• Title/Summary/Keyword: left (right) derivative

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Lr INEQUALITIES FOR POLYNOMIALS

  • Reingachan N;Mayanglambam Singhajit Singh;Nirmal Kumar Singha;Khangembam Babina Devi;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • v.29 no.2
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    • pp.451-460
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    • 2024
  • If a0 + Σnν=μ aνzν, 1 ≤ µ ≤ n, is a polynomial of degree n having no zeroin |z| < k, k ≥ 1 and p'(z) its derivative, then Qazi [19] proved $$\max_{{\left|z\right|=1}}\left|p\prime(z)\right|\leq{n}\frac{1+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|k^{{\mu}+1}}{1+k^{{\mu}+1}+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|(k^{{\mu}+1}+k^{2{\mu}})}\max_{{\left|z\right|=1}}\left|p(z)\right|$$ In this paper, we not only obtain the Lr version of the polar derivative of the above inequality for r > 0, but also obtain an improved Lr extension in polar derivative.

TURÁN-TYPE Lr-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL

  • Robinson Soraisam;Mayanglambam Singhajit Singh;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.3
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    • pp.731-751
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    • 2023
  • If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into Lr-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

Efficient Score Estimation and Adaptive Rank and M-estimators from Left-Truncated and Right-Censored Data

  • Chul-Ki Kim
    • Communications for Statistical Applications and Methods
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    • v.3 no.3
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    • pp.113-123
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    • 1996
  • Data-dependent (adaptive) choice of asymptotically efficient score functions for rank estimators and M-estimators of regression parameters in a linear regression model with left-truncated and right-censored data are developed herein. The locally adaptive smoothing techniques of Muller and Wang (1990) and Uzunogullari and Wang (1992) provide good estimates of the hazard function h and its derivative h' from left-truncated and right-censored data. However, since we need to estimate h'/h for the asymptotically optimal choice of score functions, the naive estimator, which is just a ratio of estimated h' and h, turns out to have a few drawbacks. An altermative method to overcome these shortcomings and also to speed up the algorithms is developed. In particular, we use a subroutine of the PPR (Projection Pursuit Regression) method coded by Friedman and Stuetzle (1981) to find the nonparametric derivative of log(h) for the problem of estimating h'/h.

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RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Anastassiou, George A.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1423-1433
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    • 2019
  • Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

SOME CLASSES OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS I

  • AUOF, M.K.;DARWISH, H.E.
    • Honam Mathematical Journal
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    • v.16 no.1
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    • pp.119-135
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    • 1994
  • Let $Q_{n+p-1}(\alpha)$ denote the- dass of functions $$f(z)=z^{P}-\sum_{n=0}^\infty{a_{(p+k)}z^{p+k}$$ ($a_{p+k}{\geq}0$, $p{\in}N=\left{1,2,{\cdots}\right}$) which are analytic and p-valent in the unit disc $U=\left{z:{\mid}z:{\mid}<1\right}$ and satisfying $Re\left{\frac{D^{n+p-1}f(\approx))^{\prime}}{pz^{p-a}\right}>{\alpha},0{\leq}{\alpha}<1,n>-p,z{\in}U.$ In this paper we obtain sharp results concerning coefficient estimates, distortion theorem, closure theorems and radii of p-valent close-to- convexity, starlikeness and convexity for the class $Q_{n+p-1}$ ($\alpha$). We also obtain class preserving integral operators of the form $F(z)=\frac{c+p}{z^{c}}\int_{o}^{z}t^{c-1}f(t)dt.$ c>-p $F\left(z\right)=\frac{c+p}{z^{c}}\int_{0}^{z} t^{c-1}f\left(t \right)dt. \qquad c>-p$ for the class $Q_{n+p-1}$ ($\alpha$). Conversely when $F(z){\in}Q_{n+p-1}(\alpha)$, radius of p-valence of f(z) has been determined.

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NUMBER OF VERTICES FOR POLYGONAL FUNCTIONS UNDER ITERATION

  • Li, Lin
    • The Pure and Applied Mathematics
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    • v.14 no.2 s.36
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    • pp.99-109
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    • 2007
  • Being complicated in computation, iteration of a nonlinear 1-dimensional mapping makes many interesting problems, one of which is about the change of the number of vertices under iteration. In this paper we investigate iteration of polygonal functions which each have only one vertex and give conditions under which the number of vertices either does not increase or has a bound under iteration.

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INEQUALITIES FOR COMPLEX POLYNOMIAL WITH RESTRICTED ZEROS

  • Istayan Das;Robinson Soraisam;Mayanglambam Singhajit Singh;Nirmal Kumar Singha;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.4
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    • pp.943-956
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    • 2023
  • Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z0, |z0| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

THE 3D BOUSSINESQ EQUATIONS WITH REGULARITY IN THE HORIZONTAL COMPONENT OF THE VELOCITY

  • Liu, Qiao
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.649-660
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    • 2020
  • This paper proves a new regularity criterion for solutions to the Cauchy problem of the 3D Boussinesq equations via one directional derivative of the horizontal component of the velocity field (i.e., (∂iu1; ∂ju2; 0) where i, j ∈ {1, 2, 3}) in the framework of the anisotropic Lebesgue spaces. More precisely, for 0 < T < ∞, if $$\large{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_o}^T}({\HUGE\left\|{\small{\parallel}{\partial}_iu_1(t){\parallel}_{L^{\alpha}_{x_i}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}+}{\HUGE\left\|{\small{\parallel}{\partial}_iu_2(t){\parallel}_{L^{\alpha}_{x_j}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}})dt<{{\infty}},$$ where ${\frac{2}{{\gamma}}}+{\frac{1}{{\alpha}}}+{\frac{2}{{\beta}}}=m{\in}[1,{\frac{3}{2}})$ and ${\frac{3}{m}}{\leq}{\alpha}{\leq}{\beta}<{\frac{1}{m-1}}$, then the corresponding solution (u, θ) to the 3D Boussinesq equations is regular on [0, T]. Here, (i, ${\hat{i}}$, ${\tilde{i}}$) and (j, ${\hat{j}}$, ${\tilde{j}}$) belong to the permutation group on the set 𝕊3 := {1, 2, 3}. This result reveals that the horizontal component of the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

Comparison of left and right hand peripheral vascular compliance between normal and diabetic group using the second derivative of photoplethysmogram (PPG 2차 미분을 이용한 정상인과 당뇨병 환자 간의 왼손과 오른손 말초혈관 탄성도 차이 비교)

  • Lee, Tak-Hyung;Shim, Young-Woo;No, Hyung-Wook;Kim, Deok-Won
    • Proceedings of the KIEE Conference
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    • 2008.04a
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    • pp.133-134
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    • 2008
  • 우리나라의 당뇨병 유병률은 경제발전과 식생활의 급속한 서구화로 빠른 속도로 증가하고 있다. 특히 당뇨병은 유병기간이 길수록 여러 합병증을 초래하는데 그 중 하나가 혈관경화증이다. 혈관 경화증은 신체 부위별로 그 진행 속도가 다르다. 광혈류측정법(PPG, photoplethysmogram)은 간편한 혈류량 측정방법으로 PPG 2차 미분 방법을 이용해서 혈관의 경화도를 진단할 수 있다. 이러한 PPG 2차 미분방법을 이용하여 당뇨병환자의 왼손과 오른손의 말초혈관 탄성도 차이를 정상인과 비교해 보았다. 실험결과 당뇨병환자와 정상인의 양 손가자 말초혈관 탄성도의 유의한 차이를 확인 할 수 있었다.

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POLYNOMIALLY DEMICOMPACT OPERATORS AND SPECTRAL THEORY FOR OPERATOR MATRICES INVOLVING DEMICOMPACTNESS CLASSES

  • Brahim, Fatma Ben;Jeribi, Aref;Krichen, Bilel
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1351-1370
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    • 2018
  • In the first part of this paper we show that, under some conditions, a polynomially demicompact operator can be demicompact. An example involving the Caputo fractional derivative of order ${\alpha}$ is provided. Furthermore, we give a refinement of the left and the right Weyl essential spectra of a closed linear operator involving the class of demicompact ones. In the second part of this work we provide some sufficient conditions on the inputs of a closable block operator matrix, with domain consisting of vectors which satisfy certain conditions, to ensure the demicompactness of its closure. Moreover, we apply the obtained results to determine the essential spectra of this operator.