By Pietro Cerone

ISBN-10: 1600219195

ISBN-13: 9781600219191

ISBN-10: 1606926217

ISBN-13: 9781606926215

This booklet is the 1st in a set of analysis monographs which are dedicated to featuring contemporary examine, improvement and use of Mathematical Inequalities for precise features. all of the papers included within the publication 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 targeted capabilities approximations and limits through indispensable illustration, Pietro Cerone utilises the classical Stevensen inequality and boundaries for the Ceby sev useful to acquire bounds for a few classical designated services. The method will depend on opting for bounds on integrals of goods of services. The suggestions are used to acquire novel and precious bounds for the Bessel functionality of the 1st sort, the Beta functionality, the Zeta functionality and Mathieu sequence.

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20) Ik± (p, s) := k 2 1 (np ± mp ) an am , 2 n=1 m=1 ns ms k≥1 where an ≥ 0, n ≥ 1 and s, p ∈ R. 7. 21) 2 I ± (p, s) := lim Ik± (p, s) = ψ (s − 2p) ψ (s) ± [ψ (s − p)] (≥ 0) . k→∞ Inequalities for Positive Dirichlet Series 55 Proof. We observe that k k Ik± (p, s) = = n2p ± 2np mp + m2p ns ms 1 2 n=1 m=1 1 2 k n=1 k k an ns−2p an am m=1 k k am an ±2 s m ns−p n=1 k am an + s−p m ns n=1 m=1 am . 21) is proved. 5. 22) 2 ψ (s + 2) ψ (s) − [ψ (s + 1)] = k 2 1 (n − m) lim anam ≥ 0. 8. Let α, β > 1 with α−1 +β −1 = 1.

5. 32 P. 5. The following bounds are valid for S (r) the Mathieu series. 12). Proof. 31), on noting that π 2 64 cos φ 26 3 dφ = 2 1 + (4r) 2 2 0 1 + (4r cos φ) and after some simplification. 1. 28) with µ = 1. 2) for the Mathieu series. 31). 2482358. 2) seem to be superior for the remainder of the r values. 28). 2. 28). The two figures are provided to cater for the different vertical scale. 6 Concluding Remarks In the paper the usefulness of some recent results in the analysis of inequalities, has been demonstrated through application to some special functions.

Math. , 127 (1999), 385–3996. M. Srivastava, Some families of rapidly convergent series representation for the zeta function, Taiwanese J. , 4 (2000), 569–596. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Acad. , Dordrecht/Boston/London (2001), pp. 388. F. Steffensen, On certain inequalities between mean values and their application to actuarial problems, Skandinavisk Aktuarietidskrift, (1918), 82-97. C. Titchmarsh, The Theory of the Riemann Zeta Function, Oxford Univ.

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