• Journal of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.931-940
• /
• 2007

Holomorphic mean Lipschitz space is defined in the unit ball of $\mathbb{C}^n$. The membership of the space is expressed in terms of the growth of radial derivatives, which reduced to a classical result of Hardy and Littlewood when n = 1. The membership is also expressed in terms of the growth of tangential derivatives when $n{\ge}2$.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.6
• /
• pp.1423-1438
• /
• 2022

In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish a related Schwarz type lemma for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce several versions of Schwarz and Liouville type theorems for almost holomorphic maps.

• 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.

• Korean Journal of Mathematics
• /
• v.32 no.1
• /
• pp.183-193
• /
• 2024

In this paper, a new class of bi-univalent functions that are described by Gegenbauer polynomials is presented. We obtain the estimates of the Taylor-Maclaurin coefficients |m₂| and |m₃| for each function in this class of bi-univalent functions. In addition, the Fekete-Szegö problems function new are also studied.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.325-334
• /
• 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 Korean Mathematical Society
• /
• v.50 no.1
• /
• pp.189-202
• /
• 2013

On the unit ball of $\mathbb{C}^n$, the space of those holomorphic functions satisfying the mean Lipschitz condition $${\int}_0^1\;{\omega}_p(t,f)^q\frac{dt}{t^1+{\alpha}q}\;<\;{\infty}$$ is characterized by integral growth conditions of the tangential derivatives as well as the radial derivatives, where ${\omega}_p(t,f)$ denotes the $L^p$ modulus of continuity defined in terms of the unitary transformations of $\mathbb{C}^n$.

• Journal of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.945-954
• /
• 2015

Let ${\lambda}_f(n)$ denote the n-th normalized Fourier coefficient of a primitive holomorphic form f for the full modular group ${\Gamma}=SL_2({\mathbb{Z}})$. In this paper, we are concerned with ${\Omega}$-result on the summatory function ${\sum}_{n{\leqslant}x}{\lambda}^2_f(n^2)$, and establish the following result ${\sum}_{\leqslant}{\lambda}^2_f(n^2)=c_1x+{\Omega}(x^{\frac{4}{9}})$, where $c_1$ is a suitable constant.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.1185-1196
• /
• 2016

We show a class of homogeneous polynomials of Fermat-Waring type such that for a polynomial P of this class, if $P(f_1,{\ldots},f_{N+1})=P(g_1,{\ldots},g_{N+1})$, where $f_1,{\ldots},f_{N+1}$; $g_1,{\ldots},g_{N+1}$ are two families of linearly independent entire functions, then $f_i=cg_i$, $i=1,2,{\ldots},N+1$, where c is a root of unity. As a consequence, we prove that if X is a hypersurface defined by a homogeneous polynomial in this class, then X is a unique range set for linearly non-degenerate non-Archimedean holomorphic curves.

• Bulletin of the Korean Mathematical Society
• /
• v.38 no.3
• /
• pp.487-493
• /
• 2001

$Let \Omega$ be a bounded convex domain in C^n$ with smooth boundary. Let M be a subvariety of $\Omega$ which intersects $\partial$$\Omega$ transversally. Suppose that $\Omega$ is totally convex at any point of $\partial$M in the complex tangential directions.For f $\epsilon$O(M)$\bigcap$/TEX>$C^{\infty}$($\overline{M}$/TEX>), there exists F $\epsilon$ o ($\Omega$))$\bigcap$/TEX>$C^{\infty}$($\overline{\Omega}$/TEX>) such that F(z) = f(z) for z $\epsilon$ M.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.567-578
• /
• 2017

Many scholars studied the boundedness of $Ces{\grave{a}}ro$ operators between $Q_K$ spaces and Bloch spaces of holomorphic functions in the unit disc in the complex plane, however, they did not describe the compactness. Let 0 < ${\alpha}$ < $+{\infty}$, K(r) be right continuous nondecreasing functions on (0, $+{\infty}$) and satisfy $${\displaystyle\smashmargin{2}{\int\nolimits_0}^{\frac{1}{e}}}K({\log}{\frac{1}{r}})rdr<+{\infty}$$. Suppose g is a holomorphic function in the unit disk. In this paper, some sufficient and necessary conditions for the extended $Ces{\grave{a}}ro$ operators $T_g$ between ${\alpha}$-Bloch spaces and $Q_K$ spaces in the unit disc to be bounded and compact are obtained.