April 20, 2017

Download Advances in Inequalities for Special Functions by Pietro Cerone PDF

By Pietro Cerone

This booklet is the 1st in a set of study monographs which are dedicated to proposing fresh study, improvement and use of Mathematical Inequalities for distinct features. the entire papers included within the ebook have peen peer-reviewed and canopy a number themes that come with either survey fabric of formerly released works in addition to new effects. In his presentation on designated services approximations and limits through imperative illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev useful to procure bounds for a few classical specific capabilities. The method is determined by making a choice on bounds on integrals of goods of features. The suggestions are used to acquire novel and worthy bounds for the Bessel functionality of the 1st variety, the Beta functionality, the Zeta functionality and Mathieu sequence.

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B. Conrey, The Riemann hypothesis, Notices of the AMS (2003), 341–353. [27] D. Cvijovi´c and J. Klinowski, Integral representations of the Riemann zeta function for odd-integer arguments, J. of Comput. , 142(2) (2002). 435–439. S. Dragomir, A generalisation of Gr¨ uss’ inequality in inner product spaces and applications, J. Math. Anal. , 237 (1999), 74–82. S. Dragomir, Some integral inequalities of Gr¨ uss type, Indian J. of Pure and Appl. , 31(4) (2000), 397-415. S. M. ), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, 2002.

5. Assume that m ≥ 1 and 1 2 < γ ≤ k1, . . , km ≤ Γ < ∞. 15) [ψ (kj + γ) + ψ (kj + Γ)] ≥ j=1 ψ (2kj ) + mψ (γ + Γ) . j=1 Proof. We have: 1 1 − kj nγ n 1 1 − Γ nkj n ≥0 for each j ∈ {1, . , m} and n ≥ 1. This is clearly equivalent to: 1 1 1 1 + Γ+kj ≥ 2kj + γ+Γ nγ+kj n n n for j ∈ {1, . . , m} and n ≥ 1. 16) j=1 m 1 nγ+kj m 1 + j=1 nΓ+kj ≥ j=1 m 1 + γ +Γ n2kj n for each n ≥ 1. 15). 6. 17) m− 1 2 m ψ (2kj ) ≤ j=1 + 1≤i

Km > 12 . 7) 1≤i 1. If more information about the size of kj , j = 1, . . 4. 12) ≤ m−1 2 1 2 < γ ≤ k1, . . , km ≤ Γ < ∞. Then m ψ (2kj ) − j=1 ψ (ki + kj ) 1≤i

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