By Pietro Cerone
This e-book is the 1st in a suite of analysis monographs which are dedicated to offering contemporary study, improvement and use of Mathematical Inequalities for exact features. all of the papers integrated within the e-book have peen peer-reviewed and canopy quite a number issues that come with either survey fabric of formerly released works in addition to new effects. In his presentation on distinct capabilities approximations and boundaries through indispensable illustration, Pietro Cerone utilises the classical Stevensen inequality and boundaries for the Ceby sev sensible to procure bounds for a few classical targeted features. The method depends on settling on bounds on integrals of goods of services. The thoughts are used to procure novel and worthwhile bounds for the Bessel functionality of the 1st type, the Beta functionality, the Zeta functionality and Mathieu sequence.
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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. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.  A. Elbert, Asymptotic expansion and continued fraction for Mathieu’s series, Period.
33) 2 [ζ (s)]2 ≤ 2s−1 (ln 2) 2 (2s−1 − 1)2 for s > 1. 34) s>1 which seems to be satisfied as may be seen from computer experimentation with Maple. If, however, we assume more about the positive sequence an, then we obtain some other results as follows . 8. 35) for any s > 1 and h > 0. 36) exp h · ψ (s + h) ζ (s + h) ≥ , ζ (s + h) ψ (s) for any s > 1 and h > 0. Proof. 37) exp h · ψ (s) ψ (s + h) ψ (s + h) ≥ ≥ exp h · ψ (s + h) ψ (s) ψ (s) for any s > 1 and h > 0. Observe that for s > 1 ∞ ψ (s) = − an n=1 ln n .
The investigation in the current article has restricted itself to the investigation of the Bessel function of the first kind, the Beta function, the Zeta function and Mathieu series. It may be surmised from the above investigations that the accuracy of the bounds over particular regions of parameters cannot be ascertained a priori. It has been demonstrated, however, that some useful bounds may be obtained which seem hitherto not to have been discovered. The approach of utilising developments in the field of inequalities to special functions has been shown to have the potential for further development.
Advances in Inequalities for Special Functions by Pietro Cerone