On entire functions having Taylor sections with only real zeros

Анотація

We investigate power series with positive coefficients having sections with only real zeros. For an entire function $f(z)=\displaystyle\sum_{k=0}^{\infty}a_kz^k, a_k>0$, we denote by $q_n(f):=\displaystyle\frac{a_{n-1}^2}{a_{n-2}a_n},n\geq 2$. The following problem remains open: which entire function with positive coefficients and sections with only real zeros has the minimal possible $\lim\inf_{n\to\infty}q_n(f)$ ? 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 $y_a(z):=1+\sum_{k=1}^{\infty}\frac{z^k}{(a^k-1)(a^{k-1}-1)\cdots(a-1)}, a>1$, have sections with only real zeros?