On entire functions having Taylor sections with only real zeros

Olga M. Katkova   ; Tatjana Lobova-Eisner   ; Anna M. Vishnyakova

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 4, pp. 449-469

Résumé

We investigate power series with positive coefficients having sections with only real zeros. For an entire function $f(z)=\sum_{k=0}^\infty a_kz^k$, $a_k>0$, we denote by $q_n(f):=\frac{a_{n-1}^2}{a_{n-2}a_n}$, $n\ge 2$. The following problem remains open: which entire function with positive coefficients and sections with only real zeros has the minimal possible $\liminf_{n\to \infty}q_n(f)$? We prove that the extremal function in the class of such entire functions with additional condition $\exists\,\lim_{n\to \infty}q_n(f)$ is the function of the form $f_a(z):=\sum_{k=0}^\infty\frac{z^k}{k!a^{k^2}}$. We answer also the following questions: for which $a$ do the function $f_a(z)$ and the function $y_a(z):=1+\sum_{k=1}^\infty\frac{z^k}{(a^k-1)(a^{k-1}-1)\dotsb(a-1)}$, $a>1$, have sections with only real zeros?