Refinement of some inequalities concerning to <span class="a-plus-plus inline-equation id-i-eq1"> <span class="a-plus-plus equation-source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">$B_{n}$</span> </span>-operator of polynomials with restricted zeros

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Refinement of some inequalities concerning to source format-t-e-x" xmlns:search="http://marklogic.com/appservices/search">$B_{n}$ -operator of polynomials with restricted zeros

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作者：Idrees Qasim ; A. Liman ; W. M. Shah
关键词：$B_{n}$ ; operator ; Complex polynomials ; Inequalities ; Zeros
刊名：Periodica Mathematica Hungarica
出版年：2017
出版时间：March 2017
年：2017
卷：74
期：1
页码：1-10
全文大小：
刊物类别：Mathematics and Statistics
刊物主题：Mathematics, general;
出版者：Springer Netherlands
ISSN：1588-2829
卷排序：74

文摘

If F(z) is a polynomial of degree n having all zeros in $|z|\le k,~k>0$ and f(z) is a polynomial of degree $m\le n$ such that $|f(z)|\le |F(z)|$ for $|z|=k$, then it was formulated by Rather and Gulzar (Adv Inequal Appl 2:16–30, 2013) that for every $|\delta |\le 1, |\beta |\le 1,~R>r\ge k$ and $|z|\ge 1,$$$\begin{aligned} |B[fo\sigma ](z)+\psi B[fo\rho ](z)|\le |B[Fo\sigma ](z)+\psi B[Fo\rho ](z)|, \end{aligned}$$where B is a $B_{n}$ operator, $\sigma (z){=}Rz, \rho (z){=}rz$ and $\psi {:=}\psi (R,r,\delta ,\beta ,k) {=}\beta \bigg \{\bigg (\frac{R+k}{r+k}\bigg )^{n}{-}|\delta |\bigg \}{-}\delta $. The authors have assumed that $B\in B_{n}$ is a linear operator which is not true in general. In this paper, besides discussing assumption of authors and their followers (see e.g, Rather et al. in Int J Math Arch 3(4):1533–1544, 2012), we present the correct proof of the above inequality. Moreover our result improves many prior results involving $B_{n}$ operators and a number of polynomial inequalities can also be deduced by a uniform procedure.Keywords$B_{n}$ operatorComplex polynomialsInequalitiesZerosMathematics Subject Classification30A0630A64References1.N.C. Ankeny, T.J. Rivilin, On the theorem of S. Bernstein. Pac. J. Math. 5, 849–852 (1955)CrossRefMATHGoogle Scholar2.S.N. Bernstein, Sur eordre de la meilleure approximation des functions continues par des polynomes de degre donne. Mem. Acad. R. Belg. 4, 1103 (1912)Google Scholar3.S.N. Bernstein, Sur la limitation des derives des polynomes. C. R. Acad. Sci. Paris 190, 338–340 (1930)MATHGoogle Scholar4.C. Frappier, Q.I. Rahman, St Ruscheweyh, New inequalities for polynomials. Trans. Am. Math. Soc. 288, 69–99 (1985)MathSciNetCrossRefMATHGoogle Scholar5.N.K. Govil, A. Liman, W.M. Shah, Some Inequalities concerning derivative and maximum modulus of polynomials. Aust. J. Math. Anal. Appl., 1199–1209 (2010)6.P.D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial. Bull. Am. Math. Soc. (N.S) 50, 509–513 (1944)CrossRefMATHGoogle Scholar7.A. Liman, R.N. Mohapatra, W.M. Shah, Inequalities for polynomials not vanishing in a disk. Appl. Math. Comput. 218, 949–955 (2010)MathSciNetMATHGoogle Scholar8.M. Marden, Geometry of Polynomials, vol. 3, 2nd edn., Mathematical Surveys (American Mathematical Society, Providence, 1966)MATHGoogle Scholar9.G. Polya, G. Szego, Problems and Theorems in Analysis (Springer, New York, 1972). 1 CrossRefMATHGoogle Scholar10.Q.I. Rahman, Functions of exponential type. Trans. Am. Math. Soc. 135, 295–309 (1969)MathSciNetCrossRefMATHGoogle Scholar11.Q.I. Rahman, S. Schmeisser, Analytic Theory of Polynomials (Oxford University Press, New York, 2002)MATHGoogle Scholar12.N.A. Rather, S.H. Ahanger, M.A. Shah, Inequalities concerning the B-operators. Int. J. Math. Arch. 3(4), 1533–1544 (2012)Google Scholar13.N.A. Rather, S. Gulzar, On an operator preserving inequalities between polynomials. Adv. Inequal. Appl. 2, 16–30 (2013)MATHGoogle Scholar14.W.M. Shah, A. Liman, An operator preserving inequalities between polynomials. J. Inequal. Pure Appl. Math. 9, 1–16 (2008)MathSciNetMATHGoogle ScholarCopyright information© Akadémiai Kiadó, Budapest, Hungary 2016Authors and AffiliationsIdrees Qasim1Email authorA. Liman1W. M. Shah21.Department of MathematicsNational Institute of TechnologySrinagarIndia2.Jammu and Kashmir Institute of Mathematical SciencesSrinagarIndia About this article CrossMark Publisher Name Springer Netherlands Print ISSN 0031-5303 Online ISSN 1588-2829 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; display: inline-block; line-height: 1.2; padding: 0; } .buybox__header .buybox__login:before { position: absolute; 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