Hello. How to prove that
$$\sup_{\gamma\ge 0}\int_0^1 (1+\gamma^2)^{-\frac 12} (1-t^2)^{\frac {n-4}2}( \Phi(\gamma t) +\Phi(-\gamma t))dt$$
is achived at $\gamma=0$, where
$$\Phi(\zeta)= \int_0^{\frac{\zeta +\sqrt{\zeta^2+1-r^2\left(\frac{n-2}n\right)^2}}{1+r\frac{n-2}n}} \frac{n-r (n-2) + 2 n \zeta \omega -\left(n+r(n-2)\right)\omega^2 } {(1+ \omega^2)^{\frac n2+1} \left( 1+\frac{\left(\frac{1-r}{1+r}\right)^2}{\omega^2}\right)^{\frac n2-1}}d\omega;$$ $n$ is integer $\ge 3,\ \zeta\in \mathbb R,\ 0\le r < 1$.
This problem is connected with the very important problem of the optimal poinwise estimate of the gradient of real valued harmonic functions in the unit ball; that is to find the minimal $\mathcal{K}(x)$ for the estimate
$$\left|\nabla u(x)\right|\le \mathcal{K}(x),\quad x\in B^n,$$
where $u$ is a harmonic function and $\left|u(x)\right|\le 1,\ x\in B^n$. Similar problems were treathed in the papers
G. Kresin, V. Maz'ya, Optimal estimates for the gradient of harmonic functions in the multidimensional half-space, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 425--440;
G. Kresin, V. Maz'ya, Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball, Journal of Mathematical Sciences 169 (2010).
Thank you very much for answers.
