CERTAIN TRINOMIAL EQUATIONS AND LACUNARY POLYNOMIALS

Abstract

Abstract. We estimate the positive real zeros of certain trinomial equa-tions and then deduce zeros bounds of some lacunary polynomials. 1. Introduction and statement of resultsMany of classical inequalities of analysis have been obtained from trinomialequations, and there have been a number of literatures about zero distributionsof trinomial equations and lacunary polynomials. See, for example, [1], [2], [3]and [4]. In this paper, we investigate positive real zeros distributions of certaintrinomial equations and, using this, we estimate zeros bounds for some lacunarypolynomials. While studying these, we will need a new generalized upper boundof the exponential function: for 0 ≤ x < 1 and 1 ≤ n ≤ 2 we have(1) e x ≤ U ( n,x ) = 1 − 1 n +1 n ˆ1+i1 − 1 n ¢ x 1 − xn ! n ≤ 11 −x, where U (1 ,x ) = 11 −x . For the details about this, see [5]. The first resultabout trinomial equations follows from the lemma below that will be proved inSection 2.Lemma 1. Let n be an integer ≥ 4 , and (2)12 n