Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.

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Let B denote a set of bicolorings of [n], where each bicoloring is a mapping of the points in [n] to {−1, +1}. For each B ∈ B, let YB = (B(1), . . . , B(n)). For each A ⊆ [n], let XA ∈ {0, 1} n denote the incidence vector of A. A non-empty set A is said to be an 'unbiased representative' for a bicoloring B ∈ B if XA, YB = 0. Given a set B of bicolorings, we study the minimum cardinality of a family A consisting of subsets of [n] such that every bicoloring in B has an unbiased representative in A.

We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3, . . . , n for certain t ( > 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the discfor r < a < 2, r being the greatest positive root of the equationx n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).2010 Mathematics Subject Classification: Primary 30C15; Secondary 30C10. Key words and phrases: simple zeros, zero free region, refinement, upper bound for moduli of all zeros.Communicated by Gradimir Milovanović.