• Honam Mathematical Journal
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• v.41 no.2
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• pp.381-390
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• 2019

In this paper, we introduce interesting subclasses ${\mathcal{H}}^{p,q,{\beta},{\alpha}}_{{\sigma}B}$ and ${\mathcal{H}}^{p,q,{\beta}}_{{\sigma}B}({\gamma})$ of bi-univalent functions by (p, q)-derivative operator. Furthermore, we find upper bounds for the second and third coefficients for functions in these subclasses. The results presented in this paper would generalize and improve some recent works of several earlier authors.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.631-638
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• 2021

In this paper, some sharp inequalities for ordinary derivative P'(z) and polar derivative D_αP(z) = nP(z) + (α - z)P'(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z| > k, k ≥ 1. Our results also improve the bounds of Turán's and Aziz's inequalities.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.503-520
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• 2022

We introduce a new type of fractional derivative, which we call as the right local general truncated M-fractional derivative for α-differentiable functions that generalizes the fractional derivative type introduced by Anastassiou. This newly defined operator generalizes the standard properties and results of the integer order calculus viz. the Rolle's theorem, the mean value theorem and its extension, inverse property, the fundamental theorem of calculus and the theorem of integration by parts. Then we represent a relation of the newly defined fractional derivative with known fractional derivative and in context with this derivative a physical problem, Kirchoff's voltage law, is generalized. Also, the importance of this newly defined operator with respect to the flexibility in the parametric values is described via the comparison of the solutions in the graphs using MATLAB software.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.323-329
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• 2022

Let $p(z)=\sum\limits_{\nu=0}^{n}a_{\nu}z^{\nu}$ be a polynomial of degree n and $p^{\prime}(z)$ its derivative. If $\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}$ is denoted by M(p, r). If p(z) has all its zeros on |z| = k, k ≤ 1, then it was shown by Govil [3] that $$M(p^{\prime},\;1){\leq}\frac{n}{k^n+k^{n-1}}M(p,\;1)$$. In this paper, we first prove a result concerning the sth derivative where 1 ≤ s < n of the polynomial involving some of the co-efficients of the polynomial. Our result not only improves and generalizes the above inequality, but also gives a generalization to higher derivative of a result due to Dewan and Mir [2] in this direction. Further, a direct generalization of the above inequality for the sth derivative where 1 ≤ s < n is also proved.

• The Pure and Applied Mathematics
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• v.12 no.2 s.28
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• pp.125-131
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• 2005

In this paper, we introduce the concept of derivative of the function f : $\mathbb{Q}p{\to} R$ where $\mathbb{Q}p$ is the field of the p-adic numbers and R is the set of real numbers. And some basic theorems on derivatives are given.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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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 ∈ P_n, 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.

• The Journal of Korean Medicine
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• v.19 no.1
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• pp.145-164
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• 1998

The purpose of this study is to evaluate the effect of Injinchunggantang-derivative on proliferation of hepatocyte in rats. Cell viability is studied by MTI assay. The gene related to cell replication such as p53, waf1, bcl-2 and $bcl-_{X_L}$ is quantitized by quantitative RT-PCR and the proteins coded by these genes are studied by Western blotting. The results are as follows. 1. The hepatocytes cultured in medium with lnjinchunggantang-derivative showed better viability compared with control grroup in MTI assay, and the hepatocytes cultured in medium with the Injinchunggantang-derivative-and-ethanol-mixed group showed better viability than the hepatocytes cultrued in 10% ethanol culture medium(control group), noting that Injinchunggantang-derivative has protective effect on hepatocyte injury. There was no dose- and time-dependence. 2. In quantitative RT-PCR, i) Bel-2 gene increased significantly both in Injinchunggantang-derivative group and in Injinchunggantang-derivative-and-ethanol-mixed group, while it showed no significant increase or decrease in other group. ii) $Bcl-_{X_L}$ gene increased significantly in Injinchunggantang-derivative group as well as in Injinchunggantang-deri vative-and-ethanol -mixed group. iii) P53 gene showed no significant increase or decrease in hepatocytes cultured in medium with 10% ethanol and in hepatocytes cultured in medium with Injinchunggantang-derivative-and-ethanol-mixed group, suggesting that 10% ethanol induced cell toxicity, thus increased p53 gene expression. iv) Wafl gene showed no significant increase or decrease in hepatocytes cutured in medium with Injinchtrnggantang-derivative, while increased in hepatocytes cultured in medium with 10% ethanol and in hepatocytes cultured in medium with Injinchtrnggantang-derivative-andethanol-mixed group, suggesting that 10% ethanol induced cell toxicity increased wafl gene expression. 3. In the study on protein by western blotting, the band of bcl-2 and $bcl-_{X_L}$ were widened in Injinchtrnggantang-derivative group. Especially the amount of $bcl-_{X_L}$ increased significantly compared with other groups. But in the study on p53 and wafl, there was no significant difference among those groups. Above study shows that Injinchunggantang-derivative has good effect on cell viability and that the genes resistant to cell death such as bcl-2 and $bcl-_{X_L}$ are induced by Injinchunggantang-derivative to resist to cell death by toxic agent And this is reconfirmed in protein study using' western blotting: These results suggest that Injinchunggantang-derivative has inhibitory effect on cell death as well as protective effect on hepatocyte. Therefore this prescription is recommended in various liver diseases such as chronic liver disease and-induced hepatic injury.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.141-148
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• 2022

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≤ 1, then Govil proved $$\max_{{\mid}z{\mid}=1}{\mid}p^{\prime}(z){\mid}{\leq}{\frac{n}{1+k^n}}\max_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$, provided |p'(z)| and |q'(z)| attain their maximal at the same point on the circle |z| = 1, where $$q(z)=z^n{\overline{p(\frac{1}{\overline{z}})}}$$. In this paper, we extend the above inequality to polar derivative of a polynomial. Further, we also prove an improved version of above inequality into polar derivative.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.224-261
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• 2021

The interest in implicit-explicit (IMEX) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicit-explicit second derivative linear multi-step methods (IMEX SDLMM) with error control. The proposed IMEX SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The IMEX second derivative schemes are constructed with order p ranging from p = 1 to 8. The methods are numerically validated on well-known stiff equations.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.183-194
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• 1998

Making use of a certain operator of fractional derivative, a new subclass $L_p({\alpha},{\beta},{\gamma},{\lambda})$) of analytic and p-valent functions is introduced in the present paper. Apart from various coefficient bounds, many interesting and useful properties of this class of functions are given, some of these properties involve, for example, linear combinations and modified Hadamard product of several functions belonging to the class introduced here.