2024/12/02 by Rather, N. A., Dar, Ishfaq, Gulzar, Suhail
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.01088
If all the zeros of nth degree polynomials f(z) and g(z) = ∑k=0nλk\binomnkzk respectively lie in the cricular regions |z|≤ r and |z| ≤ s|z-σ|, s>0, then it was proved by Marden \cite[p. 86]mm that all the zeros of the polynomial h(z)= ∑k=0nλk f(k)(z) ((σz)k)/(k!) lie in the circle |z| ≤ r ~ max(1,s). In this paper, we relax the condition that f(z) and g(z) are of the same degree and instead assume that f(z) and g(z) are polynomials of arbitrary degree n and m respectively, m≤ n, and obtain a generalization of this result. As an application, we also introduce a linear operator which preserve Bernstein type polynomial inequalities.