By Pietro Cerone
This e-book is the 1st in a set of study monographs which are dedicated to providing contemporary examine, improvement and use of Mathematical Inequalities for particular features. the entire papers included 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 unique capabilities approximations and boundaries through indispensable illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev practical to procure bounds for a few classical specified capabilities. The technique is determined by choosing bounds on integrals of goods of services. The strategies are used to acquire novel and important bounds for the Bessel functionality of the 1st sort, the Beta functionality, the Zeta functionality and Mathieu sequence.
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Additional resources for Advances in inequalities for special functions
B. Conrey, The Riemann hypothesis, Notices of the AMS (2003), 341–353.  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.
M. Fink, A treatise on Gr¨ uss’ inequality, Analytic and Geometric Inequalites and Applications, Math. , 478 (1999), Kluwer Academic Publishers, Dordrecht, 93114.  I. Gavrea, Some remarks on Mathieu’s series, Mathematical Analysis and Approximation Theory, 113-117, Burg Verlag, 2002. -N. Guo, Note on Mathieu’s inequality, RGMIA Res. Rep. , 3(3) (2000), Article 5. html]. ¨  G. Gr¨ uss, Uber das Maximum des absoluten Betrages von 1 (b−a)2 b b a f(x)dx a g(x)dx, 1 b−a b a f(x)g(x)dx − Math.
2 > 0. Utilising the relationship Proof. 37). 3. 39) β (m) = 22m+1 − 1 Γ (2m + 1) and 2 γ (m) = Γ (m + 1) . 40) ζ (2m + 1) < α (m) ζ (2m) − γ (m) ζ 2 (m + 1) . 28). 42) ζ (4k + 1) < α (2k) ζ (4k) − γ (2k) ζ 2 (2k + 1) , β (2k) k = 1, 2, . .. 38) with m = 2k there are two zeta functions with odd arguments. There are a number of possibilities for resolving this, but firstly it should be noticed that ζ (x) is monotonically decreasing for x > 1 so that ζ (x1) > ζ (x2) for 1 < x1 < x2. 26). 43) ζ L (4k + 1) < α (2k) ζ (2k) − γ (2k) L22 (2k) , β (2k) where we have used the fact that L2 (x) < ζ (x + 1) .
Advances in inequalities for special functions by Pietro Cerone