Some inequalities for polynomials and transcendental entire functions of exponential type

Abstract

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.