Let $A$ and $B$ be two (non commuting) self-adjoint bounded operator acting on a Hilbert space and let $p,q>1$ such that $\frac1p+\frac1q=1$
Do we have a Young-type inequality such as $ \frac12|AB+BA| \leq \frac{|A|^p}{p}+ \frac{|B|^q}{q}$ in the sense of quadratic form ? It is of course obvious for $p=q=2$.
As a general criterion, these kind of inequalities hold at the level of singular values (which implies a version of the inequality where one of the sides is conjugated by a unitary/partial isometry), but they tend to fail at the plain operator level.
Here is a counterexample for $p=3$, $q=3/2$. Take $$ A=\begin{bmatrix}1&0\\0&3\end{bmatrix},\qquad\qquad B=\begin{bmatrix}1&1\\1&1\end{bmatrix}. $$ Then $$ \frac{A^3}3+\frac{2B^{3/2}}3=\frac13\,\Big(\begin{bmatrix}1&0\\0&27\end{bmatrix} +\begin{bmatrix}2\sqrt2&2\sqrt2\\2\sqrt2&2\sqrt2\end{bmatrix}\Big) =\frac13\,\begin{bmatrix}1+2\sqrt2&2\sqrt2\\2\sqrt2&27+2\sqrt2\end{bmatrix}. $$ We have $$ AB+BA=\begin{bmatrix}2&21\\21&40\end{bmatrix}, $$ so $$ (AB+BA)^2=\begin{bmatrix} 20&32\\32&52\end{bmatrix} $$ and then Wolfram Alpha tells us that $$ |AB+BA|=\begin{bmatrix} 20&32\\32&52\end{bmatrix}^{1/2}=\frac2{\sqrt5}\,\begin{bmatrix}3& 4\\ 4& 7 \end{bmatrix}. $$
Then, using Wolfram Alpha again, we see that $$ \frac{A^3}3+\frac{2B^{3/2}}3-\frac12\,|AB+BA| $$ is not positive, as it has a negative eigenvalue.
