Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I am wondering if there exists a log-Sobolev inequality for the uniform distribution $\mu$, in the sense that $$\sum_{n=0}^N \mu_n f^2_n \log f^2_n \leq C\sum_{n=0}^{N-1} \mu_n(f_{n+1}-f_n)^2 \label{1}\tag{1}$$ for some universal constant $C$, where $f = (f_0,\ldots,f_N) \in \mathbb{R}^{N+1}_+$ satisfies $\sum_{n=0}^N \mu_n f^2_n =1$. Of course, \eqref{1} is merely a discrete analog of the log-Sobolev inequality for the uniform measure on a one dimensional compact interval, which takes the following form: $$ \int_0^N f^2 \log f^2 \mathrm{d}\mu \leq C \int_0^N |\nabla f|^2 \mathrm{d}\mu$$ where $\mu$ is the uniform distribution on the interval $[0,N]$ and $f \colon [0,N] \to \mathbb{R}_+$ satisfies the constraint that $\int_0^N f^2 \mathrm{d}\mu = 1$. I am aware of some literatures on log Sobolev inequalities for uniform measures on compact intervals (or finite Lebesgue measure) such as this reference, but it seems very hard to locate the precise reference for the corresponding result (for discrete state space) stated as in the form of \eqref{1}. Any help or pointer to related references are greatly appreciated (and I am not interested in the optimal/sharp log-Sobolev constant) !

Remark: when $N=1$, the log-Sobolev inequality \eqref{1} boils down to the well-known log-Sobolev inequality for (symmetric) Bernoulli distribution due to Leonard Gross (1975) that can be found in here for instance.

Edit: Here $N$ is fixed and for sure $C$ can depend on $N$, but $C$ cannot depend on the specific choice of $f$

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