• Title/Summary/Keyword: Bessel 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.

SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

  • Guo, Bai-Ni;Qi, Feng
    • Communications of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.355-363
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    • 2016
  • By employing a refined version of the $P{\acute{o}}lya$ type integral inequality and other techniques, the authors establish some inequalities and absolute monotonicity for modified Bessel functions of the first kind with nonnegative integer order.

COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS

  • Qi, Feng;Zhang, Xiao-Jing
    • The Pure and Applied Mathematics
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    • v.21 no.2
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    • pp.141-145
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    • 2014
  • In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function $e^{1/t}$ and the trigamma function ${\psi}^{\prime}(t)$ on (0, ${\infty}$).

BOUNDS FOR EXPONENTIAL MOMENTS OF BESSEL PROCESSES

  • Makasu, Cloud
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1211-1217
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    • 2019
  • Let $0<{\alpha}<{\infty}$ be fixed, and let $X=(X_t)_{t{\geq}0}$ be a Bessel process with dimension $0<{\theta}{\leq}1$ starting at $x{\geq}0$. In this paper, it is proved that there are positive constants A and D depending only on ${\theta}$ and ${\alpha}$ such that $$E_x\({\exp}[{\alpha}\;\max_{0{\leq}t{\leq}{\tau}}\;X_t]\){\leq}AE_x\({\exp}[D_{\tau}]\)$$ for any stopping time ${\tau}$ of X. This inequality is also shown to be sharp.

TURÁN-TYPE INEQUALITIES FOR GAUSS AND CONFLUENT HYPERGEOMETRIC FUNCTIONS VIA CAUCHY-BUNYAKOVSKY-SCHWARZ INEQUALITY

  • Bhandari, Piyush Kumar;Bissu, Sushil Kumar
    • Communications of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1285-1301
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    • 2018
  • This paper is devoted to the study of $Tur{\acute{a}}n$-type inequalities for some well-known special functions such as Gauss hypergeometric functions, generalized complete elliptic integrals and confluent hypergeometric functions which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. We also apply these inequalities for some sample of interest such as incomplete beta function, incomplete gamma function, elliptic integrals and modified Bessel functions to obtain their corresponding $Tur{\acute{a}}n$-type inequalities.

ON GENERALIZED EXTENDED BETA AND HYPERGEOMETRIC FUNCTIONS

  • Shoukat Ali;Naresh Kumar Regar;Subrat Parida
    • Honam Mathematical Journal
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    • v.46 no.2
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    • pp.313-334
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    • 2024
  • In the current study, our aim is to define new generalized extended beta and hypergeometric types of functions. Next, we methodically determine several integral representations, Mellin transforms, summation formulas, and recurrence relations. Moreover, we provide log-convexity, Turán type inequality for the generalized extended beta function and differentiation formulas, transformation formulas, differential and difference relations for the generalized extended hypergeometric type functions. Also, we additionally suggest a generating function. Further, we provide the generalized extended beta distribution by making use of the generalized extended beta function as an application to statistics and obtaining variance, coefficient of variation, moment generating function, characteristic function, cumulative distribution function, and cumulative distribution function's complement.