Modern Scientific Press

ABSTRACT: Let ∆(α) denote the fractional difference operator. In this paper, we define new difference sequence spaces c₀(Γ, ∆(α), u) and c(Γ, ∆(α), u). Also, the β−dual of the spaces c₀(Γ, ∆(α), u) and c(Γ, ∆(α), u) are determined and calculated their Schauder basis. Furthermore, we characterize the classes (µ(Γ, ∆(α), u) : λ) for µ ∈ {c₀, c} and λ ∈ {c₀, c, ℓ_∞, ℓ₁} .

ABSTRACT: : It was proved by Joyal, Labelle and Rahman [A. Joyal, G. Labelle and Q. I. Rahman, On the location of zeros of polynomials, Canad. Math. J., Bull., 10(1967), 53-63] that if p>1, then all the zeros of P(z)= zⁿ + a_(n-1) z^(n-1) + … + a₁z + a₀ are contained in the circle |z| ≤k, where k ≥max(1,|a_(n-1) |) is a root of the equation (|z|- |a_n |)^q (|z|^q- 1)-B^q=0, p^(-1)+ q^(-1)=1. In this paper, we not only generalize the above result but a verity of interesting results can be deduced from it.