On a theorem of S. Dancs and P. Turán

Abstract

Choosing r so that the abovementioned circle covers the chosen square we obtain the required upper bound for the number of zeros of F(z) in the square. Moreover this upper bound will prove to be independent of the coefficients of the polynomials pk(Z) in F(z). We easily show that this implies that our upper bound is one for the number of zeros of F(z) in any square of side S in the complex plane. Our starting point is Lemma 1, which is Theorem 4 of [2]. We refer the reader to that paper for the proof. Throughout we employ the following notation: