• 제목/요약/키워드: inequalities of Bernstein

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BERNSTEIN-TYPE INEQUALITIES PRESERVED BY MODIFIED SMIRNOV OPERATOR

  • Shah, Wali Mohammad;Fatima, Bhat Ishrat Ul
    • Korean Journal of Mathematics
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    • 제30권2호
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    • pp.305-313
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    • 2022
  • In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.

SOME INEQUALITIES ON POLAR DERIVATIVE OF A POLYNOMIAL

  • N., Reingachan;Robinson, Soraisam;Barchand, Chanam
    • Nonlinear Functional Analysis and Applications
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    • 제27권4호
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    • pp.797-805
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    • 2022
  • Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ Pn, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

A Note on Exponential Inequalities of ψ-Weakly Dependent Sequences

  • Hwang, Eunju;Shin, Dong Wan
    • Communications for Statistical Applications and Methods
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    • 제21권3호
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    • pp.245-251
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    • 2014
  • Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called ${\psi}$-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The ${\psi}$-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

INEQUALITIES AND COMPLETE MONOTONICITY FOR THE GAMMA AND RELATED FUNCTIONS

  • Chen, Chao-Ping;Choi, Junesang
    • 대한수학회논문집
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    • 제34권4호
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    • pp.1261-1278
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    • 2019
  • It is well-known that if ${\phi}^{{\prime}{\prime}}$ > 0 for all x, ${\phi}(0)=0$, and ${\phi}/x$ is interpreted as ${\phi}^{\prime}(0)$ for x = 0, then ${\phi}/x$ increases for all x. This has been extended in [Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336 (2007), 812-822]. In this paper, we extend the above result to the very general cases, and then use it to prove some (logarithmically) completely monotonic functions related to the gamma function. We also establish some inequalities for the gamma function and generalize some known results.

SOME Lq INEQUALITIES FOR POLYNOMIAL

  • Chanam, Barchand;Reingachan, N.;Devi, Khangembam Babina;Devi, Maisnam Triveni;Krishnadas, Kshetrimayum
    • Nonlinear Functional Analysis and Applications
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    • 제26권2호
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    • pp.331-345
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    • 2021
  • Let p(z)be a polynomial of degree n. Then Bernstein's inequality [12,18] is $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;n\;{\max_{{\mid}z{\mid}=1}{\mid}(z){\mid}}$$. For q > 0, we denote $${\parallel}p{\parallel}_q=\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}$$, and a well-known fact from analysis [17] gives $${{\lim_{q{\rightarrow}{{\infty}}}}\{{\frac{1}{2{\pi}}}{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{0}}^{2{\pi}}}\;{\mid}p(e^{i{\theta}}){\mid}^qd{\theta}\}^{\frac{1}{q}}={\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. Above Bernstein's inequality was extended by Zygmund [19] into Lq norm by proving ║p'║q ≤ n║p║q, q ≥ 1. Let p(z) = a0 + ∑n𝜈=𝜇 a𝜈z𝜈, 1 ≤ 𝜇 ≤ n, be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then for 0 < r ≤ R ≤ k, Aziz and Zargar [4] proved $${\max\limits_{{\mid}z{\mid}=R}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{nR^{{\mu}-1}(R^{\mu}+k^{\mu})^{{\frac{n}{\mu}}-1}}{(r^{\mu}+k^{\mu})^{\frac{n}{\mu}}}\;{\max\limits_{{\mid}z{\mid}=r}}\;{\mid}p(z){\mid}}$$. In this paper, we obtain the Lq version of the above inequality for q > 0. Further, we extend a result of Aziz and Shah [3] into Lq analogue for q > 0. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.