For $c : [a,b] \to M$ a minimizing geodesic of endpoints $x=c(a)$ and $y=c(b)$ in a Riemannian manifold, I need an upper bound of the integral $\int _a ^b t \| c'(t) \|^2 \ \mathrm d t$. Is it possible to compute it explicitly?
Using the Cauchy-Schwarz inequality, I can write
$$d(x,y)^2 = \left( \int \limits _a ^b \| c'(t) \| \ \mathrm d t \right)^2 = \left( \int \limits _a ^b \frac 1 {\sqrt t} \cdot \sqrt t \| c'(t) \| \ \mathrm d t \right)^2 \le \int \limits _a ^b \frac 1 t \ \mathrm d t \ \int \limits _a ^b t \| c'(t) \|^2 \ \mathrm d t$$
whence
$$\int \limits _a ^b t \| c'(t) \|^2 \ \mathrm d t \ge \frac {d(x,y)^2} {\log t - \log s}$$
but I need an upper, not a lower bound. Since $c$ is a minimizing geodesic, can I hope that the inequality is, in fact, equality?
Of course, I could use the rough bound $\int _a ^b t \| c'(t) \|^2 \ \mathrm d t \le t d(x,y)^2$, but I am hoping for something subtler if possible.