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Let $D=\{(x_1,x_2)\mid x_1\gt0,x_2\gt0,x_1+x_2\lt1\}$. For every $a=(a_1,a_2,a_3)$ such that $a_i\gt0$ and $a_1+a_2+a_3=1$, consider the function $u_a$ defined on $D$ by $$ u_a(x_1,x_2)=\frac{x_1^{n_1-1} x_2^{n_2-1} (1-x_1-x_2)^{n_3-1}}{(a_1 x_1 + a_2 x_2 +a_3 (1-x_1-x_2))^c}, $$ and the ratio of integrals $$ R(a)=\iint_Dx_1x_2u_a(x_1,x_2)\mathrm dx_1\mathrm dx_2 \,{\LARGE /} \iint_Du_a(x_1,x_2)\mathrm dx_1\mathrm dx_2. $$ Assume that $c\gt0$ and that each $n_i$ is an integer such that $n_i\gt c$.

The question is to prove that $R(a)$ is maximal at $a=(0,0,1)$ and minimal either at $a=(1,0,0)$ if $n_1\lt n_2$ or at $a=(0,1,0)$ if $n_2\lt n_1$.

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  • $\begingroup$ Maybe if you tell the origin of the problem, it would be easier to give an advice $\endgroup$ – Ilya Jul 18 '13 at 8:34
  • $\begingroup$ It is the expectation of $x_1x_2$ with respect to the above probability density function. I need to find the minimum and maximum of the expectation at the varying of $a_i$. Note that if $a_1=a_2=a_3$, the probability density function becomes a Dirichlet distribution. Maybe this can help. $\endgroup$ – vatna Jul 18 '13 at 8:41
  • $\begingroup$ Rewrote the thing in a way I can follow. If you prefer it that way, delete the part below the line, otherwise delete the part above the line. $\endgroup$ – Did Jul 18 '13 at 8:58
  • $\begingroup$ Why the condition $n_i\gt c$? $\endgroup$ – Did Jul 18 '13 at 9:02
  • $\begingroup$ The denominator in the big fraction is the normalization constant of the PDF. $n_i-c>0$ guarantees that the PDF is always integrable, even for $a_3=1,a_1=a_2=0$. $\endgroup$ – vatna Jul 18 '13 at 9:17

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