Archive - AMS Classification 30C15

For an entire function \(f\) with non-negative Maclaurin coefficients, a region is obtained which is defined in terms of Hayman’s function \(b(r) = r (rf^{\prime} (r)/f(r))^{\prime}\), and which is free of all zeros of \(f\) and those of all its sections. The new region defined improves on previous results. In particular, it is shown that when \(\underset{n\rightarrow \infty}{\limsup}\, b(r) = A^2/4\), \(A>0\), then the zeros \(r_n \exp (i\theta_n)\) of \(f\) satisfy the inequality, \(\underset{n\rightarrow \infty}{\liminf}\, |\theta_n| \geq 4\sin^{-1} (1/A\sqrt{2})\), which is very close to being optimal.

Etant donnée une function entière \(f\) avec des coéfficients positifs, on trouve une région définie en termes de la fonction \(b(r) = r (rf^{\prime}(r)/f(r))^{\prime}\) de Hayman, dépourvue des zéros de \(f\) et de ceux de toutes ses sections. Particulièrement, on démontre qu’au cas où \(\underset{n\rightarrow \infty}{\limsup}\, b(r) = A^2/4\), \(A>0\), les zéros \(r_n \exp (i\theta_n)\) de \(f\) satisfont l’inégalité \(\underset{n\rightarrow\infty}{\liminf} \, |\theta_n| \geq 4\sin^{-1} (1/A\sqrt{2})\), qui est presque optimale.