On the equation of an improper convex affine sphere: a generalization of a theorem of Jörgens

It is proved that if a function $\varphi(t)$ of variable $t>0$ belongs to the class $C^{3,\alpha}$ and satisfies for sufficiently small positive $\varepsilon$ ($\varepsilon<10^{-4}$) the conditions
\begin{gather*} 1-\varepsilon\leqslant\varphi(t)\leqslant1+\varepsilon,\qquad t>0, \\ \begin{alignedat}{2} |\varphi'(t)|&\leqslant\varepsilon\frac{\varphi(t)}t\,,&\qquad t&\geqslant 2\sqrt{1-\varepsilon}, \\ |\varphi''(t)|&\leqslant\varepsilon\frac{\varphi(t)}{t^2}\,,&\qquad t&\geqslant2\sqrt{1-\varepsilon}, \\ |\varphi'''(t)|&\leqslant\varepsilon\frac{\varphi(t)}{t^3}\,,&\qquad t&\geqslant2\sqrt{1-\varepsilon}, \end{alignedat} \end{gather*}
then every complete solution $z(x,y)$ of the equation $z_{xx}z_{yy}-z_{xy}^2=\varphi(z_{xx}+z_{yy})$ is a quadratic polynomial.