• Title, Summary, Keyword: partial sums

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Partial Sums of Starlike Harmonic Univalent Functions

  • Porwal, Saurabh;Dixit, Kaushal Kishore
    • Kyungpook Mathematical Journal
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    • v.50 no.3
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    • pp.433-445
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    • 2010
  • Although, interesting properties on the partial sums of analytic univalent functions have been investigated extensively by several researchers, yet analogous results on partial sums of harmonic univalent functions have not been so far explored. The main purpose of the present paper is to establish some new and interesting results on the ratio of starlike harmonic univalent function to its sequences of partial sums.

DUI DUO SHU in LEE SANG HYUK's IKSAN and DOUBLE SEQUENCES of PARTIAL SUMS (이상혁(李尙爀)(익산(翼算))의 퇴타술과 부분합 복수열)

  • Han, Yong-Hyeon
    • Journal for History of Mathematics
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    • v.20 no.3
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    • pp.1-16
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    • 2007
  • In order to generalize theory of series in Iksan(翼算), we introduce a concept of double sequence of partial sums and elementary double sequence of partial sums, which play a dominant role in the study of double sequences of partial sums. We introduce a concept of finitely generated double sequence of partial sums and find a necessary and sufficient condition for those double sequences. Finally we prove a multiplication theorem for tetrahedral numbers and for 4 dimensional tetrahedral numbers.

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NORM CONVERGENT PARTIAL SUMS OF TAYLOR SERIES

  • YANG, JONGHO
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1729-1735
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    • 2015
  • It is known that the partial sum of the Taylor series of an holomorphic function of one complex variable converges in norm on $H^p(\mathbb{D})$ for 1 < p < ${\infty}$. In this paper, we consider various type of partial sums of a holomorphic function of several variables which also converge in norm on $H^p(\mathbb{B}_n)$ for 1 < p < ${\infty}$. For the partial sums in several variable cases, some variables could be chosen slowly (fastly) relative to other variables. We prove that in any cases the partial sum converges to the original function, regardlessly how slowly (fastly) some variables are taken.

ON SOME SOLUTIONS OF A FUNCTIONAL EQUATION RELATED TO THE PARTIAL SUMS OF THE RIEMANN ZETA FUNCTION

  • Martinez, Juan Matias Sepulcre
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.29-41
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    • 2014
  • In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)+${\cdots}$+f(nx) = 0, with $n{\geq}2$, which are related to the partial sums of the Riemann zeta function. These curves ${\alpha}$-densify a large class of compact sets of the plane for arbitrary small ${\alpha}$, extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the $n^{th}$ power of the density approaches the Jordan content of the compact set which the curve densifies.

THE WEAK LAW OF LARGE NUMBERS FOR RANDOMLY WEIGHTED PARTIAL SUMS

  • Kim, Tae-Sung;Choi, Kyu-Hyuck;Lee, Il-Hyun
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.273-285
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    • 1999
  • In this paper we establish the weak law of large numbers for randomly weighted partial sums of random variables and study conditions imposed on the triangular array of random weights {$W_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}n{\geq}1$} and on the triangular array of random variables {$X_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}{\geq}1$} which ensure that $\sum_{j=1}^{n}{\;}W_{nj}{\mid}X_{nj}{\;}-{\;}B_{nj}{\mid}$ converges In probability to 0, where {$B_{nj}{\;}:{\;}1{\;}{\leq}{\;}j{\;}{\leq}{\;}n,{\;}n{\;}{\geq}{\;}1$} is a centering array of constants or random variables.

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On Convergence in p-Mean of Randomly Indexed Partial Sums and Some First Passage Times for Random Variables Which Are Dependent or Non-identically Distributed

  • Hong, Dug-Hun
    • Journal of the Korean Statistical Society
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    • v.25 no.2
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    • pp.175-183
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    • 1996
  • Let $S_n,n$ = 1, 2,... denote the partial sums of not necessarily in-dependent random variables. Let N(c) = min${ n ; S_n > c}$, c $\geq$ 0. Theorem 2 states that N (c), (suitably normalized), tends to 0 in p-mean, 1 $\leq$ p < 2, as c longrightarrow $\infty$ under mild conditions, which generalizes earlier result by Gut(1974). The proof follows by applying Theorem 1, which generalizes the known result $E$\mid$S_n$\mid$^p$ = o(n), 0 < p< 2, as n .rarw..inf. to randomly indexed partial sums.

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