Inequalities Involving the Derivative of Rational Functions With Prescribed Poles.

This paper gives an upper bound of a modulus of the derivative of rational functions. rz=z−z0shz/wz∈Rm,n, where r(z) has exactly n poles a1, a2, ⋯, an and all the zeros of r(z) lie in Dk∪Dk+,k ≥ 1 except the zeros of order s at z0, |z0| < k. Moreover, we give an upper bound of a modulus of the derivative of rational functions. rz=z−zvsvz−zv−1sv−1⋯z−z0s0hz/wz, where r(z) has the zeros z0, z1, ⋯, zv with |zi| < k for 0 ≤ i ≤ v and the remaining n − (s0 + s1+⋯+sv) zeros lie in Dk∪Dk+. Additionally, we provide proofs for several results that not only generalize certain inequalities for rational functions with restricted zeros but also offer refinements of certain polynomial inequalities as special cases. [ABSTRACT FROM AUTHOR]