Approximation of entire functions of zero order (Q1097370)

Let \[ f(z)=\sum^{\infty}_{n=0}a_ nz\quad n(a_ n\geq 0) \] be an entire function of order zero such that \[ \lim_{r\to \infty} \sup \log \log M(r)/\log \log r=\rho (1<\rho <\infty), \] \[ \limsup (\inf)(\log M(r))(\log r)^{-\rho}=T(t) \] where \(M(r)=\max_{| z| =r}| f(z)|\). \textit{A. R. Reddy} [J. Approximation Theory 15, 206-208 (1975; Zbl 0347.41010)] showed that if \(\lambda_{0,n}\) denotes the degree of approximation of 1/f on the positive real axis in the uniform norm by means of reciprocals of polynomials of degree n, then, for \(0<t\leq T<\infty\), one has \[ \exp [-(\rho -1)/\rho (\rho T)^{1/(\rho -1)}]\leq \lim_{n\to \infty} \sup (\lambda_{0,n})n^{-\rho /(\rho -1)}<1. \] The author, in the present paper, improves and generalizes the above estimates. For instance, improvement, obtained by him for functions satisfying \(0<t=T<\infty\), reads as \[ \exp [-(\rho -1)/\rho (\rho t)^{1/(\rho -1)}]\leq \liminf_{n\to \infty}(\lambda_{0,n})n^{-\rho /(\rho -1)} \] \[ \leq \limsup_{n\to \infty}(\lambda_{0,n})n^{-\rho /(\rho -1)} \] \[ \leq \exp [-(\rho -1)/\rho (\rho T)^{1/(\rho -1)}\{1- 2^{-1/(\rho -1)}\}]. \]