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Simplify the following system of inequities: $$\frac{d^2y}{dx^2}<\frac{dy}{xdx}$$ $$\frac{d^2x}{dy^2}<\frac{dx}{ydy}$$

Known that $F(x,y)=0$. $F$ is monotonic.


I got something like this:

$$\frac{F_{yy}}{F_y}-\frac{F_{xy}}{F_x}<1/x$$ $$\frac{F_{xx}}{F_x}-\frac{F_{xy}}{F_y}<1/y$$

Not sure if it is correct or it could be further simplified.

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