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of lnequal. & Appl., 1999, Vol. 3, pp. 45-49
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J.
Equivalence of Markov’s and Schur’s
Inequalities on Compact Subsets of
the Complex Plane
L. BIALAS-CIE2*
Department of Mathematics, Jagiellonian University, 30-059 KrakOw,
Reymonta 4, Poland
(Received 24 March 1997; Revised 3 September 1997)
We prove that, on an arbitrary compact subset of the complex plane, Markov’s and
Schur’s inequalities are equivalent.
Keywords." Markov inequality; Schur inequality
AMS Subject Classification." 41A17
We recall first the two classical inequalities of the title.
Markov’s inequality ([5], 1889): for any polynomial P
max{lP’(x)l"
x
[-1,1]} <_ (degP) 2 max{lP(x)l"
[-1, 11}.
x
Schur’s inequality ([8], 1919): for any polynomial P
max{lP(x)l"
x
[-1, 1]} _< (1 +degP)max{lxP(x)l"
x
[-1, 1]}.
These inequalities are extensively used in approximation theory and
have been widely generalized in many ways (see e.g. [1,4,6,7]).
Fax: (48) (12) 632 43 72. E-mail: bialas@im.uj.edu.pl.
45
L.
46
BIALAS-CIE
For example in the one-dimensional case, it has been proved for some
compact subset E of C that
I[P’IILq(E) C (deg p)ml ]lpl}zq(E)
(M)
]le[ltq(E) C2(dege)m2l{(x-Xo)P(x)]ILq(E),
(s)
where Xo E C and C1, ml, C2, m2 are positive constants depending only
on E and q,
[]fllLq(E)
[f(x)l q dx
IlfllL() max{[f(x)l"
q
x
[1, +),
E}.
If E [-1,1], it is not difficult to show that (S) can be established
using (M) (see [2, Lemma 2]); moreover (S) implies (M).
In this note we show that for an arbitrary compact set of C, (M) and
(S) are equivalent:
PROPOSITION Let E be a compact subset of R (or C) and q [1, +x].
Then the following conditions are equivalent:
(i) There exist two positive constants C, m, depending only on E and
q such that for any polynomial P 79(R) (resp. 79(C)),
IIP’IIq(E) C1 (deg p)m, lipiltq(E)
(ii) There exist two positive constants C2, m2, depending only on E
and q such that for any polynomial P 79(R) (resp. 79(C)) and any
Xo R (resp. C),
[IPII.cE/ C2(degp)m2[l(x-Xo)P(x)]ILq(E),
(iii) There exist two positive constants C3, m3, depending only on E
and q such that for any polynomial P 79(R) (resp. 79(C)) and any
a, b, c R (resp. C),
[{(ax2+ bx + c)P’(x)IIL,,(E <_ C3(degp)m3ll(ax2+ by + c)P(x)[IL,,(E ),
EQUIVALENCE OF MARKOV’S AND SCHUR’S INEQUALITIES
47
(iv) There exist two positive constants C4, m4, depending only on E and
q such that for any polynomials P and any polynomial R E P(R)
(resp. 79(C)),
IIRP’IILq(E) <_ C4(deg P -+- degR)m411RPIILq(E ).
Inequalities of types (iii) and (iv) were investigated by many authors
(see e.g. Nevai [6, Chapter 9] and Goetgheluck [1]).
Proof We will give the proof only for the real case; the complex case
is similar.
(1) Inequalities (i) and (ii) are equivalent.
The implication (i)=> (ii) is due to Goetgheluck [3].
(ii) :=> (i). We can write e(x) (x x)(x x2)... (x xk)(x2 +
bx + Cl) (x2 + b2x + c2)... (x2 + blx + cl), where x, x2,..., x:, b,
b2,...,bt, c,c2,...,c1 ER and ba2. < 4cj for everyj {1,2,...,1}.
Then we have
Thus
IIP’IIL ,(e) <-- C2k(degp)m llPllL,,( ) + 21C2(degp)mllPllL,()
C2(deg p)m
+l
L.
48
BIALAS-CIE
(2) Inequalities (i), (iii) and (iv) are equivalent.
(i) (iv). Fix an arbitrary unitary polynomial R. We have R(x)-(X- Xl) (X- X2)... (X Xk) (X2 + blx + cl) (x2 -I- b2x + c:)... (x2
btx + ct), for some x, x2,..., xk, b, b2,..., bt, c, ez,..., ct, R with
b} < 4c for every j { 1,2,..., l}. Then
=
II(RP)’IIq>
+
()
"=
+
(+
(-
()( + )
+
kn=l
kj=l,ji
Lq(E)
(+
(- )
"=
+
kn=l,ni
Lq(E)
By (i) which is equivalent to (ii), we have
C (deg P + deg R) m’
+ C2k(deg P + deg R)m[IRPI[q(e
+ 2c2(eg e + deg R) m IIRPII Lq(E)
2max{G, C2)(eg P + degR)max{m"m+}l[RP]lcq(e
The obvious implications (iv) (iii) and (iii) (i) complete the
proof.
Remark For the complex case it is easily seen that m
m2
m3
m4.
Acknowledgments
The author wishes to express her gratitude to Professor Pierre
Goetgheluck for his interest in this work and stimulating discussions.
The author wishes to thank the University Paris-Sud, where the
paper was written.
References
[1] P. Goetgheluck, Polynomial inequalities and Markov’s inequality in weighted
LP-spaces, Acta Math. Acad. Sci. Hungar. 33 (1979), 325-331.
EQUIVALENCE OF MARKOV’S AND SCHUR’S INEQUALITIES
49
[2] P. Goetgheluck, Une ingalitb polynomiale en plusieurs variables, J. Approx. Theory
40 (1984), 161-172.
[3] P. Goetgheluck, Polynomial inequalities on general subsets of RN, Coll. Math. 57
(1989), 127-136.
[4] A. Jonsson and H. Wallin, Function Spaces on Subset of Rn, Mathematical Reports,
Vol. 2, Part 1, Harwood Academic, London, 1984.
[5] A.A. Markov, On a problem posed by D.I. Mendeleev, Izv. Akad. Nauk St.
Petersbourg 62 (1889), 1-24 (in Russian).
[6] P. Nevai, Orthogonal polynomials, Mere. Amer. Math. Soc. 213 (1979), 1-185.
[7] W. Paqcjucki and W. Pleniak, Markov’s inequality and C functions on sets with
polynomial cusps, Math. Ann. 275(3) (1986), 467-480.
[8] I. Schur, (Ober das Maximum des absoluten Betrages eines Polynoms in einen
gegebenen Intervall, Math. Z. 4 (1919), 271-287.