• The Pure and Applied Mathematics
• /
• v.14 no.1 s.35
• /
• pp.23-35
• /
• 2007

We investigate the modified Hyers-Ulam-Rassias stability for the following mixed type functional equation, i.e, cubic or quadratic type functional equation : 9f(x+y)-9f(x-y)+f(6y)=3f(x+3y)-3f(x-3y)+9f(2y).

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.3
• /
• pp.603-608
• /
• 2021

Fix k ≥ 3. If a multiplicative function f satisfies f(x₁ + x₂ + ⋯ + x_k) = f(x₁) + f(x₂) + ⋯ + f(x_k) for arbitrary positive triangular numbers x₁, x₂, …, x_k, then f is the identity function. This extends Chung and Phong's work for k = 2.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.1281-1289
• /
• 2014

In this paper, we consider the differential equation $$F^{\prime}-Q_1=Re^{\alpha}(F-Q_2)$$, where $Q_1$ and $Q_2$ are polynomials with $Q_1Q_2{\neq}0$, R is a rational function and ${\alpha}$ is an entire function. We consider solutions of the form $F=f^n$, where f is an entire function and $n{\geq}2$ is an integer, and we prove that if f is a transcendental entire function, then $\frac{Q_1}{Q_2}$ is a polynomial and $f^{\prime}=\frac{Q_1}{nQ_2}f$. This theorem improves some known results and answers an open question raised in [16].

• Communications of the Korean Mathematical Society
• /
• v.27 no.1
• /
• pp.107-116
• /
• 2012

Here, by using the symbolical method introduced by Burchnall and Chaundy, we aim at constructing certain expansion formulas for the generalized hypergeometric function $_pF_q$. In addition, using our expansion formulas for $_pF_q$, we present formulas of an analytic continuation of the Clausen hypergeometric function $_3F_2$, which are much simpler than an earlier known result. We also give some integral representations for $_3F_2$.

• Communications of the Korean Mathematical Society
• /
• v.36 no.4
• /
• pp.743-757
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.8 no.1
• /
• pp.111-117
• /
• 1995

A function $f:{\Omega}{\rightarrow}X$ is called intrinsically-separable valued if there exists $E{\in}{\Sigma}$ with ${\mu}(E)=0$ such that $f({\Omega}-E)$ is a separable in X. For a given Dunford integrable function $f:{\Omega}{\rightarrow}X$ and a weakly compact operator T, we show that if f is intrinsically-separable valued, then f is Pettis integrable, and if there exists a sequence ($f_n$) of Dunford integrable and intrinsically-separable valued functions from ${\Omega}$ into X such that for each $x^*{\in}X^*$, $x^*f_n{\rightarrow}x^*f$ a.e., then f is Pettis integrable. We show that a function f is Pettis integrable if and only if for each $E{\in}{\Sigma}$, F(E) is $weak^*$-continuous on $B_{X*}$ if and only if for each $E{\in}{\Sigma}$, $M=\{x^*{\in}X^*:F(E)(x^*)=O\}$ is $weak^*$-closed.

• Journal of applied mathematics & informatics
• /
• v.32 no.3_4
• /
• pp.331-341
• /
• 2014

In this article we introduce the sequence spaces $_2c^I_0(f,p)$, $_2c^I(f,p)$ and $_2l^I_{\infty}(f,p)$ for a modulus function f, where $p=(p_k)$ is a sequence of positive reals and study some of the properties of these spaces.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.4
• /
• pp.699-704
• /
• 2010

In this paper, we prove that the Hilbert 2-class field tower of an imaginary quadratic function field $F=k({\sqrt{D})$ is infinite if $r_2({\mathcal{C}}(F))=4$ and exactly one monic irreducible divisor of D is of odd degree, except for one type of $R{\acute{e}}dei$ matrix of F. We also compute the density of such imaginary quadratic function fields F.

• Korean Journal of Mathematics
• /
• v.31 no.4
• /
• pp.433-444
• /
• 2023

Given a function f analytic in open disk centred at origin of radius unity and satisfying the condition |f(z)/g(z) - 1| < 1 for a analytic function g with certain prescribed conditions in the unit disk, radii constants R are determined for the values of Rzf'(Rz)/f(Rz) to lie inside the domain enclosed by the curve |w² - 1| = 1 (lemniscate of Bernoulli). This, in turn, provides a partial solution to the conjectures and problems for determination of sharp bounds R for such functions f.

• Communications of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.115-129
• /
• 2016

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. [9] evaluated the fractional integrals of the product of Appell function of the third kernel $F_3$ and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function $F_3$ and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, $Erd\acute{e}lyi$-Kober and Saigo operators, associated with multivariable H-function and Appell function $F_3$ are shown to be deduced as special cases of our findings.