Precise inequalities for norms of functions, third partial, second mixed, or directional derivatives

For functions $f$ which are bounded throughout the plane $R^2$ together with the partial derivatives $f^{(3,0)}$, $f^{(0,3)}$, inequalities
\begin{gather*} \|f^{(1,1)}\|\le\sqrt[3]3\|f\|^{1/3}\|f^{(3,0)}\|^{1/3}\|f^{(0,3)}\|^{1/3}, \\ \|f_e^{(2)}\|\le\sqrt[3]3\|f\|^{1/3}(\|f^{(3,0)}\|^{1/3}|e_1|+\|f^{(0,3)}\|^{1/3}|e_2|)^2, \end{gather*}
are established, where $\|\cdot\|$ the upper bound on $R^2$ of the absolute values of the corresponding function, andf $f_e^{(2)}$ is the second derivative in the direction of the unit vector $e=(e_1,e_2)$. Functions are exhibited for which these inequalities become equalities.