• 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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PARTIAL SUMS AND NEIGHBORHOODS OF JANOWSKI-TYPE SUBCLASSES OF MEROMORPHIC FUNCTIONS

  • Abdullah Alatawi;Maslina Darus
    • Korean Journal of Mathematics
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    • v.31 no.3
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    • pp.259-267
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    • 2023
  • The paper presents the introduction of a novel linear derivative operator for meromorphic functions that are linked with q-calculus. Using the linear derivative operator, a new category of meromorphic functions is generated in the paper. We obtain sufficient conditions and show some properties of functions belonging to these subclasses. The partial sums of its sequence and the q-neighborhoods problem are solved.

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.

PARTIAL SUMS AND INCLUSION RELATIONS FOR STARLIKE FUNCTIONS ASSOCIATED WITH AN EVOLUTE OF A NEPHROID CURVE

  • Gurpreet Kaur ;Sumit Nagpal
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.6
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    • pp.1477-1496
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    • 2023
  • A class of normalized univalent functions f defined in an open unit disk of the complex plane is introduced and studied such that the values of the quantity zf'(z)/f(z) lies inside the evolute of a nephroid curve. The inclusion relations of the newly defined class with other subclasses of starlike functions and radius problems concerning the second partial sums are investigated. All the obtained results are sharp.

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.

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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