On the Frequent Universality of Universal Taylor Series in the Complex Plane

Abstract

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

Polynomial inequalities and universal Taylor series

“…Furthermore the same argument does not work for Nestoridis universal series, because of their nonzero radius of convergence. However the Turán inequality allowed us to show that every universal Taylor series is 1-upper frequently universal and then we were able to conclude with a combinatorial argument [24]. In the present paper we refine this proof in order to prove that the previous result holds for more general densities.…”

“…To prove the second assertion, it suffices to use the same combinatorial argument as that of the proof of [24,Theorem 3.4]. This finishes the proof.…”

“…To do this, we refine the statements and the proofs of [24]. Indeed in [24] we showed that the universal Taylor series in the complex plane are never frequently universal (see Definition 1.2). In other hand, we proved that all these universal Taylor series are 1-upper frequently universal (see Definition 1.3).…”

“…Proof To obtain the first assertion, we proceed as in the proof of Theorem 3.3 of [24]. Set N 0 = 0 and μ (0) 0 = 0.…”

“…In [24] we showed that there cannot exist Seleznev frequently universal series. The proof of this fact is rather easy and is essentially based on a polynomial inequality due to Turán (see Lemma 2.2 below).…”