• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.27-41
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• 2006

A counterpart of the famous Bessel's inequality for orthornormal families in real or complex inner product spaces is given. Applications for some Gruss type inequalities are also provided.

• 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.

• 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.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.279-292
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• 2006

A reverse of Bessel's inequality in 2-inner product spaces and companions of $Gr\ddot{u}ss$ inequality with applications for determinantal integral inequalities are given.

• East Asian mathematical journal
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• v.22 no.1
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• pp.1-16
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• 2006

Some new reverses of the Schwarz inequality in inner product spaces and applications for vector-valued sequnces and integrals are given.

• 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.

• 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 $_2S_{p,-1}$.