The polar derivative of a polynomial p(z) of degree n with respect to a complex number α is a polynomial np(z)+α-zp′(z), denoted by Dαp(z). Let 1≤R≤k. For a polynomial p(z) of degree n having all its zeros in z≤k, we investigate a lower bound of modulus of Dαp(z) on z=R. Furthermore, we present an upper bound of modulus of Dαp(z) on z=R for a polynomial p(z) of degree n having no zero in z<k. In particular, our results in case R=1 generalize some well-known inequalities.

We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the form ( ) = ( ( ) + ( )). Considering a convergent power series ( ) of an auxiliary equation, we obtain an analytic solution ( ). Furthermore, we characterize a polynomial solution when ( ) is a polynomial.

.], Kumar and Lal provided an upper bound of a derivative for polynomial degree n having some of zeros at the origin and rest of zeros lying on or outside the boundary of a prescribed disk. In this paper, we present an upper bound of a derivativen having zeros z 0 ,... ,z m with |z j | < 1 for 0 j m and the remaining n − (t m + ··· +t 0 ) zeros are outside {z : |z| < k} where k 1.Mathematics subject classification (2010): 30A10, 30C10, 30C15.

No abstract