Given a probability measure, a Poincaré inequality says that the "energy" - in the sense of L2 norm - of any centered (and regular enough) function is controlled uniformly by the energy of its derivative. In global sensitivity analysis, under the assumption of independent inputs, Poincaré inequalities connect total sensitivity indices (energy of the function) to derivative-based sensitivity indices (energy of the derivative). The derivative-based indices are less informative but require less computations, especially when the gradient is provided, and are thus attractive when the computational budget is limited. This motivates the research of sharpest bounds in Poincaré inequalities for the one-dimensional probability measures used by practitioners, such as the triangular distribution or truncated distributions. We first consider log-concave distributions, and show some facts coming from transport techniques: the usual explicit upper bound obtained by double-exponential transport and its improvement with logistic transport. In a second time, we recall the spectral interpretation and derive optimal bounds for the triangular distribution and the truncated normal distribution. Finally we apply these results to a real case study that was tackled using the log-concave assumption, and observe that the optimal constants are 6 times lower.