If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$ $$ d_H( Log(x_0,x),Log(y,x_0) ) < d_M^2(x,y)<\|Log(x_0,y)-Log(x_0,x)\|_2^2 , $$ where $d_H$ is the hyperbolic metric on $\mathbb{R}^d$, $d_M$ is the metric induced by the Riemannian metric tensor $g$ and $Log(x_0,\cdot)$ is the Riemannian Log map on $M$ about $x_0$?

If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$ $$ d_H( Log(x_0,x),Log(y,x_0) ) \leq d_M^2(x,y) \leq \|Log(x_0,y)-Log(x_0,x)\|_2^2 , $$ where $d_H$ is the hyperbolic metric on $\mathbb{R}^d$, $d_M$ is the metric induced by the Riemannian metric tensor $g$ and $Log(x_0,\cdot)$ is the Riemannian Log map on $M$ about $x_0$?

If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$ $$ d_H( Log(x_0,x),Log(y,x_0) ) < d_M^2(x,y)<\|Log(x_0,y)-Log(x_0,x)\|_2^2 , $$ where $d_H$ is the hyperbolic metric on $\mathbb{R}^d$, $d_M$ is the metric induced by the Riemannian metric tensor $g$ and $Log(x_0,\cdot)$ is the Riemannian Log map on $M$ about $x_0$.

If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$ $$ d_H( Log(x_0,x),Log(y,x_0) ) < d_M^2(x,y)<\|Log(x_0,y)-Log(x_0,x)\|_2^2 , $$ where $d_H$ is the hyperbolic metric on $\mathbb{R}^d$, $d_M$ is the metric induced by the Riemannian metric tensor $g$ and $Log(x_0,\cdot)$ is the Riemannian Log map on $M$ about $x_0$?