A Generalization of Ankeny and Rivlin's Result on the Maximum Modulus of Polynomials not Vanishing in the Interior of the Unit Circle

V. K. JAIN

Makale detayları

A Generalization of Ankeny and Rivlin's Result on the Maximum Modulus of Polynomials not Vanishing in the Interior of the Unit Circle

Yazarlar:

V. K. JAIN

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue 1, pp.89-94

Belge Tipi:

Makale

Yayın Tarihi:

12/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748474

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Anahtar kelimeler:

Özet:

 

For an arbitrary entire function f(z), let    M(f,r) = max|z|=r|f(z)|.    For a polynomial p(z) of degree n, it is known that    M(p,R) \leq Rn M(p,1), R > 1.    By considering the polynomial p(z) with no zeros in |z|  1.    By considering the polynomial p(z) with no zeros in |z| < k, (k \geq 1)  and simultaneously thinking of s\rm th derivative (0 \leq s < n)  of the polynomial, we have obtained the generalization  \begin{displaymath}  M(p(s),R) \leq \left\{(1/2)dsdRs(Rn+kn)(2/(1+k))nM(p,1),R≥k, (1/(Rs+ks))[dsdxs(1+xn)x=1]((R+k)/(1+k))nM(p,1),1≤R≤k,(1/2)dsdRs(Rn+kn)(2/(1+k))nM(p,1),R≥k, (1/(Rs+ks))[dsdxs(1+xn)x=1]((R+k)/(1+k))nM(p,1),1≤R≤k,\begin{array}{l} (1/2){\frac{ds}{dRs}(Rn  + kn)}(2/(1+k))nM(p,1), R \geq k,\  (1/(Rs+ks))[{\frac{ds}{dxs}(1+xn)}x=1]((R+k)/(1+k))nM(p,1),  1 \leq R \leq k,\end{array}\right.  of Ankeny and Rivlin's result.

 

For an arbitrary entire function f(z), let    M(f,r) = max|z|=r|f(z)|.    For a polynomial p(z) of degree n, it is known that    M(p,R) \leq Rn M(p,1), R > 1.    By considering the polynomial p(z) with no zeros in |z|  1.    By considering the polynomial p(z) with no zeros in |z| < k, (k \geq 1)  and simultaneously thinking of s\rm th derivative (0 \leq s < n)  of the polynomial, we have obtained the generalization  \begin{displaymath}  M(p(s),R) \leq \left\{(1/2)dsdRs(Rn+kn)(2/(1+k))nM(p,1),R≥k, (1/(Rs+ks))[dsdxs(1+xn)x=1]((R+k)/(1+k))nM(p,1),1≤R≤k,(1/2)dsdRs(Rn+kn)(2/(1+k))nM(p,1),R≥k, (1/(Rs+ks))[dsdxs(1+xn)x=1]((R+k)/(1+k))nM(p,1),1≤R≤k,\begin{array}{l} (1/2){\frac{ds}{dRs}(Rn  + kn)}(2/(1+k))nM(p,1), R \geq k,\  (1/(Rs+ks))[{\frac{ds}{dxs}(1+xn)}x=1]((R+k)/(1+k))nM(p,1),  1 \leq R \leq k,\end{array}\right.  of Ankeny and Rivlin's result.

Makale detayları

A Generalization of Ankeny and Rivlin's Result on the Maximum Modulus of Polynomials not Vanishing in the Interior of the Unit Circle

Yazarlar:

V. K. JAIN

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue 1, pp.89-94

Belge Tipi:

Makale

Yayın Tarihi:

12/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748474

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Polynomial

maximum modulus principle

not vanishing in the interior of unit circle

generalization

sth derivative

Sayı

Makale detayları

A Generalization of Ankeny and Rivlin's Result on the Maximum Modulus of Polynomials not Vanishing in the Interior of the Unit Circle

Yazarlar: V. K. JAIN

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2007 Vol.31 Issue 1, pp.89-94

Belge Tipi: Makale

Yayın Tarihi: 12/2006

Belge Dili: Türkçe

Document ID (Org No.): 2000748474

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Polynomial

maximum modulus principle

not vanishing in the interior of unit circle

generalization

sth derivative

Sayı

Yazarlar:

V. K. JAIN

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue 1, pp.89-94

Belge Tipi:

Makale

Yayın Tarihi:

12/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748474

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098