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Let P(z) be a polynomial of degree nn and for any complex number aa, let DαP(z)=nP(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study polynomial functional equations of the form af(p(x))+bf(q(x))=g(x), where p(x),q(x) are given polynomials and g(x) is a given function. Theorems 2.1 and 2.2 contain sufficient conditions under which the functional equation has a solution of the special form. In Section 3 we present an algorithm of constructing polynomial solutions of the functional equations. Other non-polynomial solutions depend on solutions of the homogeneous equation af(p(x)) + bf(q(x)) = 0. That case is analyzed in Section 4. Finally, we present a simple method of constructing examples with desirable properties. [ABSTRACT FROM AUTHOR]