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By Pietro Cerone
This booklet is the 1st in a set of analysis monographs which are dedicated to providing fresh learn, improvement and use of Mathematical Inequalities for exact capabilities. the entire papers included within the ebook have peen peer-reviewed and canopy various issues that come with either survey fabric of formerly released works in addition to new effects. In his presentation on distinctive features approximations and limits through fundamental illustration, Pietro Cerone utilises the classical Stevensen inequality and boundaries for the Ceby sev sensible to procure bounds for a few classical designated capabilities. The technique is determined by identifying bounds on integrals of goods of capabilities. The innovations are used to acquire novel and worthy bounds for the Bessel functionality of the 1st sort, the Beta functionality, the Zeta functionality and Mathieu sequence.
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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. Cerone and S. S. au Abstract. In this paper we survey some recent results of the authors concerning inequalities for Dirichlet series with positive terms.
5) ψ (a) ψ (a + b + c) ≥ (≤) ψ (a + b) ψ (a + c) . Proof. Consider the sequence αn := nb , n ≥ 1, b ∈ R. It is clear that αn is increasing if b > 0 and decreasing if b < 0. Therefore, the sequences n1b , n1c are synchronous if bc ≥ 0 and asynchronous when bc < 0. 5) is proved. 2. 6) provided the real numbers a, b are such that a, a + b, a + 2b > 1. 3). 3. Assume that m ≥ 2 and k1, . . , km > 12 . 7) 1≤i 3) ψ 2 (s + 1) ≤ ψ (s) ψ (s + 2) , provided s > 1. 4) ζ (s + 1) ζ (s + 2) ≤ ζ (s) ζ (s + 1) for s > 1. This inequality is an improvement of a recent result due to Laforgia and Natalini  who proved that ζ (s + 1) s + 1 ζ (s + 2) ≤ · for s > 1. ζ (s) s ζ (s + 1) Their arguments make use of an integral representation of the Zeta function and Tur´ antype inequalities. 4) shows that ζ (2n + 1) ≤ ζ (2n) ζ (2n + 2), demonstrating that Zeta at the odd integers is bounded above by the geometric mean of its immediate even Zeta values.
Advances in Inequalities for Special Functions by Pietro Cerone
3) ψ 2 (s + 1) ≤ ψ (s) ψ (s + 2) , provided s > 1. 4) ζ (s + 1) ζ (s + 2) ≤ ζ (s) ζ (s + 1) for s > 1. This inequality is an improvement of a recent result due to Laforgia and Natalini  who proved that ζ (s + 1) s + 1 ζ (s + 2) ≤ · for s > 1. ζ (s) s ζ (s + 1) Their arguments make use of an integral representation of the Zeta function and Tur´ antype inequalities. 4) shows that ζ (2n + 1) ≤ ζ (2n) ζ (2n + 2), demonstrating that Zeta at the odd integers is bounded above by the geometric mean of its immediate even Zeta values.