Some inequalities for polynomials and transcendental entire functions of exponential type

Tariq, Qazi M

Mathematical Communications, Tome 18 (2013) no. 1, p. 457-477 / Harvested from Mathematical Communications

Résumé

Let $p$ be a polynomial of degree $n$ such that $|p(z)|\leq M$ ($|z|=1$). The Bernstein's inequality for polynomials states that $|p^\prime(z)|\leq Mn $ ($|z|=1$). A polynomial $p$ of degree $n$ that satisfies the condition $p(z)\equiv z^n p(1/z)$ is called a self-reciprocal polynomial. If $p$ is a self-reciprocal polynomial, then $f(z)=p({\rm e}^{\rm i z})$ is an entire function of exponential type $n$ such that $f(z)={\rm e}^{\rm i n z} f(-z)$. Thus the class of entire functions of exponential type $\tau$ whose elements satisfy the condition $f(z)={\rm e}^{\rm i \tau z} f(-z)$ is a natural generalization of the class of self-reciprocal polynomials. In this paper we present some Bernstein's type inequalities for self-reciprocal polynomials and related entire functions of exponential type under certain restrictions on the location of their zeros.

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Publié le : 2013-11-12
Classification:  Polynomials, Bernsteins inequality, entire functions of exponential type, LP inequality,  Primary 41A17, Secondary 30A10, 30D15, 41A10, 41A17, 42A05, 42A16, 42A75