Bound on Jensen's gap for bivariate functions

Question

I have recently been reading on Jensen's gap for univariate functions. Specifically, here, page 6, Theorem 2.1, an upper bound on Jensen's gap is provided that is based on absolute central moments. I was wondering if there are any extensions to the bivariate case, i.e., $f:R^2 \rightarrow R$ in the mentioned theorem. Assume $x$ and $y$ are the inputs to the function $f$ with means $\mu_x$, $\mu_y$, and variances $\sigma^2_x$, $\sigma^2_y$ and covariance $\sigma_{xy}$. I thought if I replace the conditions 2 and 3 respectively with: $$|f(x,y)-f(\mu_x, \mu_y)| =O(|(x-\mu_x)(y-\mu_y)|^\alpha), \qquad \alpha>0$$ and $$|f(x,y)| =O(|xy|^n), \qquad n\geq\alpha$$ and define $$M=\text{sup} \frac{|f(x,y)-f(\mu_x, \mu_y)|}{|(x-\mu_x)(y-\mu_y)|^\alpha + |(x-\mu_x)(y-\mu_y)|^n}$$ would do the job, but it didn't! Appreciate your time!