Is the following true?
Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$
$$ \beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}t_r $$
Claim: If the binomial mean $\beta_{t_n,a}$ approaches a finite positive constant and $f$ is continious on the interval $[0,1]$ then, $$ \frac{1}{2^n}\sum_{r^a \le n} \binom{n}{r^a}f\Big(\frac{t_r}{t_n}\Big) \sim f(\beta_{t_n,a}) $$
Note: The case $a = 1, t_n = n$ is true and is a special case of the approximation of $f$ using Bernstein polynomials as mentioned in the answer to the related question Binomial analogue of Riemann sum for definite integral
Example: Let $a = 2, t_n = p_n$ be the $n$-th prime and $f(x) = \log(x)$ for $n = 10^6$, LHS =$-0.7426766$, RHS =$-0.7426760$, and the error is $< 5.65 \times 10^{-7}$
Note: This question was posted in MSE last month. It got upvotes but no answer hence posting it in MO
