LIMITED POLYNOMIALS T. AGAMA Abstract. In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign. 1. Introduction and motivation The sendov conjecture is the assertion that any complex coefficient polynomial P n (x) of degree n ≥ 2 with sufficiently small zeros must lie in the same unit disk with some zero of P ′ n (x). More formally if |a i | < 1 such that P n (a i ) = 0, then there exist some b k with P ′ n (b k ) = 0 such that |a i - b k | < 1. There has been and still is a flurry of research devoted to this problem and mani- festly the current literature contains dozens of papers just for the problem. There has really been substantive progress ever since it was posed. For instance, It has been shown in [5] that the conjecture holds for zeros near the unit circle. In [1], the conjecture has been verified for degree at most six. This was improved further to polynomials of degree at most seven in [2] and polynomials of degree at most eight in [4]. The best result thus far concerning sendov conjecture is found in [3], where it was verified to hold for sufficiently large degree polynomials. In the sequel, we will study a particular class of complex polynomial. It turns out that such polynomial almost satisfies the sendov conjecture. 2. Limited complex-valued polynomials In this section we introduce the concept of limited polynomials. We study various settings under which polynomials of this forms are preserved. Definition 2.1. Let P n (z) be a complex-valued polynomial of degree n and let Z (P n (z)) = {a 1 ,a 2 , ··· ,a n } be the set of zeros of P n (z). By the measure of P n (z) denoted M(P n (z)), we mean the value M(P n (z)) = n  i=1 |a i |. Date : July 13, 2020. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Sendov; critical; zeros; limited. 1