• The Pure and Applied Mathematics
• /
• v.26 no.2
• /
• pp.59-73
• /
• 2019

In this paper, a version of the boundary Schwarz Lemma for the holomorphic function belonging to $\mathcal{N}$(${\alpha}$) is investigated. For the function $f(z)=z+c_2z^2+C_3z^3+{\cdots}$ which is defined in the unit disc where $f(z){\in}\mathcal{N}({\alpha})$, we estimate the modulus of the angular derivative of the function f(z) at the boundary point b with $f(b)={\frac{1}{b}}\int\limits_0^b$ f(t)dt. The sharpness of these inequalities is also proved.

• Kyungpook Mathematical Journal
• /
• v.55 no.2
• /
• pp.449-472
• /
• 2015

Let X be a nonempty set and $\mathcal{F}$(X) be the set of nonempty finite subsets of X. The paper deals with the extended metrics ${\tau}:\mathcal{F}(X){\rightarrow}\mathbb{R}$ recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets A with ${\mid}A{\mid}{\leqslant}n$ make possible to consider "distance functions" with n variables and related them quantities. In particular, we study such type generalized diameters $diam_{{\tau}^n}$ and find conditions under which $B{\mapsto}diam_{{\tau}^n}B$ is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction ${\tau}$ to the set of $A{\in}\mathcal{F}(X)$ with ${\mid}A{\mid}{\leqslant}3$ is a symmetric G-metric. An infinitesimal analog for extended by Balk metrics is constructed.

• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.781-791
• /
• 2016

Let $d_*(1,w)^2 ={\mathbb{R}}^2$ with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on $d_*(1,w)^2$. We also show that the unit sphere of the space of symmetric bilinear forms on $d_*(1,w)^2$ is the disjoint union of the sets of smooth points, extreme points and the set A as follows: $$S_{{\mathcal{L}}_s(^2d_*(1,w)^2)}=smB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}extB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}A$$, where the set A consists of $ax_1x_2+by_1y_2+c(x_1y_2+x_2y_1)$ with (a = b = 0, $c={\pm}{\frac{1}{1+w^2}}$), ($a{\neq}b$, $ab{\geq}0$, c = 0), (a = b, 0 < ac, 0 < ${\mid}c{\mid}$ < ${\mid}a{\mid}$), ($a{\neq}{\mid}c{\mid}$, a = -b, 0 < ac, 0 < ${\mid}c{\mid}$), ($a={\frac{1-w}{1+w}}$, b = 0, $c={\frac{1}{1+w}}$), ($a={\frac{1+w+w(w^2-3)c}{1+w^2}}$, $b={\frac{w-1+(1-3w^2)c}{w(1+w^2)}}$, ${\frac{1}{2+2w}}$ < c < ${\frac{1}{(1+w)^2(1-w)}}$, $c{\neq}{\frac{1}{1+2w-w^2}}$), ($a={\frac{1+w(1+w)c}{1+w}}$, $b={\frac{-1+(1+w)c}{w(1+w)}}$, 0 < c < $\frac{1}{2+2w}$) or ($a={\frac{1=w(1+w)c}{1+w}}$, $b={\frac{1-(1+w)c}{1+w}}$, $\frac{1}{1+w}$ < c < $\frac{1}{(1+w)^2(1-w)}$).

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.593-605
• /
• 2017

In this paper, we consider some normality criteria concerning multiple values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain D. Let k be a positive integer and ${\psi}(z){\not\equiv}0$, ${\infty}$ be a meromorphic function in D. If, for each $f{\in}\mathcal{F}$ and $z{\in}D$, (1) $f(z){\neq}0$, and all of whose poles are multiple; (2) all zeros of $f^{(k)}(z)-{\psi}(z)$ have multiplicities at least k + 3 in D; (3) all poles of ${\psi}(z)$ have multiplicities at most k in D, then $\mathcal{F}$ is normal in D.

• Communications of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.543-555
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1389-1413
• /
• 2013

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.

• The Pure and Applied Mathematics
• /
• v.23 no.4
• /
• pp.339-345
• /
• 2016

The set of priors in the representation of Choquet expectation is expressed as the one of equivalent martingale measures under some conditions. We show that the set of priors, $\mathcal{Q}_c$ in (1.1) is the same set of $\mathcal{Q}^{\theta}$ in (1.3).

• Journal of applied mathematics & informatics
• /
• v.30 no.3_4
• /
• pp.623-633
• /
• 2012

Let {${\Omega}$, $\mathcal{F}$, P} be a probability space and {$X_n|n{\geq}1$} be a sequence of random variables defined on it. A finite sequence of random variables {$X_n|n{\geq}1$} is said to be conditionally negatively associated given $\mathcal{F}$ if for every pair of disjoint subsets A and B of {1, 2, ${\cdots}$, n}, $Cov^{\mathcal{F}}(f_1(X_i,i{\in}A),\;f_2(X_j,j{\in}B)){\leq}0$ a.s. whenever $f_1$ and $f_2$ are coordinatewise nondecreasing functions. We extend the H$\grave{a}$jek-R$\grave{e}$nyi-type inequality from negative association to conditional negative association of random variables. In addition, some corollaries are given.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.1
• /
• pp.165-173
• /
• 2018

In this paper we obtain the bounds of the third Hankel determinants for the classes $\mathcal{S}^*({\alpha})$ of starlike functions of order ${\alpha}$ and $\mathcal{K}({\alpha}$) of convex functions of order ${\alpha}$. Moreover,we derive the sharp bounds for functions in these classes which are additionally 2-fold or 3-fold symmetric.

• Journal of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.493-502
• /
• 2012

In this article, we give a characterization theorem for a class of corank-1 Butler groups of the form $\mathcal{G}$($A_1$, ${\ldots}$, $A_n$). In particular, two groups $G$ and $H$ are quasi-isomorphic if and only if there is a label-preserving bijection ${\phi}$ from the edges of $T$ to the edges of $U$ such that $S$ is a circuit in T if and only if ${\phi}(S)$ is a circuit in $U$, where $T$, $U$ are quasi-representing graphs for $G$, $H$ respectively.