On the diagonal Pade approximants of meromorphic functions

Let f be meromorphic in C, and analytic at 0, and let E(nn)(r) denote the error of best rational approximation of f by rational functions of type (n,n) on some small disc {z:\z\less than or equal to r}. We prove: (I) If lim sup(n)-->infinity E(nn)(r)(1/n2) < 1, then the Baker-Gammel-Wills conjecture is true for f. (II) If E(nn)(r)(1/(2n+1)) is non-increasing in n, then (a) [n/n] has less than or equal to 2l + o(n(2)/\log E(n-l,n-1)(r)\) poles in \z\ less than or equal to r if f has l poles there. (b) If lim(n)-->infinity inf E(nn)(r)(1/n2) < 1, then the Baker-Gammel-Wills conjecture is true for f. (c) If f is entire and lim(n)-->infinity sup E(nn)(r)(1/n2) < 1, then the full diagonal sequence {[n/n]} converges pointwise to f. We also discuss some extensions and consequences of these results.