Let α, β and γ be three positive real numbers. Let f(x)=α x^5+β x^3+γ x, x ∈ R and g: R → R be such that g(f(x))=x for all x ∈ R. If a_1, a_2, a_3,…

Let $\displaystyle \alpha, \beta$ and $\displaystyle \gamma$ be three positive real numbers. Let $\displaystyle f(x)=\alpha x^5+\beta x^3+\gamma x, x \in \mathbf{R}$ and $\displaystyle g: \mathbf{R} \rightarrow \mathbf{R}$ be such that $\displaystyle g(f(x))=x$ for all $\displaystyle x \in \mathbf{R}$. If $\displaystyle \mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots, \mathrm{a}_{\mathrm{n}}$ be in arithmetic progression with mean zero, then the value of $\displaystyle f\left(g\left(\frac{1}{\mathrm{n}} \sum_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_i\right)\right)\right)$ is equal to :