• Communications for Statistical Applications and Methods
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• v.8 no.1
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• pp.229-235
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• 2001

Let {X$\_$n/, n$\geq$1] be a stationary sequence of negatively associated random variables with distribution function F(x)=P(X$_1$$\leq$x). The empirical distribution function F$\_$n/(x) based on X$_1$, X$_2$,....., X$\_$n/ is proposed as an estimator for F$\_$n/(x). Strong consistency and asymptotic normality of F$\_$n/(x) are studied. We also apply these ideas to estimation of the survival function.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.235-243
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• 2009

In this paper, we deal with the problem of uniqueness of meromorphic functions that share two small functions with their derivatives, and obtain the following result which improves a result of Yao and Li: Let f(z) be a nonconstant meromorphic function, k > 5 be an integer. If f(z) and g(z) = $a_1(z)f(z)+a_2(z)f^{(k)}(z)$ share the value 0 CM, and share b(z) IM, $\overline{N}_E(r,f=0=F^{(k)})=S(r)$, f${\equiv}$g, where $a_1(z)$, $a_2(z)$ and b(z) are small functions of f(z).

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.29-38
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• 2016

For a transcendental entire function f(z) with zero order, the purpose of this article is to study the value distributions of q-difference polynomial $f(qz)-a(f(z))^n$ and $f(q_1z)f(q_2z){\cdots}f(q_mz)-a(f(z))^n$. The property of entire solution of a certain q-difference equation is also considered.

• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.31-34
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• 1995

Let R$^n$be n-th Euclidean space. Let be the n-th spere embeded as a subspace in R$\^$n+1/ centered at the origin. In this paper, we are going to consider the function space F = {f│f : S$^n$\longrightarrow S$^n$} metrized by as follow D(f,g)=d(f($\chi$), g($\chi$)) where f, g $\in$ F and d is the metric in S$^n$. Finally we want to find certain relation these spaces.(omitted)

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.543-550
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• 2007

In this paper, we define the ap-Denjoy integral and investigate some properties od the ap-Denjoy integral. In particular, we show that a function f : [a,b]${\rightarrow}\mathbb{R}$ is ap-Denjoy integrable on [a,b] if and only if there exists an $ACG_s$ function F on [a,b] such that $F^{\prime}_{ap}=f$ almost everywhere on [a,b].

• East Asian mathematical journal
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• v.17 no.2
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• pp.233-237
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• 2001

The authors aim at presenting summation formulas of Appell's function $F_1$: $$F_1(a;b,b';1+a+b-b'+i;1,-1)\;(i=0,\;{\pm}1,\;{\pm}2,\;{\pm}3,\;{\pm}4,\;{\pm}5)$$, which, for i=0, yields a known result due to Srivastava.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.105-120
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• 2021

In this paper, we extend some results related to the growth rates of entire functions by introducing the relative logarithmic order ����g(f) of a nonconstant entire function f with respect to another nonconstant entire function g. Next we investigate some theorems related the behavior of ����g(f). We also define the relative logarithmic proximate order of f with respect to g and give some theorems on it.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.243-258
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• 2023

A normalized analytic function f defined on the unit disk 𝔻 is starlike of reciprocal order α, 0 ≤ α < 1, if Re(f(z)/(zf'(z))) > α for all z ∈ 𝔻. Such functions are starlike and therefore univalent in 𝔻. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order α. Furthermore, a few conditions are derived for a function f to belong to a subclass of reciprocal starlike functions, satisfying |f(z)/(zf'(z)) - 1| < 1 - α.

• Journal of Korean Society for Quality Management
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• v.29 no.2
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• pp.37-45
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• 2001

A survival variable is a non-negative random variable X with distribution function F(t) satisfying F(0) : 0 and a survival function F(t): 1-F(t). This variable is said to be New Better than Used of specified age t$_{0}$ if F(x+ t$_{0}$)$\leq$F(x).F(t$_{0}$) for all x$\geq$0 and a fixed t$_{0}$. We propose the test for H$_{0}$ : F(x+t$_{0}$)=F(x).F(t$_{0}$) for all x$\geq$0 against H$_1$: F(x+t$_{0}$) $\leq$ F(x).F(t$_{0}$) for all x$\geq$0 when the specified age to is unknown but can be estimated from the data when t$_{0}$=ζ$_{p}$, the pth percentile of F. This test statistic, which is based on the normalized spacings between the ordered observations, is readily applied in the case of small sample. Also, our test is more simple than Ahmad's test (1998). Finally, the performance of our test is presented.our test is presented.