• Title, Summary, Keyword: Bessel's inequality

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On Bessel's and Grüss Inequalities for Orthonormal Families in 2-Inner Product Spaces and Applications

  • Dragomir, Sever Silverstru;Cho, Yeol-Je;Kim, Seong-Sik;Kim, Young-Ho
    • Kyungpook Mathematical Journal
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    • v.48 no.2
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    • pp.207-222
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    • 2008
  • A new counterpart of Bessel's inequality for orthonormal families in real or complex 2-inner product spaces is obtained. Applications for some Gr$\"{u}$ss inequality for determinantal integral inequalities are also provided.

SOME NEW RESULTS RELATED TO BESSEL AND GRUSS INEQUALITIES IN 2-INNER PRODUCT SPACES AND APPLICATIONS

  • DRAGOMIR S.S.;CHO, Y.J.;KIM, S.S.
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.591-608
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    • 2005
  • Some new reverses of Bessel's inequality for orthonormal families in real or complex 2-inner product spaces are pointed out. Applications for some Gruss type inequalities and for determinantal integral inequalities are given as well.

MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS

  • Ali, Rosihan M.;Mondal, Saiful R.;Nisar, Kottakkaran S.
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.575-598
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    • 2017
  • This paper introduces and studies a generalization of the classical Struve function of order p given by $$_aS_{p,c}(x):=\sum\limits_{k=0}^{\infty}\frac{(-c)^k}{{\Gamma}(ak+p+\frac{3}{2}){\Gamma}(k+\frac{3}{2})}(\frac{x}{2})^{2k+p+1}$$. Representation formulae are derived for $_aS_{p,c}$. Further the function $_aS_{p,c}$ is shown to be a solution of an (a + 1)-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function $_aS_{p,c}$ are investigated, particulary for the case c = -1. As a consequence, $Tur{\acute{a}}n$-type inequalities are established. For a = 2 and c = -1, dominant and subordinant functions are obtained for the Struve function $_2S_{p,-1}$.