BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS

Indian J Pure Appl Math., 46(3): 337-348, June 2015 c Indian National Science Academy DOI: 10.1007/s13226-015-0133-8 BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS Idrees Qasim and A Liman Department of Mathematics, National Institute of Technology Srinagar 190 006, India e-mails: idreesf3@gmail.com, abliman@rediffmail.com (Received 27 November 2013; accepted 11 August 2014) In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions Key words : Rational function; polynomials; inequalities; poles; zeros I NTRODUCTION Let Pn denote the space of complex polynomials of degree at most n and T := {z : |z| = 1} we denote by D− the region inside T and by D+ the region outside T If P ∈ Pn , then concerning the estimate of |P (z)| on the unit circle T , we have the following well known result which relates the norm of a polynomial to that of its derivative due to Bernstein [9] max |P (z)| ≤ n max |P (z)| z∈T z∈T (1.1) The inequality (1.1) is sharp and equality holds for polynomials having all zeros at the origin The inequality (1.1) was improved by Malik [6] In fact he proved: If P ∈ Pn and Q(z) = z n P ( z1¯ ), then max |P (z)| + max |Q (z)| ≤ n max |P (z)| z∈T z∈T z∈T (1.2) If we consider the class of polynomials P ∈ Pn having no zero in D− , then the bounds in inequality (1.1) can be considerably improved In fact, Erdăos conjectured and later Lax [4] verified that if P (z) does not vanish in D− , then (1.1) can be replaced by max |P (z)| ≤ z∈T n max |P (z)| z∈T (1.3) 338 IDREES QASIM AND A LIMAN Tur´an [10] reversed the hypothesis of the result proved by Erdăos-Lax and showed that if P Pn and P (z) = in D+ , then max |P (z)| ≥ z∈T n max |P (z)| z∈T (1.4) In 1988, Mohapatra, O’Hara and Rodrigues [7] proved that, if z1 , z2 , , z2n are any 2n equally spaced points on T listed in order, say zk = ue P ∈ Pn max |P (z)| ≤ z∈T kπi n , where u ∈ T and k = 1, 2, , 2n, then for n [max |P (zk )| + max |P (zk )|] k even k odd (1.5) R ATIONAL F UNCTIONS Let α1 , α2 , , αn be n given points in D+ Consider the following space of rational functions with prescribed poles and with finite limit at infinity Rn = p(z) : p ∈ Pn , w(z) where n w(z) = (z − αj ) j=1 The inequalities of Bernstein and Erdăos-lax have been extended to the rational functions ([2], [5]) by replacing the polynomial p(z) by a rational function r(z) and z n by Blaschke product B(z) defined by B(z) = z n w( z1¯ ) w∗ (z) = = w(z) w(z) n j=1 1−α ¯j z z − αj Besides other things they proved, for any r ∈ Rn |r (z)| ≤ |B (z)|||r|| (2.1) Furthermore, the inequality (2.1) is sharp and the equality holds if r(z) = αB(z) with α ∈ T If we assume r ∈ Rn does not vanish in D− , then for z ∈ T , the inequality (2.1) can be strengthened to |r (z)| ≤ |B (z)|||r|| (2.2) The inequality is sharp and equality holds if r(z) = αB(z) + β with α, β ∈ T Also, if r(z) does not vanish in D+ , then |r (z)| ≥ |B (z)|||r|| (2.3) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 339 In this paper we consider a more general class of rational functions r(s(z)), defined by (ros)(z) = r(s(z)) = p(s(z)) , w(s(z)) where s(z) is a polynomial of degree m and r(z) is a rational function of degree n, so that r(s(z)) ∈ Rmn , and mn w(s(z)) = (z − aj ) j=1 Hence, Balachke product is given by w(s( z1¯ )) w∗ (s(z)) B(z) = = = w(s(z)) w(s(z)) mn j=1 1−a ¯j z z − aj Thereby prove the following results which in turn generalizes the above inequalities M AIN R ESULTS From now on, we shall always assume that all poles a1 , a2 , , amn of r(s(z)) lie in D+ For the case when all poles are in D− , we can obtain analogous results with suitable modification Theorem — If r(s(z)) ∈ Rmn and z ∈ T , then B (z)r(s(z)) − s (z)r (s(z))[B(z) − λ] = B(z) z mn ck r(s(tk )) k=1 B(z) − λ , z − tk (3.1) where ck = ck (λ) is defined for k = 1, 2, 3, , mn by mn |aj |2 − |tk − aj |2 c−1 k = j=1 Furthermore, for z ∈ T mn zB (z) = B(z) ck k=1 (3.2) B(z) − λ , z − tk (3.3) where tk , k = 1, 2, 3, , mn are defined as in Lemma (to be mentioned later) Corollary — Let ck and tk (for k = 1, 2, 3, , mn) be defined as in Theorem If u1 , u2 , , umn are the roots of B(z) = −λ and dk is defined as ck with tk replaced by uk , for k = 1, 2, , mn If |s(z)| = m (3.4) z∈T and all zeros of s(z) lie in T ∪ D− , then for z ∈ T |r (s(z))| ≤ |B (z)| 2mm max |r(s(tk ))| + max |r(s(uk ))| 1≤k≤mn 1≤k≤mn (3.5) 340 IDREES QASIM AND A LIMAN The inequality is sharp and equality holds for r(s(z)) = uB(z) with u ∈ T , where s(z) = z m Corollary immediately yields the following generalization of inequality (2.1) Corollary — If r(s(z)) ∈ Rmn and all zeros of s(z) lie in T ∪ D− , then |r (s(z))| ≤ |B (z)|||r(s)||, mm (3.6) where m is defined by equation (3.4) and ||r(s)|| = maxz∈T |r(s(z))| The inequality is sharp in the sense that equality is obtained when r(s(z)) = uB(z) with u ∈ T , where s(z) = z m As a generalization of inequality (1.2), we prove: Theorem — If r(s(z)) ∈ Rmn and all zeros of s(z) lie in T ∪ D− then for z ∈ T , |r∗ (s(z))| + |r (s(z))| ≤ |B (z)| ||r(s)||, mm (3.7) where r∗ (s(z)) = B(z)r(s( z1¯ )) Also equality holds for r(s(z)) = uB(z) with u ∈ T , where s(z) = z m We next present the following generalization of inequality (2.2) Theorem — Let r(s(z)) ∈ Rmn be such that r(s(z)) = in D− and all zeros of s(z) lie in T ∪ D− If |s(z)| = m , z∈T then for z ∈ T , we have |r (s(z))| ≤ |B (z)|||r(s)|| 2mm (3.8) The inequality is sharp and equality holds for the rational functions of the form r(s(z)) = αB(z) + β with α, β ∈ T where s(z) = z m Theorem — Let r(s(z)) ∈ Rmn and r(s(z)) = in D+ If max |s(z)| = M , z∈T (3.9) then for z ∈ T , we have |r (s(z))| ≥ 2mM |B (z)| − m(n − n ) |r(s(z))|, where mn and mn are respectively number of zeros and poles of r(s(z)) (3.10) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 341 The inequality is sharp and equality holds for rational functions of the form r(s(z)) = αB(z)+β with α, β ∈ T where s(z) = z m If r(s(z)) has exactly mn zeros then n = n and we get the following generalization of inequality (2.3) Corollary — Let r(s(z)) ∈ Rmn and r(s(z)) = in D+ If max |s(z)| = M z∈T and r(s(z)) has exactly mn zeros, then for z ∈ T , we have |r (s(z))| ≥ |B (z)|r(s(z))| 2mM (3.11) The inequality is sharp and equality holds for rational functions of the form r(s(z)) = αB(z) + β with α, β ∈ T where s(z) = z m L EMMAS For the proofs of these Theorems we need the following lemmas The first two lemmas are due to Li, Mohapatra and Rodrigues [5] Lemma — Suppose λ ∈ T Then the equation B(z) = λ has exactly mn simple roots, say t1 , t2 , , tmn and all lie on the unit circle T Moreover tk B (tk ) = λ mn j=1 |aj |2 − f or k = 1, 2, 3, , mn |tk − aj |2 (4.1) Lemma — If |u| = |v| = 1, then (u − v)2 = −uv|u − v|2 (4.2) Next lemma is due to Aziz and Dawood [1] Lemma — If p(z) is a polynomial of degree n, having all zeros in T ∪ D− , then |p (z)| ≥ n |p(z)| z∈T z∈T (4.3) The inequality is sharp and equality holds for polynomials having all zeros at the origin Lemma — If z ∈ T , then zB (z) = B(z) mn j=1 |aj |2 − = |B (z)| |z − aj |2 (4.4) 342 IDREES QASIM AND A LIMAN P ROOF : We have B(z) = w∗ (s(z)) = w(s(z)) This gives zB (z) = B(z) mn mn j=1 1−a ¯j z z − aj −z¯ aj z − 1−a ¯ j z z − aj j=1 Hence for z ∈ T , we have zB (z) = B(z) mn j=1 |aj |2 − |z − aj |2 Since |aj | > ∀ ≤ j ≤ mn, it follows from above that for z ∈ T , we have zB (z) B(z) is real and positive Therefore zB (z) zB (z) = = |B (z)| B(z) B(z) This completes the proof of Lemma Lemma — Let r(s(z)) ∈ Rmn If all zeros of r(s(z)) lie in T ∪ D+ , then for z ∈ T and r(s(z)) = Re z(r(s(z))) r(s(z)) ≤ |B (z)| (4.5) P ROOF : If p(z) has n zeros and s(z) has m zeros, then p(s(z)) has mn zeros Let b1 , b2 , , bmn be the zeros of p(s(z)), mn ≤ mn Now r(s(z)) = p(s(z)) w(s(z)) This gives z (r(s(z))) = r(s(z)) mn j=1 z − z − bj mn j=1 z z − aj (4.6) Since all zeros of p(s(z)) lie in T ∪ D+ , therefore for z ∈ T with z = bk , we have z z ≤ − f or j = 1, 2, 3, , mn z − bj z − bj Using the fact that Re(z) ≤ Re if and only if |z| ≤ |z − 1|, we get from inequality (4.7) z z − bj ≤ f or j = 1, 2, , mn (4.7) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 343 Hence from equation (4.6), we have Re z mn (r(s(z))) r(s(z)) ≤ j=1 − mn ≤ Re j=1 mn = j=1 mn Re j=1 z z − aj z − z − aj |aj |2 − 2|z − aj |2 This with the help of equation (4.4) gives Re z (r(s(z))) r(s(z)) ≤ |B (z)| This completes the proof of Lemma Lemma — Let r(s(z)) ∈ Rmn If all zeros of r(s(z)) lie in T ∪ D− , then for z ∈ T and r(s(z)) = 0, we have Re z(r(s(z))) r(s(z)) |B (z)| − m(n − n )| , ≥ (4.8) where mn and mn are respectively the number of zeros and poles of r(s(z)) P ROOF : Suppose all the zeros of r(s(z)) lie in T ∪ D− and z ∈ T with z = bj ∀ ≤ j ≤ mn Then as in lemma 5, we obtain Re z z − bj ≥ for j = 1, 2, , mn Using equation (4.6), we get Re z mn (r(s(z))) r(s(z)) ≥ j=1 mn = Re j=1 mn = j=1 = This completes the proof of Lemma − mn Re j=1 z − z − aj z z − aj − (mn − mn ) |aj |2 − m − (n − n ) 2|z − aj |2 |B (z)| − m(n − n ) 344 IDREES QASIM AND A LIMAN P ROOFS OF T HEOREMS P ROOF OF T HEOREM : Let q(z) = w∗ (s(z)) − λ(w(s(z))) Since the solution of B(z) = λ is same as polynomial equation w∗ (s(z)) − λw(s(z)) = which has degree exactly mn, it follows that it has exactly mn roots counting multiplicities If these roots are denoted by t1 , t2 , , tmn , then mn q(z) = w(s(z))[B(z) − λ] = K (z − tk ) k=1 For r(s(z)) = p(s(z)) w(s(z)) ∈ Rmn , let p(s(z)) = µz mn + , then p(s(z)) − µ q(z) ∈ Pmn−1 K The numbers t1 , t2 , , tmn are distinct, so by Lagrange interpolation formula we obtain p(s(z)) − µ q(z) = K mn k=1 p(s(tk ))q(z) q (tk )(z − tk ) Dividing both sides by q(z) and differentiating, we get p(s(z)) q(z) mn = k=1 q (tk )(z − tk )(p(s(tk ))) − p(s(tk ))q (tk ) (q (tk )(z − tk ))2 mn =− k=1 p(s(tk )) q (tk )(z − tk )2 (5.1) Next recall that, q(z) = w(s(z))[B(z) − λ] and p(s(z)) = w(s(z))r(s(z)) Hence q (tk ) = w(s(tk ))B (tk ) and p(s(tk )) = w(s(tk ))r(s(tk )) Moreover, since tk are the zeros of B(z) = λ Therefore q(tk ) = Using these in equation (5.1), we get r(s(z)) B(z) − λ mn =− k=1 r(s(tk )) B (tk )(z − tk )2 (5.2) Which implies [B(z) − λ](r(s(z))) − r(s(z))B (z) =− [B(z) − λ]2 mn k=1 r(s(tk )) B (tk )(z − tk )2 Multiplying both sides by −[B(z) − λ]2 , we get mn r(s(z))B (z) − s (z)r (s(z))[B(z) − λ] = k=1 r(s(tk ))[B(z) − λ]2 B (tk )(z − tk )2 (5.3) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 345 For z ∈ T , |B(z)| = and |λ| = Therefore by virtue of lemma 2, we obtain [B(z) − λ]2 = −B(z)λ|B(z) − λ|2 Similarly, (z − tk )2 = −ztk |z − tk |2 Hence it follows from equation (5.3) B(z) B (z)r(s(z)) − s (z)r (s(z))[B(z) − λ] = z mn k=1 λr(s(tk )) B(z) − λ tk B (tk ) z − tk (5.4) B(z) − λ z − tk (5.5) Using Lemma (1) and definition of ck , we get B (z)r(s(z)) − s (z)r (s(z))[B(z) − λ] = B(z) z mn ck r(s(tk )) k=1 Which completely proves Theorem P ROOF OF C OROLLARY : Applying Theorem after replacing λ by −λ, we get B (z)r(s(z)) − s (z)r (s(z))[B(z) + λ] = B(z) z mn dk r(s(uk )) k=1 B(z) + λ z − uk (5.6) Subtract (5.5) and (5.6), we get zs (z)r (s(z))[B(z) + λ − B(z) + λ] = B(z) mn k=1 B(z) − λ ck r(s(tk )) − z − tk mn dk r(s(uk )) k=1 B(z) + λ z − uk Hence for z ∈ T , we have mn |2s (z)r (s(z))| ≤ k=1 B(z) − λ |ck ||r(s(tk ))| z − tk mn |ck | ≤ max |r(s(tk ))| k k=1 mn |dk ||r(s(uk ))| + k=1 B(z) − λ z − tk B(z) + λ z − uk mn |dk | + max |r(s(uk ))| k k=1 B(z) + λ z − uk Since both ck and dk are positive by definition, therefore using (3.3) we get |2s (z)r (s(z))| ≤ zB (z) B(z) max |r(s(tk ))| + max |r(s(uk ))| 1≤k≤n 1≤k≤n Finally, by virtue of lemma and lemma 4, we obtain |r (s(z))| ≤ |B (z)| 2mm max |r(s(tk ))| + max |r(s(uk ))| , 1≤k≤n 1≤k≤n where m is defined by equation (3.4) This proves Corollary completely P ROOF OF T HEOREM : From Theorem 1, we have for z ∈ T |B (z)r(s(z)) − s (z)r (s(z))[B(z) − λ]| = B(z) z mn ck r(s(tk )) k=1 B(z) − λ z − tk 2 346 IDREES QASIM AND A LIMAN B(z) ≤ z mn |r(s(tk ))| ck k=1 ≤ max |r(s(z))| z∈T B(z) − λ z − tk zB (z) B(z) = |B (z)|||r(s)|| Since right hand side is independent of λ, therefore we can suitably choose λ such that |B (z)r(s(z)) − s (z)r (s(z))B(z)| + |s (z)r (s(z))| ≤ |B (z)|||r(s)|| Next recall that (5.7) r∗ (s(z)) = B(z)r(s( )) z¯ So that 1 1 (r∗ (s(z))) = B (z)r(s( )) − B(z)r (s( )).s ( ) z¯ z z¯ z¯ Which implies 1 1 (r∗ (s(z))) = B (z)r(s( )) − B(z)r (s( ))s ( ) z¯ z z¯ z¯ Hence for z ∈ T , we have (r∗ (s(z))) = z Using the fact that zB (z) B(z) B (z) r(s(z)) − zr (s(z)).s (z) B(z) is real, we get (r∗ (s(z))) = z B (z) r(s(z)) − zr (s(z)).s (z) B(z) = |B (z)r(s(z)) − r (s(z))s (z)B(z)| Hence we have from inequality (5.7) |r∗ (s(z))s (z)| + |r (s(z))s (z)| ≤ |B (z)|||r(s)|| Which gives the desired result by use of Lemma P ROOF OF T HEOREM : From equation (4.4), we have z B (z) = |B (z)| > B(z) (5.8) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 347 Hence for z ∈ T , with r(s(z)) = 0, we have from (5.8) (r∗ (s(z))) = |B (z)r(s(z)) − r (s(z))s (z)B(z)| = z B (z) r(s(z)) − zr (s(z))s (z) B(z) = ||B (z)|r(s(z)) − zr (s(z))s (z)| = zr (s(z))s (z) − |B (z)r(s(z))| |B (z)|r(s(z)) From lemma 5, we have Re zr (s(z))s (z) |B (z)|r(s(z)) (5.9) ≤ Which further implies zr (s(z))s (z) zr (s(z))s (z) −1 ≤ |B (z)|r(s(z)) |B (z)|r(s(z)) Using in (5.9), we get |(r∗ (s(z))) | ≥ zr (s(z))s (z) |B (z)r(s(z))| |B (z)|r(s(z)) Which further implies |(r∗ (s(z))) | ≥ |(r(s(z))) | Hence Theorem yields |(r(s(z))) | ≤ |B (z)|||r(s)|| Lemma is thereby allowing us to write |r (s(z))| ≤ |B (z)|||r(s(z))|| 2mm This proves the theorem for r(s(z)) = Since the above inequality is trivially true for r(s(z)) = Therefore we conclude that the theorem is true for all z ∈ T P ROOF OF T HEOREM : Let r(s(z)) = Since z ∈ T , therefore we have by use of lemma (r(s(z))) (r(s(z))) ≥ Re z r(s(z)) r(s(z)) ≥ |B (z)| − m(n − n ) Which yields by the use of inequality (1.1) |r (s(z))| ≥ 2mM |B (z)| − m(n − n ) |r(s(z))| (5.10) This proves Theorem for r(s(z)) = Since inequality (5.10) is trivially true for r(s(z)) = Therefore Theorem holds for all z ∈ T 348 IDREES QASIM AND A LIMAN R EFERENCES A Aziz and Q M Dawood, Inequalities for a polynomial and its derivative, J Approx Theory, 54 (1988), 119-122 P Borwein and T Erdelyi, Sharp extension of Bernstein inequalities to rational spaces, Mathematika, 43 (1996), 413-423 C Frappier, Q I Rahman and St Derivatives of polynomials, Approx Theory, 66 (1991), 29-35 P D Lax, Proof of a conjecture of P Erdăos on the derivative of a polynomial, Bull Amer Math Soc (N.S), 50 (1944), 509-513 Xin Li, R N Mohapatra and R S Rodgriguez, Bernstein inequalities for rational functions with prescribed poles, J London Math Soc., 51 (1995), 523-531 M A Malik, On the derivative of the polynomial, J London Math Soc., (1969), 57-60 R N Mohapatra, P J O’Hara and R S Rodrigues, Simple proofs of Bernstein 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D+ , then |r (z)| ≥ |B (z)|||r|| (2.3) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 339 In this paper we consider a more general class of rational functions r(s(z)), defined by (ros)(z)... k=1 r(s(tk ))[B(z) − λ]2 B (tk )(z − tk )2 (5.3) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS 345 For z ∈ T , |B(z)| = and |λ| = Therefore by virtue of lemma 2, we obtain [B(z) − λ]2 =... then for z ∈ T , we have |r (s(z))| ≥ 2mM |B (z)| − m(n − n ) |r(s(z))|, where mn and mn are respectively number of zeros and poles of r(s(z)) (3.10) BERNSTEIN TYPE INEQUALITIES FOR RATIONAL FUNCTIONS