• Korean Journal of Mathematics
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• 제31권3호
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• pp.243-258
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• 2023

A normalized analytic function f defined on the unit disk 𝔻 is starlike of reciprocal order α, 0 ≤ α < 1, if Re(f(z)/(zf'(z))) > α for all z ∈ 𝔻. Such functions are starlike and therefore univalent in 𝔻. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order α. Furthermore, a few conditions are derived for a function f to belong to a subclass of reciprocal starlike functions, satisfying |f(z)/(zf'(z)) - 1| < 1 - α.

• 호남수학학술지
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• 제24권1호
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• pp.67-73
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• 2002

In this paper we discuss on the existence of general solution of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z})+G(z,\;w,\;\bar{w})$ in the Sololev Space $W_{1,p}(D)$, that is generalization of a first order Non-linear Elliptic System of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z}).$

• Journal of the Korean Data and Information Science Society
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• 제10권1호
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• pp.243-250
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• 1999

We consider $Y=X{\lambda}Z,\;{\lambda}>0$, where X and Z are independent random variables, and Y is the length biased distribution or the equilibrium distribution of X. The purpose of this paper is to consider the distribution of X or Y when the distribution of Z is given and the distribution of Z when the distribution of X or Y is given, In particular, we obtain that the necessary and sufficient conditions for X to be $X^{2}({\upsilon})\;is\;Z{\sim}X^{2}(2)\;and\;for\;Z\;to\;be\;X^{2}(1)\;is\;X{\sim}IG({\mu},\;{\mu}^{2}/{\lambda})$, where $IG({\mu},\;{\mu}^{2}/{\lambda})$ is two-parameter inverse Gaussian distribution. Also we show that X is smaller than Y in the reverse Laplace transform ratio order if and only if $X_{e}$ is smaller than $Y_{e}$ in the Laplace transform ratio order. Finally, we can get the results that if X is smaller than Y in the Laplace transform ratio order, then $Y_{L}$ is smaller than $X_{L}$ in the Laplace transform order, and that if X is smaller than Y in the reverse Laplace transform ratio order, then $_{\mu}X_{L}$ is smaller than $_{\nu}Y_{L}$ in the Laplace transform order.

• 대한환경공학회지
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• 제31권11호
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• pp.941-946
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• 2009

본 연구는 상용 합성 제올라이트(Z-WK), 석탄계 비산재를 이용한 합성 제올라이트(Z-C1) 및 비산재(F-C1)를 이용하여 Lagergen 1차 및 2차 속도식과 Langmuir, Freundlich, Redlich-Peterson 및 Koble-Corrigan 식에 의해 코발트(Co)의 흡착속도와 평형흡착 특성을 평가하였다. 석탄계 비산재를 이용하여 Si/Al 비가 1.51인 제올라이트(Z-C1)를 제조하였으며, 흡착제 종류에 따른 Co 흡착량은 Z-C1(94.15mg/g) > F-C1(92.94mg/g) > Z-WK(88.56mg/g)순이었다. Z-WK와 Z-C1의 Co 흡착속도는 유사 2차 반응식으로서 정확한 예측이 가능하였다. Z-WK와 Z-C1의 Co 평형흡착량은 Langmuir 식과 Redlich-Peterson 식으로 잘 예측할 수 있었다. Z-C1은 수중의 Co 제거를 위한 유용한 흡착제로 사용이 가능할 것이다.

• 대한수학회논문집
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• 제36권4호
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• pp.743-757
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• 2021

We consider a different version of Schwarz Lemma for λ-spirallike function of complex order at the boundary of the unit disc D. We estimate the modulus of the angular derivative of the function $\frac{zf^{\prime}(z)}{f(z)}$ from below for λ-spirallike function f(z) of complex order at the boundary of the unit disc D by taking into account the zeros of the function f(z)-z which are different from zero. We also estimate the same function with the second derivatives of the function f at the points z = 0 and z = z₀ ≠ 0. We show the sharpness of these estimates and present examples.

• 대한수학회보
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• 제59권4호
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• pp.827-841
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• 2022

In this paper, we study the uniqueness of two finite order transcendental meromorphic solutions f(z) and g(z) of the following complex difference equation A₁(z)f(z + 1) + A₀(z)f(z) = F(z)e^𝛼(z) when they share 0, ∞ CM, where A₁(z), A₀(z), F(z) are non-zero polynomials, 𝛼(z) is a polynomial. Our result generalizes and complements some known results given recently by Cui and Chen, Li and Chen. Examples for the precision of our result are also supplied.

• 대한수학회논문집
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• 제37권2호
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• pp.445-454
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• 2022

Using the concept of the strong differential subordination introduced in [2], we find conditions on the functions θ, 𝜑, G, F such that the first order strong subordination θ(p(z)) + $\frac{G(\xi)}{\xi}$zp'(z)𝜑(p(z)) ≺≺ θ(q(z)) + F(z)q'(z)𝜑(q(z), implies p(z) ≺ q(z), where p(z), q(z) are analytic functions in the open unit disk 𝔻 with p(0) = q(0). Corollaries and examples of the main results are also considered, some of which extend and improve the results obtained in [1].

• 대한수학회보
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• 제51권6호
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• pp.1735-1748
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• 2014

In this paper, we investigate the finite order transcendental meromorphic solutions of complex difference equation of Malmquist type $$\prod_{i=1}^{n}f(z+c_i)=R(z,f)$$, where $c_1,{\ldots},c_n{\in}\mathbb{C}{\backslash}\{0\}$, and R(z, f) is an irreducible rational function in f(z) with meromorphic coefficients. We obtain some results on deficiencies of the solutions. Using these results, we prove that the growth order of the finite order solution f(z) is 1, if f(z) has Borel exceptional values $a({\in}\mathbb{C})$ and ${\infty}$. Moreover, we give the forms of f(z).

• 대한수학회보
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• 제49권5호
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• pp.911-921
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• 2012

In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

• 대한수학회보
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• 제58권6호
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• pp.1495-1506
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• 2021

The growth of solutions of second order complex differential equations f" + A(z)f' + B(z)f = 0 with transcendental entire coefficients is considered. Assuming that A(z) has a finite deficient value and that B(z) has either Fabry gaps or a multiply connected Fatou component, it follows that all solutions are of infinite order of growth.