By Pietro Cerone

This e-book is the 1st in a set of study monographs which are dedicated to proposing fresh learn, improvement and use of Mathematical Inequalities for specified features. all of the papers included within the e-book have peen peer-reviewed and canopy a variety of subject matters that come with either survey fabric of formerly released works in addition to new effects. In his presentation on certain services approximations and limits through essential illustration, Pietro Cerone utilises the classical Stevensen inequality and boundaries for the Ceby sev practical to acquire bounds for a few classical designated services. The method depends upon choosing bounds on integrals of goods of capabilities. The ideas are used to procure novel and invaluable bounds for the Bessel functionality of the 1st sort, the Beta functionality, the Zeta functionality and Mathieu sequence.

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Pure Appl. , 4(2) (2003), In press. html. -L. -H. Wang, A refinement of the Mathieu inequality, Univ. Beograd. Publ. Elektroteh. Fak. Ser. Mat. No. 716-734 (1981), 22-24. N. , Cambridge University Press. T. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1978. E. , In: Advances in Inequalities for Special Functions Editors: P. Cerone and S. S. Dragomir, pp. 37–65 ISBN 978-1-60021-919-1 c 2008 Nova Science Publishers, Inc. Inequalities for Positive Dirichlet Series P.

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. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974. [32] A. Elbert, Asymptotic expansion and continued fraction for Mathieu’s series, Period.

5). 6). 6). The following lemma examines the behaviour of χµ (r) (see also [9]). 26 P. 3. 319846901 . .. Proof. 13) x µ− 12 Jµ− 1 (rx) dx. 16), and Watson [56, p. 17). 18) χµ (r) − · √ 2 π · Γ (µ) 2 r + 14 µ · π2 6 ∞ x = 0 1 e− 2 xµ− 2 Jµ− 12 (rx) − K π2 x − −x 12 e −e 2 x 2 dx. 19) 1 (2r) χµ (r) − · √ 2 π Γ (µ) 2 r + 14 · µ π2 6 · ∞ x ≤ 0 1 2 2 1 e− 2 xµ− 2 Jµ− 12 (rx) − K dx ∞ π2 x x x − 12 e 2 − e− 2 −x 2 × e 0 1 2 2 dx . 17), namely µ− 1 (2r) 2 Γ (µ) K = K∗ = √ µ. 19). 21) 0 ∞ 2 x x e − e− 2 x e− 2 dx = x 2 0 π2 12 dx − 2 x 2 2 .