right\} \int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}2\left(Q-x\right)\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2QF\left(Q,y\right)\,\int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}\left[2\left(Q-x\right)-2QF\left(Q,y\right)\right]\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}\left[2\left(Q-x\right)-2QF\left(Q,y\right)\right]\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx<0
$$
If the below condition holds,
$$
\int_{0}^{Q}2\left(Q-x\right)\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx<\int_{0}^{Q}2QF\left(Q,y\right)\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$
$$
\int_{0}^{Q}2\left(Q-x\right)\,dx>\int_{0}^{Q}2QF\left(Q,y\right)\,dx
$$
$$
\left|\left(Qx-\frac{x^{2}}{2}\right)\right|_{0}^{Q}>QF\left(Q,y\right)\left|x\right|_{0}^{Q}
$$
$$
\frac{Q^{2}}{2}>Q^{2}F\left(Q,y\right)
$$
$$
F\left(Q,y\right)<\frac{1}{2}
$$

Alternately,

$$
g\left(y\right)\geq\int_{0}^{Q}2Q\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2\left\{ \int_{0}^{Q}F\left(x,y\right)\,dx\right\} \int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\geq\int_{0}^{Q}\left[2Q-2\left\{ \int_{0}^{Q}F\left(x,y\right)\,dx\right\} \right]\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

Am unable to simplify this alternate condition further.

Related Question:

Inference Regarding Definite Integral of Product of Two Functions

Solutions Collecting From Web of "Is this expression positive, negative or both under different regions of positive real numbers?"

Steps Tried,

$$
g\left(y\right)\leq\int_{0}^{Q}2\left(Q-x\right)\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2F\left(Q,y\right)\,\left\{ \int_{0}^{Q}dx\right\} \int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}2\left(Q-x\right)\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2QF\left(Q,y\right)\,\int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}\left[2\left(Q-x\right)-2QF\left(Q,y\right)\right]\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\leq\int_{0}^{Q}\left[2\left(Q-x\right)-2QF\left(Q,y\right)\right]\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx<0
$$
If the below condition holds,
$$
\int_{0}^{Q}2\left(Q-x\right)\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx<\int_{0}^{Q}2QF\left(Q,y\right)\,\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$
$$
\int_{0}^{Q}2\left(Q-x\right)\,dx>\int_{0}^{Q}2QF\left(Q,y\right)\,dx
$$
$$
\left|\left(Qx-\frac{x^{2}}{2}\right)\right|_{0}^{Q}>QF\left(Q,y\right)\left|x\right|_{0}^{Q}
$$
$$
\frac{Q^{2}}{2}>Q^{2}F\left(Q,y\right)
$$
$$
F\left(Q,y\right)<\frac{1}{2}
$$

Alternately,

$$
g\left(y\right)\geq\int_{0}^{Q}2Q\frac{\partial F\left(x,y\right)}{\partial y}\,dx-2\left\{ \int_{0}^{Q}F\left(x,y\right)\,dx\right\} \int_{0}^{Q}\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

$$
g\left(y\right)\geq\int_{0}^{Q}\left[2Q-2\left\{ \int_{0}^{Q}F\left(x,y\right)\,dx\right\} \right]\frac{\partial F\left(x,y\right)}{\partial y}\,dx
$$

Am unable to simplify this alternate condition further.