  For rational functions, we use simple but elegant techniques to strengthen
generalizations of certain results which extend some widely known polynomial
inequalities of Erd\"os-Lax and Tur\'an to rational functions R. In return
these reinforced results, in the limiting case, lead to the corresponding
refinements of the said polynomial inequalities. As an illustration and as an
application of our results, we obtain some new improvements of the Erd\'os-Lax
and Tur\'an type inequalities for polynomials. These improved results take into
account the size of the constant term and the leading coefficient of the given
polynomial. As a further factor of consideration, during the course of this
paper we shall demonstrate how some recently obtained results due to S. L. Wali
and W. M. Shah, [Some applications of Dubinin's lemma to rational functions
with prescribed poles, J. Math.Anal.Appl.450 (2017) 769-779], could have been
proved without invoking the results
