This question was inspired by Carnot's great theorem on the efficiency of reversible thermodynamic cycles.
We have two continuous positive functions $T(u)$ and $S(u)$ of the parameter $u\in [0,1]$. Assume that both are continuously differentiable wrt $u$ except for a finite number of points and $T(0)=T(1)$ and $S(0)=S(1)$. In general, the closed curve represented by $\mathcal C := [T(u),S(u)]$ can cross itself several times.
Let $$\rm{min}_{u\in [0,1]} T(u)=T_{min}; \quad \rm{max}_{u\in [0,1]} T(u)=T_{max}\\ \rm{min}_{u\in [0,1]} S(u)=S_{min}; \quad \rm{max}_{u\in [0,1]} S(u)=S_{max}$$ Now consider the contour integral $$ \mathcal I = \oint_{\mathcal C} TdS = \int_0^1 T(u)\frac{dS}{du} du \tag{1}\label{1}$$ and break it up in two pieces of area depending on whether $S$ is increasing or decreasing function of $u$, that is $$\rm{min}_{u\in [0,1]} T(u)=T_0; \quad \rm{max}_{u\in [0,1]} T(u)=T_1\\ \rm{min}_{u\in [0,1]} S(u)=S_0; \quad \rm{max}_{u\in [0,1]} S(u)=S_1$$ Now consider the contour integral $$ \mathcal J = \sum_{dS>0}\int_{\mathcal C} TdS+\sum_{dS<0}\int_{\mathcal C} TdS \\ = \mathcal J_+ + \mathcal J_-\tag{2}\label{2}$$ where $$\mathcal J_+= \sum_{\frac{dS}{du}>0}\int T(u)\frac{dS}{du} du\\ \mathcal J_-=\sum_{\frac{dS}{du}<0}\int T(u)\frac{dS}{du} du \tag{3}\label{3}$$
In the special case in which the contour is simple (does not cross itself) and there are exactly two continuous pieces on which $S$ is monotonic, that is, either $dS<0$ or $dS>0$, then a simple geometric argument shows that $$\frac{T_0}{T_1}\le \frac{|\mathcal J_-|}{\mathcal J_+} \tag{4}\label{4}.$$ The proof of this simple case is illustrated below where the areas are defined as: $$\mathbf E = \mathcal J; \quad \mathbf F = T_0(S_1-S_0)\\ \mathbf A +\mathbf B = T_1(S_1-S_0)-\mathcal J_+; \quad \mathbf C +\mathbf D + \mathbf F =- \mathcal J_-.$$ Without changing this pictorial argument we can make the picture more complicated by adding a finite number loops of similar structure and still get the inequality $\eqref{4}$ for each loop, and thus for the ratios of the sum of the (+) and (-) loop segments. But I am unsure if I can just add wiggling self-intersecting paths before the picture to get unmanageably complicated especially if I wish this inequality $\eqref{4}$ be generalized for $T(u),S(u)$ functions that may have a countable infinite number of monotonic segments. Is it even true then?
