Integral mean estimates for the polar derivative of polynomials whose zeros are within a circle
نویسندگان
چکیده
منابع مشابه
Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin
Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$, let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$. Dewan et al proved that if $p(z)$ has all its zeros in $|z| leq k, (kleq 1),$ with $s$-fold zeros at the origin then for every $alphainmathbb{C}$ with $|alpha|geq k$, begin{align*} max_{|z|=...
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For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
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Let Q be a real polynomial of degree N all of whose zeros lie in the half-plane Re z < 0. Let M(r, Q) be the maximum of | Q(z) | on \z\= r and n(r,0) the counting function of the zeros of Q. It is shown that the inequality M(r, Q') « (2r)"'{A' + n(r,0)}M(r, Q) holds for r > 0. It is also shown that Bernstein's inequality characterizes polynomials.
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ژورنال
عنوان ژورنال: Journal of Inequalities and Applications
سال: 2013
ISSN: 1029-242X
DOI: 10.1186/1029-242x-2013-307