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) =∑^∞_k=0a_kz^k, a_k > 0; we denote by q_n(f) :=(a²_n-1)/(a_n-2^a_n) n ≥ 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→∞ q_n(f)? We prove that the ex- tremal function in the class of such entire functions with additional condi tion $ lim _n→∞ q_n(f) is the function of the form f_a(z) := ∑^∞_k=0(z^k)/k! a^k2. We answer also the following questions: for which a do the function f_a(z) and the function y_a(z) := 1+∑^∞_k=1(z^k)/((a^k-1)(a^k-1-1)...(a-1)), a>1, have sections with only real zeros?