J. Comp. & Math. Sci. Vol. 1(4), 419-423 (2010).
Note on a theorem of S. Bernstein M. S. PUKHTA*, ABDULLAH MIR** and T A Raja*** *Division of Agri.Engineering Sher-e-Kashmir University of Agricultural Sciences and Technology Kashmir,191121 (India) **Department of Mathematics, University of Kashmir, 191121 (India) ***Div. of Agri.Statistics , SKUAST-Kashmir, 191121 (India) ABSTRACT nv=
Let
a polynomial of degree n and let M(P,r)=
,
be
|P(z)|. If P(z) has all
its zeros on |z|=k,k1, then it was shown by Dewan & Hans7
In this paper we have improved the above inequality by involving some or all the coefficients of . Our result also generalizes a known result of Dewan & Mir5. Key words: Polynomial, Inequality, Derivatives, Zeros.
1. INTRODUCTION AND STATEMENT OF RESULTS Let P(z) be a polynomial of degree n and let then according to Bernstein’s inequality
in (1.1) is obtained for
,
.
If we restrict ourselves to the class of polynomials not vanishing in , then the Erdos conjectured and Lax3 proved
(1 .1)
(1 .2)
The result is best possible and equality
Inequality (1.2) is best possible
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
420
M.S. Pukhta et al., J.Comp.&Math.Sci. Vol.1(4), 419-423 (2010) is a polynomial of degree n having all
and the extremal polynomial is with .
its zeros on
then
As a generalization of (1.2), Malik4 proved Theorem 1.1. If be a polynomial of degree n which does not vanish in , then (1 .3)
(1.5) Further as a generalization of Govil1, Dewan & Hans7 proved the following.
, The result is best possible and equality holds for .
nv=
Theorem 1.4. If
is a polynomial of
degree n, having all its zeros on , then
While trying to obtain inequality analogous to (1.3) for polynomials not vanishing in , Govil1 was able to prove the following:
In this paper we shall generalize Theorem 1.3 for the polynomial of the
nv=0
Theorem 1.2. If be a polynomial of degree . If its zeros on
and let
has all then (1.4)
Dewan and Abdullah following. Theorem 1.3.If
5
(1.6)
proved the
nv=
type
,
, and which in turn refines Theorem 1.4 as well. More precisely, we prove Theorem. If , of degree n having
a polynomial all its zeros on
then nv=0
(1.7) Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
M.S. Pukhta et al., J.Comp.&Math.Sci. Vol.1(4), 419-423 (2010) The bound obtained in the above theorem is better than the bound obtained in Theorem 1.4
421
the above theorem then the inequality (1.7) reduces to the inequality (1.5). 2. Lemmas
To prove that the bound obtained in the above theorem is better than the bound obtained in Theorem 1.4, we show that
We need the following lemmas for the proof of the theorem. Lemma 2.1. If ,
, is a polynomial
of degree having no zeros in then Which is equivalent to (2 .1) Where here and throughout this paper Which implies
. The above lemma is due to Dewan & Hans7. Lemma 2.2. If ďƒĽ nv=0 be a polynomial of degree n, then on |z|=1 (2 .2)
Or
The above lemma is a special case of a result due to Govil and Rahman2.
Or
Lemma 2.3. If
Or
, a polynomial of degree which does not vanish in then for |z|=1,
Or Which is always true see lemma 2.4. Remark 1. If we take Âľ=1 in
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
(2.3)
422
M.S. Pukhta et al., J.Comp.&Math.Sci. Vol.1(4), 419-423 (2010)
.
, where
and
which implies
(2.4) for
.
where
The above lemma follows easily on using arguments similar to that used in [6, lemma 1].
and Equivalently
Lemma 2.4. If , a polynomial of degree n having all its zeros in then , for
(2.5)
and
(2.6)
for . Which completes the proof of lemma. 3. Proof of the Theorem : Let z0 be a point on |z|=1 such that 2.2 it follows that
, then by Lemma
.
where
(3 .1)
Proof of Lemma 2.4. Since all t he zeros of P(z) lie in , then all the zeros of q(z) lie in
Which when combined with Lemma 2.4 gives
.
Hence applying Lemma 2.3 to the polynomial and using the fact that
Which is equivalent to
we get
for and
(3.2) Inequality (3.1) when combined with Lemma 2.1, gives
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)
M.S. Pukhta et al., J.Comp.&Math.Sci. Vol.1(4), 419-423 (2010)
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Which implies
This completes the proof of the theorem.
4.
REFERENCES 5. 1. N. K. Govil, On a theorem of S. Bernstein, Jour. Math. Phy. Sci. 2, 183-187 (1980). 2. N.K. Govil and Q.I. Rahman, Functions of exponential type not vanishing in half plane and related polynomials. Trans. Amer. Math. Soc. 137, 501- 517 (1969). 3. P.D. Lax Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc.
6.
7.
50, 509-513 (1944). M.A. Malik. On the derivative of a polynomial. Jour. London Math. Soc. 1, 57-60 (1969). K.K. Dewan & Abdullah Mir. Note on a Theorem of S. Bernstein, Southeast Assian Bulletin of Mathematics 30, 1-5 (2006). M.A. Qazi, On the maximum modulus of polynomials. Proc. Amer.Math. Soc. 115, 337-343 (1992). K.K.Dewan & Sunil Hans. On extremal properties for the derivative of polynomial, Mathematica Balkanica, vol. 23, 2009, Fasc. 1-2.
Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)