Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2023-03-31
Hankel determinants for starlike functions with respect to symmetrical points
Nak Eun Cho, Young Jae Sim, Derek K. ThomasAbstract : We prove sharp bounds for Hankel determinants for starlike functions $f$ with respect to symmetrical points, i.e., $f$ given by $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ for $z\in \mathbb{D}$ satisfying $$ Re\dfrac{zf'(z)}{f(z)-f(-z)}>0, \quad z\in \mathbb{D}. $$ We also give sharp upper and lower bounds when the coefficients of $f$ are real.
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2023-05-31
Some results on meromorphic solutions of q-difference differential equations
Lingyun Gao, Zhenguang Gao, Manli LiuAbstract : In view of Nevanlinna theory, we investigate the meromorphic solutions of q-difference differential equations and our results give the estimates about counting function and proximity function of meromorphic solutions to these equations. In addition, some interesting results are obtained for two general equations and a class of system of q-difference differential equations.
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2023-05-31
There are no numerical radius peak $n$-linear mappings on $c_0$
Sung Guen KimAbstract : For $n\geq 2$ and a real Banach space $E$, ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself. Let $$\Pi(E)=\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}~\}.$$ An element $[x^*, (x_1, \ldots, x_n)]\in \Pi(E)$ is called a {\em numerical radius point} of $T\in {\mathcal L}(^n E:E)$ if $|x^{*}(T(x_1, \ldots, x_n))|=v(T)$, where the numerical radius $v(T)=\sup_{[y^*, y_1, \ldots, y_n]\in \Pi(E)}\Big|y^{*}\Big(T(y_1, \ldots,y_n)\Big)\Big|$. For $T\in {\mathcal L}(^n E:E)$, we define \begin{align*} {Nradius}({T})=&\ \{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): [x^*, (x_1, \ldots, x_n)]\\ &\quad \mbox{is a numerical radius point of}~T\}. \end{align*} $T$ is called a {\em numerical radius peak $n$-linear mapping} if there is a unique $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that ${Nradius}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}$. In this paper we present explicit formulae for the numerical radius of $T$ for every $T\in {\mathcal L}(^n E:E)$ for $E=c_0$ or $l_{\infty}$. Using these formulae we show that there are no numerical radius peak mappings of ${\mathcal L}(^n c_0:c_0)$.
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2023-03-31
Some integral inequalities for the Laplacian with density on weighted manifolds with boundary
Fanqi ZengAbstract : In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincar\'{e}-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.
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2023-03-31
Some estimates for generalized commutators of multilinear Calder\'on-Zygmund operators
Honghai Liu, Zengyan Si, Ling WangAbstract : Let $T$ be an $m$-linear Calder\'on-Zygmund operator. $T_{\vec{b},S}$ is the generalized commutator of $T$ with a class of measurable functions $\{b_{i}\}_{i=1}^\infty$. In this paper, we will give some new estimates for $T_{\vec{b},S}$ when $\{b_{i}\}_{i=1}^\infty$ belongs to Orlicz-type space and Lipschitz space, respectively.
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2023-01-31
Representations by quaternary quadratic forms with coefficients $1$, $2$, $11$ and $22$
B\"{u}lent K\"{o}kl\"{u}ce
Abstract : In this article, we find bases for the spaces of modular forms $M_{2}(\Gamma _{0}(88),\big( \frac{d}{\cdot }\big) )$ for $d=1,8,44\text{ and }88$. We then derive formulas for the number of representations of a positive integer by the diagonal quaternary quadratic forms with coefficients $1,2,11$ and $ 22 $.
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2022-11-30
Boundedness and continuity for variation operators on the Triebel--Lizorkin spaces
Feng Liu, Yongming Wen, Xiao ZhangAbstract : In this paper, we establish the boundedness and continuity for variation operators for $\theta$-type Calder\'{o}n--Zygmund singular integrals and their commutators on the Triebel--Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.
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2023-03-31
Hyperbolic and spherical power of a circle
Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Heayong Shin, Seong-Deog YangAbstract : Suppose that a line passing through a given point $P$ intersects a given circle $\mathcal{C}$ at $Q$ and $R$ in the Euclidean plane. It is well known that $|PQ||PR|$ is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.
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2023-03-31
A note on comparison principle for elliptic obstacle problems with $L^{1}$-data
Kyeong Song, Yeonghun YounAbstract : In this note, we study a comparison principle for elliptic obstacle problems of $p$-Laplacian type with $L^1$-data. As a consequence, we improve some known regularity results for obstacle problems with zero Dirichlet boundary conditions.
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2023-01-31
Stability and topology of translating solitons for the mean curvature flow with the small $L^m$ norm of the second fundamental form
Eungmo Nam, Juncheol PyoAbstract : In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.
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B\'ezout rings and weakly B\'ezout rings
Haitham El~Alaoui
Bull. Korean Math. Soc. 2022; 59(4): 843-852
https://doi.org/10.4134/BKMS.b210492 -
A sharp integral inequality for compact linear Weingarten hypersurfaces
Henrique F. de~Lima
, F\'{a}bio R. dos~Santos, Lucas S. RochaBull. Korean Math. Soc. 2022; 59(3): 789-799
https://doi.org/10.4134/BKMS.b210479 -
Application of Rothe's method to a nonlinear wave equation on graphs
Yong Lin, Yuanyuan Xie
Bull. Korean Math. Soc. 2022; 59(3): 745-756
https://doi.org/10.4134/BKMS.b210445 -
Bisectors in the Heisenberg group I
Gaoshun Gou, Yueping Jiang, Ioannis D. Platis
Bull. Korean Math. Soc. 2023; 60(1): 225-235
https://doi.org/10.4134/BKMS.b220059
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Estimate for bilinear Calder\'{o}n-Zygmund operator and its commutator on product of variable exponent spaces
Guanghui Lu, Shuangping Tao
Bull. Korean Math. Soc. 2022; 59(6): 1471-1493
https://doi.org/10.4134/BKMS.b210839 -
Representations over Green algebras of weak Hopf algebras based on Taft algebras
Liufeng Cao
Bull. Korean Math. Soc. 2023; 60(6): 1687-1695
https://doi.org/10.4134/BKMS.b220845 -
Certain properties of the class of univalent functions with real coefficients
Milutin Obradovic, Nikola Tuneski
Bull. Korean Math. Soc. 2023; 60(5): 1253-1263
https://doi.org/10.4134/BKMS.b220643 -
Differences of weighted composition operators on Bergman spaces induced by doubling weights
Jiale Chen
Bull. Korean Math. Soc. 2023; 60(5): 1201-1219
https://doi.org/10.4134/BKMS.b220578
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