• 대한수학회논문집
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• 제38권2호
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• pp.389-400
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• 2023

In this paper, some estimations will be given for the analytic functions belonging to the class 𝓡(α). In these estimations, an upper bound and a lower bound will be determined for the first coefficient of the expansion of the analytic function h(z) and the modulus of the angular derivative of the function ${\frac{zh^{\prime}(z)}{h(z)}}$, respectively. Also, the relationship between the coefficients of the analytical function h(z) and the derivative mentioned above will be shown.

• Korean Journal of Mathematics
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• 제28권4호
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• pp.699-715
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• 2020

In this paper, we give estimates of the Hankel determinant H₂(1) in a novel class 𝓝 (𝜀) of analytical functions in the unit disc. In addition, the relation between the Fekete-Szegö function H₂(1) and the module of the angular derivative of the analytical function p(z) at a boundary point b of the unit disk will be given. In this association, the coefficients in the Hankel determinant b₂, b₃ and b₄ will be taken into consideration. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제27권4호
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• pp.157-169
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• 2020

In this paper, we improve a new boundary Schwarz lemma, for analytic functions in the unit disk. For new inequalities, the results of Rogosinski's lemma, Subordinate principle and Jack's lemma were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

• Korean Journal of Mathematics
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• 제27권4호
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• pp.1027-1041
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• 2019

In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Jack's lemma and Hankel determinant were used. We will get a sharp upper bound for Hankel determinant H₂(1). Also, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권2호
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• pp.169-178
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• 2015

In this paper, a boundary version of Carathéodory’s inequality is investigated. Also, new inequalities of the Carathéodory’s inequality at boundary are obtained and the sharpness of these inequalities is proved.

• 대한수학회보
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• 제50권6호
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• pp.2053-2059
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• 2013

In this paper, a boundary version of the Schwarz lemma is investigated. We obtain more general results at the boundary. Also, new inequalities of the Schwarz lemma at boundary is obtained and the sharpness of these inequalities is proved.

• Interaction and multiscale mechanics
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• 제4권4호
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• pp.291-311
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• 2011

To simulate the self excited torsional vibrations of rotating drill strings (DSs) in vertical bore-holes, the nonlinear wave models of homogeneous and sectional torsional pendulums are formulated. The stated problem is shown to be of singularly perturbed type because the coefficient appearing before the second derivative of the constitutive nonlinear differential equation is small. The diapasons ${\omega}_b\leq{\omega}\leq{\omega}_l$ of angular velocity ${\omega}$ of the DS rotation are found, where the torsional auto-oscillations (of limit cycles) of the DS bit are generated. The variation of the limit cycle states, i.e. birth (${\omega}={\omega}_b$), evolution (${\omega}_b<{\omega}<{\omega}_l$) and loss (${\omega}={\omega}_l$), with the increase in angular velocity ${\omega}$ is analyzed. It is observed that firstly, at birth state of bifurcation of the limit cycle, the auto-oscillation generated proceeds in the regime of fast and slow motions (multiscale motion) with very small amplitude and it has a relaxation mode with nearly discontinuous angular velocities of elastic twisting. The vibration amplitude increases as ${\omega}$ increases, and then it decreases as ${\omega}$ approaches ${\omega}_l$. Sectional drill strings are also considered, and the conditions of the solution at the point of the upper and lower section joints are deduced. Besides, the peculiarities of the auto-oscillations of the sectional DSs are discussed.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권3호
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• pp.263-273
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• 2015

In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z) = a + c_pz^p + c_p+₁z^p+1 + ... defined in the unit disc satisfying |f(z) − 1| < 1, where 0 < a < 2, we estimate a module of angular derivative at the boundary point b, f(b) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

• 대한수학회논문집
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• 제34권2호
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• pp.543-555
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• 2019

In this paper, we give some results on ${\frac{zf^{\prime}(z)}{f(z)}}$ for the certain classes of holomorphic functions in the unit disc $E=\{z:{\mid}z{\mid}<1\}$ and on ${\partial}E=\{z:{\mid}z{\mid}=1\}$. For the function $f(z)=z^2+c_3z^3+c_4z^4+{\cdots}$ defined in the unit disc E such that $f(z){\in}{\mathcal{A}}_{\alpha}$, we estimate a modulus of the angular derivative of ${\frac{zf^{\prime}(z)}{f(z)}}$ function at the boundary point b with ${\frac{bf^{\prime}(b)}{f(b)}}=1+{\alpha}$. Moreover, Schwarz lemma for class ${\mathcal{A}}_{\alpha}$ is given. The sharpness of these inequalities is also proved.

• Transactions on Control, Automation and Systems Engineering
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• 제1권2호
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• pp.94-98
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• 1999

Optical scanning systems use glavanometers to point the laser beam to the desired position on the workpiece. The angular speed of a galvanometer is typically controlled using Proportional+Integral+Derivative(PID) control algorithms. However, natural variations in the dynamics of different galvanometers due to manufacturing, aging, and environmental factors(i.e., process uncertainty) impose a hard limit on the bandwidth of the galvanometer control system. In general, the control bandwidth translates directly into efficiency of the system response. Since the optical scanning system must have rapid response, the higher control bandwidth is required. Auto-tuning PID algorithms have been accepted in this area since they could overcome some of the problems related to process uncertainty. However, when the galvanometer is attached to a larger mechanical system, the combined dynamics often exhibit resonances. It is well understood that PId algorithms may not have the capacity to increase the control bandwidth in the face of such resonances. This paper compares the achieable performance and robustness of a galvanometer control system using a PID controller tuned by the Ziegler-Nichols method and a controller designed by the Quantitative Feedback Theory(QFT) method. The results clearly indicate that-in contrast to PID designs-QFT can deliver a single, fixed controller which will supply high bandwidth design even when the dynamics is uncertain and includes mechanical resonances.