Sharper upper bounds on maximum modulus for rational functions

We study upper bounds for rational functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>r</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>w</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:math> r(z)=p(z)/w(z) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>w</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mo form="prefix" movablelimits="false">∏</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace class="tmspace" width="2.77695pt"/> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">|</m:mo> <m:mo>></m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:math> w(z)=∏ ⱼ₌₁ⁿ(z-λ ⱼ), λ ⱼ >1, and p ( z ) is a polynomial of degree at most n . The Maximum Modulus Principle gives the trivial estimate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo stretchy="false">|</m:mo> <m:mi>r</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mi>z</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>≤</m:mo> <m:msub> <m:mrow> <m:mi>max</m:mi> </m:mrow> <m:mrow> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo stretchy="false">|</m:mo> <m:mi>r</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:math> r(μ z) ≤ max_z∈ T₁ r(r) for 0 ≤ μ ≤ 1 and z ∈ T 1 , the unit circle | z | = 1. In this paper, we derive a new inequality that strengthens this estimate and serves as a reverse analogue of Varga’s inequality [J. Soc. Indust. Appl. Math., 5(2), 1957, 39 − 46] for rational functions. Furthermore, the classical Ankeny-Rivlin inequality is refined by establishing sharper estimates for | r ( γz )|, where γ ≥ 1 and z ∈ T 1 . These results strengthen and extend several earlier works, providing new insights into the extremal behavior of rational functions. In addition numerical examples are presented using WOLFRAM MATHEMATICA to illustrate the superiority of our results both numerically and graphically over theirs.