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Using the condition $\int_{S^1}dx=1$ and Jensen's inequality, we have $$\sup_{t\in S^1} \int_{S^1}dx\,|f(x + t)-f(x)| \ge\int_{S^1}dt\, \int_{S^1}dx\,|f(x + t)-f(x)| \\ =\int_{S^1}dx\, \int_{S^1}dt\,|f(x)-f(x + t)| \ge\int_{S^1}dx\, \Big|f(x)-\int_{S^1}dt\,f(x + t)\Big| \\ =\int_{S^1}dx\, \Big|f(x)-\int_{S^1}dy\,f(y)\Big|. $$ So, your conjectured inequality holds with $C=1$. $\quad\Box$