• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.727-749
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• 2015

We obtain the conditions on ${\beta}$ so that $1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$ implies p(z) ${\prec}$ (2+z)/(2-z), $1+(1-{\alpha})z$, $(1+(1-2{\alpha})z)/(1-z)$, ($0{\leq}{\alpha}$<1), exp(z) or ${\sqrt{1+z}}$. Similar results are obtained by considering the expressions $1+{\beta}zp^{\prime}(z)/p(z)$, $1+{\beta}zp^{\prime}(z)/p^2(z)$ and $p(z)+{\beta}zp^{\prime}(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition ${\mid}log(zf^{\prime}(z)/f(z)){\mid}$ < 1 or ${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.479-487
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• 2009

Let $S(z_1,z_2),\;S_1(z_1,z_2)$ and $S_2(z_1,z_2)$ be power series with operator coefficients such that $S_(z_1,\;z_2)=S_1(z_1,z_2)S_2(z_1,z_2)$. Assume that the multiplications by $S_1(z_1,z_2)$ and $S_2(z_1,z_2)$ are contractive transformations in H($\mathbb{D}^2,\;\mathcal{C}$). Then the factorizations of spaces $\mathcal{D}(\mathbb{D},\;\tilde{S})$ and $\mathcal{D}(\mathbb{D}^2,\mathcal{S})$ are well-behaved.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.915-937
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• 2021

In this paper we deal with the open problem posed by Lü, Li and Yang [10]. In fact, we prove the following result: Let f(z) be a transcendental meromorphic function of finite order having finitely many poles, c₁, c₂, …, c_n ∈ ℂ\{0} and k, n ∈ ℕ. Suppose fⁿ(z), f(z+c₁)f(z+c₂) ⋯ f(z+c_n) share 0 CM and fⁿ(z)-Q₁(z), (f(z+c₁)f(z+c₂) ⋯ f(z+c_n))^(k) - Q₂(z) share (0, 1), where Q₁(z) and Q₂(z) are non-zero polynomials. If n ≥ k+1, then $(f(z+c_1)f(z+c_2)\;{\cdots}\;f(z+c_n))^{(k)}\;{\equiv}\;{\frac{Q_2(z)}{Q_1(z)}}f^n(z)$. Furthermore, if Q₁(z) ≡ Q₂(z), then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where c, λ ∈ ℂ \ {0} such that e^{λ(c₁+c₂+⋯+c_n)} = 1 and λ^k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.15 no.1
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• pp.449-459
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• 2014

The proposed 2+1 view integrated metamodel defined formerly with a graphical class model can not be guaranteed the syntactic clarity and accuracy precisely for the metamodel due to the informal specification. This paper specifies the syntactic semantics formally for the 2+1 view integrated metamodel using Z and Object-Z and checks the accuracy of the metamodel with Z/Eves tool. The formal specification is expressed in Z and Object-Z schema separately for syntax and statics semantics of the 2+1 view integrated metamodel, which applying the converting rule between class model and Z/Object-Z. The accuracy of the Z specification for the metamodel is verified using Z/Eves tool, which can check the syntax, type, and domain of the Z specification. The transformation specification and checking of the 2+1 view integrated metamodel can help establish more accurate the syntactic semantics of its construct and check the accuracy of the metamodel.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.185-196
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• 2024

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), b_i(z) for i = 0, 1, 2 and d_j (z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d₀(z) + 1, or, deg a(z) = deg d₀(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.461-468
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• 2004

Let W(z) be a power series with operator coefficients such that multiplication by W(z) is contractive in C(z). The overlapping space $L(\varphi)$ of H(W) in C(z) is a Herglotz space with Herglotz function $\varphi(z)$ which satisfies $\varphi(z)+\varphi^＊(z^{-1})=2[1-W^{*}(z^{-1})W(z)]$. The identity ${}_{L(\varphi)}={-}_{H(W)}$ holds for every f(z) in $L(\varphi)$ and for every vector c.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.221-231
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• 1995

Let $Z_1, Z_2, \ldots, Z_l$ be random set functions or intergrals. Then it is possible to discuss their products. In the case of random integrals, $Z_i$ is a random set function indexed y a family, $G_i$ say, of real valued functions g on $S_i$ for which the integrals $Z_i(g) = \smallint gdZ_i$ are well defined. If $g_i = \in g_i (i = 1, 2, \ldots, l) and g_1 \otimes \cdots \otimes g_l$ denotes the tensor product $g(s) = g_1(s_1)g_2(s_2) \cdots g_l(s_l) for s = (s_1, s_2, \ldots, s_l) and s_i \in S_i$, then we can defined $Z(g) = (Z_1 \times Z_2 \times \cdots \times Z_l)(g) = Z_1(g_1)Z_2(g_2) \cdots Z_l(g_l)$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.839-850
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• 2020

For functions f(z) = z + a₂z² + a₃z³ + ⋯ belonging to particular classes, this paper finds sharp bounds for the initial coefficients a₂, a₃, a₄, as well as the sharp estimate for the second order Hankel determinant H₂(2) = a₂a₄ - a₂³. Two classes are treated: first is the class consisting of f(z) = z + a₂z² + a₃z³ + ⋯ in the unit disk 𝔻 satisfying $$\|\(\frac{z}{f(z)}\)^{1+{\alpha}}\;f^{\prime}(z)-1\|<{\lambda},\;0<{\alpha}<1,\;0<{\lambda}{\leq}1.$$ The second class consists of Bazilevič functions f(z) = z+a₂z²+a₃z³+⋯ in 𝔻 satisfying $$Re\{\(\frac{f(z)}{z}\)^{{\alpha}-1}\;f^{\prime}(z)\}>0,\;{\alpha}>0.$$

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.325-334
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• 2016

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$ holomorphic in the unit disc and $\|\frac{f(z)}{{\lambda}f(z)+(1-{\lambda})z}-{\alpha}\|$ < ${\alpha}$ for ${\mid}z{\mid}$ < 1, where $\frac{1}{2}$ < ${\alpha}$ ${\leq}{\frac{1}{1+{\lambda}}}$, $0{\leq}{\lambda}$ < 1. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $c_{p+1}$, $c_{p+2}$ and zeros of f(z) - z. We obtain a sharp lower bound of ${\mid}f^{\prime}(b){\mid}$ at the point b, where ${\mid}b{\mid}=1$.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.2
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• pp.87-98
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• 2001

In this paper, we study hash function, which will take a message of arbitrary length and produce a massage digest of a specified size. The message digest will then be signed. We have to be careful that the use of a hash function h does not weaken the security of the signature scheme, for it is the message digest that is signed, not the message. It will be necessary for h to satisfy certain properties in order to prevent various forgeries. In order to prevent various type of attack, we require that hash function satisfy collision-free property. In section 1, we introduce some definitions and collision-free properties of hash function. In section 2, we study a discrete log hash function and introduce the main theorem as follows : Theorem Suppose $h:X{\rightarrow}Z$ is a hash function. For any $z{\in}Z$, let $$h^{-1}(z)={\lbrace}x:h(x)=z{\rbrace}$$ and denote $s_z={\mid}h^{-1}(z){\mid}$. Define $$N={\mid}{\lbrace}{\lbrace}x_1,x_2{\rbrace}:h(x_1)=h(x_2){\rbrace}{\mid}$$. Then (1) $\sum\limits_{z{\in}Z}s_z={\mid}x{\mid}$ and the mean of the $s_z$'s is $\bar{s}=\frac{{\mid}X{\mid}}{{\mid}Z{\mid}}$ (2) $N=\sum\limits_{z{\in}Z}{\small{s_z}}C_2=\frac{1}{2}\sum\limits_{z{\in}Z}S_z{^2}-\frac{{\mid}X{\mid}}{2}$. (2) $\sum\limits_{z{\in}Z}(S_z-\bar{s})^2=2N+{\mid}X{\mid}-\frac{{\mid}X{\mid}^2}{{\mid}Z{\mid}}$.