Strongly nonlinear Gagliardo-Nirenberg inequality in Orlicz spaces and Boyd indices

Given a N-function $A$ and a continuous function $h$ satisfying certain assumptions, we derive the inequality \begin{eqnarray*} \int_\r A(|f^{'}(x)|h(f(x)))dx\leq C_1\int_\r A\left(C_2 \sqrt[p]{ |{\cal M}f^{''}(x){\cal T}_{h,p}(f,x)| }\cdot h(f(x)) \right)dx, \end{eqnarray*} with constants $C_1,C_2$ independent of $f$, where $f\ge 0$ belongs locally to the Sobolev space $W^{2,1}(\r)$, { $f^{'}$ has} compact support, $p>1$ is smaller than the lower Boyd index of $A$, ${\cal T}_{h,p}(\cdot)$ is certain nonlinear transform depending of $h$ but not of $A$ and ${\cal M}$ denotes the Hardy-Littlewood maximal function. Moreover, we show that when $h\equiv 1$, then ${\cal M}f^{''}$ can be improved by $f^{''}$. This inequality generalizes previous result by third author and Peszek, which was dealing with $p=2$.