Some Inequalities for Rational Functions with Fixed Poles
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and Complex Analysis
Some Inequalities for Rational Functions with Fixed Poles Abdullah Mir* University of Kashmir, Srinagar, India Received March 23, 2018; revised March 29, 2019; accepted April 25, 2019
Abstract—By using lemmas of Dubinin and Osserman some results for rational functions with fixed poles and restricted zeros are proved. The obtained results strengthen some known results for rational functions and, in turn, produce refinements of some polynomial inequalities as well. MSC2010 numbers : 30A10, 30C10, 26D10 DOI: 10.3103/S1068362320020077 Keywords: rational function, polynomial, poles, zeros.
1. INTRODUCTION Let Pn denote the class of all complex polynomials of degree at most n. If P ∈ Pn , then concerning the estimate of |P (z)| on |z| = 1, we have |P (z)| ≤ n max |P (z)|. |z|=1
(1.1)
The inequality (1.1) is a famous result due to Bernstein [3], who proved it in 1912. It is worth mentioning that in (1.1) equality holds if and only if P (z) has all its zeros at the origin. So, it is natural to seek improvements under appropriate assumption on the zeros of P (z). If we restrict ourselves to the class of polynomials P (z) having no zeros in |z| < 1, then (1.1) can be replaced by n (1.2) max |P (z)| ≤ max |P (z)|, 2 |z|=1 |z|=1 whereas, if P (z) has no zeros in |z| > 1, then by max |P (z)| ≥ |z|=1
n max |P (z)|. 2 |z|=1
(1.3)
¨ and later it was verified by Lax [7], whereas the inequality The inequality (1.2) was conjectured by Erdos ´ [10]. (1.3) is due to Turan In 1997, Jain [6] had used a parameter β and proved an interesting generalization of (1.3). More precisely, Jain proved that if P ∈ Pn and P (z) has all its zeros in |z| ≤ 1, then for every β with |β| ≤ 1, we have nβ n P (z)| ≥ {1 + Re(β)} max |P (z)|. (1.4) max |zP (z) + 2 2 |z|=1 |z|=1 Li, Mohapatra and Rodriguez [12] gave a new perspective to the above inequalities (1.1)–(1.3), and extended them to rational functions with fixed poles. Essentially, in these inequalities they replaced the polynomial P (z) by a rational function r(z) with poles a1 , a2 , ..., an all lying in |z| > 1, and z n was replaced by a Blaschke product B(z). Before proceeding towards their results, we first introduce the set of rational functions involved. *
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105
106
MIR
For aj ∈ C with j = 1, 2, ..., n, we define W (z) :=
n
(z − aj ); B(z) :=
j=1
n 1 − aj z j=1
z − aj
=
W ∗ (z) , W (z)
where 1 W ∗ (z) = z n W ( ) z and
Rn := Rn (a1 , a2 , ..., an ) =
P (z) : P ∈ Pn . W (z)
Then Rn is defined to be the set of rational functions with poles a1 , a2 , ..., an at most and with finite P (z) limit at ∞. Note that B(z) ∈ Rn and |B(z)| = 1 for |z| = 1. Also, for r(z) = W (z) ∈ Rn , the conjugate transpose r ∗ of r is defined by r ∗ (z) = B(z)r( 1z ). In the past few years several papers pertaining to Bernstein-type inequalities for rational functions have appeared in the study of rational approximations (see, e.g., [2, 4, 11–13]). For r ∈ Rn , Li, Mohapatra and Rodriguez [12] proved the following, similar to (1.1), inequ
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