Some Inequalities for the Derivative of Polynomials

If is a polynomial of degree , having no zeros in , then Aziz (1989) proved , where . In this paper, we consider a class of polynomial of degree , defined as and present certain generalizations of above inequality and some other well-known results. 1. Introduction and Statement of the Results Let denote the space of all complex polynomials of degree . If , that is, , then according to the famous result of Bernstein (see [1]) and a simple deduction from the maximum modulus principle (see [2, page 346]) Both inequalities (1) and (2) are sharp and the equality holds if and only if has all its zeros at the origin. If we restrict ourselves to the class of polynomial having no zeros in , then (1) can be sharpened. In fact, P. Erd？s conjectured and later Lax [3] proved that if in , then It was shown by Frappier et al. [4, Theorem 8] that if , then and clearly (4) represents a refinement of (1), since the maximum of on may be larger than maximum of taken over th roots of unity as one can show by taking a simple example , . In this connection, Aziz [5] improved inequality (4) by showing that if , then for every real and inequality (2) for where Inequality (3) has also been improved by Aziz [5] by using the same argument as in inequality (5) and he proved that if and in , then for every real and also improved inequality (6) for every real and where is same as defined in (7). Recently, Rather and Shah [6] considered the class of polynomials having no zeros in , and and improved the inequalities (8) and (9) of Aziz [5] by showing the following. Theorem A. If , in , , and , then for every real where is defined in (7). Theorem B. If , in , , and , then for every real and where is defined in (7). In this paper, we first present the following result, which is a generalization of inequality (10) on class of polynomial of degree defined as , and having no zeros in , . Theorem 1. If , in , , and , then for every real where is defined in (7). On taking in Theorem 1, the following result has been obtained, which was shown by Rather and Shah [6]. Corollary 2. If , in and , then for every real where is defined in (7). If we consider that some zeros of the polynomial are on , then and the following result has been obtained from Theorem 1. Corollary 3. If and in , , then for every real where is defined in (7). Remark 4. For , Corollary 3 reduces to inequality (8) due to Aziz [5]. Taking in inequality (14) of Corollary 3, we get the following result. Corollary 5. If and in , , then for every real where is defined in (7). Next result is the generalization of inequality (11) of