Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_0}{2}+\frac{x_1}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_0,x_1,\ldots,x_{n-1})= \log \left[ \prod_{k=0}^{n-1}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{(k+j)~\text{mod}~n}}\right) \right], 0<q<1. $$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_0}{2}+\frac{x_1}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit1: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_0,x_1,\ldots,x_{n-1})= \log \left[ \prod_{k=0}^{n-1}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{(k+j)~\text{mod}~n}}\right) \right], 0<q<1. $$

Edit2: I changed the notation and started the indices from zero to make the general case accurate.

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_1,x_2,x_3)= \log \left[ \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_3}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_3}{2}+\frac{x_1}{4}}\right) \left(1+\frac{x_3}{1+x_3+\frac{x_1}{2}+\frac{x_2}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_1,x_2,\ldots,x_n)= \log \left[ \prod_{k=1}^{n}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{k+j}}\right) \right], 0<q<1. $$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_0}{2}+\frac{x_1}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_0,x_1,\ldots,x_{n-1})= \log \left[ \prod_{k=0}^{n-1}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{(k+j)~\text{mod}~n}}\right) \right], 0<q<1. $$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_1,x_2,x_3)= \log \left[ \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_3}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_3}{2}+\frac{x_1}{4}}\right) \left(1+\frac{x_3}{1+x_3+\frac{x_1}{2}+\frac{x_2}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_1,x_2,x_3)= \log \left[ \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_3}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_3}{2}+\frac{x_1}{4}}\right) \left(1+\frac{x_3}{1+x_3+\frac{x_1}{2}+\frac{x_2}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_1,x_2,\ldots,x_n)= \log \left[ \prod_{k=1}^{n}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{k+j}}\right) \right], 0<q<1. $$