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

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.