Let $f\colon (1,2)\times (1,2) \to \mathbb{R}$ be a Lebesgue measurable, bounded and non-negative function such that $$ \int_1^2 f(x,y) dy = 1, \qquad x \in (1,2). $$ Moreover, assume that for any measurable $A \subset (1,2)$ with $0 < \mu(A) < 1$ ($\mu$ is the Lebesgue measure on $(1,2)$) we have $$ \int_A \int_{(1,2) \setminus A} f(x,y) dy dx > 0. $$

Let now $x_0 \in (1,2)$ and define a sequence $(a_n)_{n \geq 2}$ of real numbers by $$ \int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n. $$ Thus $$ a_2 = \int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 f(x_0,x_1) f(x_1,x_2) dx_2 dx_1, $$ $$ a_3 = \int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \int_{(4/2-1/x_1-1/x_2)^{-1}}^2 f(x_0,x_1) f(x_1,x_2) f(x_2,x_3) dx_3 dx_2 dx_1, $$ and so on.

I would like to prove that it is possible to fix $x_0$ in such a way that there exist constants $c > 0$ and $q \in (0,1)$ such that $$\int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n \geq cq^n$$ for all $n \geq 2$.


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