Let f = Σ j = 0 ∞ a j z j f = \Sigma _{j = 0}^\infty {a_j}{z^j} be an entire function which satisfies \[ | a j − 1 a j + 1 / a j 2 | ⩽ ρ 2 , j = 1 , 2 , 3 , … , |{a_{j - 1}}a{ _{j + 1}}/a_j^2| \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots , \] where 0 > ρ > ρ 0 0 > \rho > {\rho _0} and ρ 0 = 0.4559 … {\rho _0} = 0.4559 \ldots is the positive root of the equation 2 Σ j = 1 ∞ ρ j 2 = 1 2\Sigma _{j = 1}^\infty {\rho ^{{j^2}}} = 1 . Let r > 0 r > 0 be fixed. Let W L M {W_{LM}} denote the rational function of type ( L , M ) (L,M) of best approximation to f f in the uniform norm on | z | ⩽ r |z| \leqslant r . We show that for any sequence of nonnegative integers { M L } L = 1 ∞ \{ {M_L}\} _{L = 1}^\infty that satisfies M L ⩽ 10 L , L = 1 , 2 , 3 , … {M_L} \leqslant 10L,\,L = 1,2,3, \ldots , the rational approximations W L M L {W_{L{M_L}}} converge to f f throughout C {\mathbf {C}} as L → ∞ L \to \infty . In particular, convergence takes place for the diagonal sequence and for the row sequences of the Walsh array for f f .