By Pietro Cerone

This ebook is the 1st in a set of study monographs which are dedicated to proposing contemporary examine, improvement and use of Mathematical Inequalities for designated features. all of the papers included within the e-book have peen peer-reviewed and canopy a variety of themes that come with either survey fabric of formerly released works in addition to new effects. In his presentation on unique services approximations and limits through imperative illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev useful to procure bounds for a few classical targeted services. The method is determined by settling on bounds on integrals of goods of services. The innovations 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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59) ζ (s) ζ (s) ≥ 2 ζ (s) 2 which is equivalent with the fact that 1/ζ is concave on the interval ζ −1 (e) , ∞ . 58) is trivially satisfied on this interval as well. Figure 1: The plot for ζ (x) ζ (x) − 2 ζ (x) 2 on (1, ζ −1 (e)). 10 that ζ (s + 1) is bounded by the harmonic mean of ζ (s) and ζ (s + 2) . 13. 60) ζ (s + 1) ≤ 2ζ (s) ζ (s + 2) ζ (s) + ζ (s + 2) ≤ ζ (s) ζ (s + 2) . 50 P. Cerone and S. S. 1 2ζ (2n) ζ (2n + 2) ζ (2n) + ζ (2n + 2) ≤ ζ (2n) ζ (2n + 2) . 1). We assume that the series which defines ψ is uniformly convergent for s > 1.

54) ln ζ (s) = (see for example [3, Eq. 2)]) ∞ Λ (n) , s ln n n n=2 (see [3, Eq. 55) ζ (s) ζ (s) − ζ (s) [ζ (s)] 2 ∞ = 2 Λ (n) ln n , ns n=2 (see [3, Eq. 4). Utilising the Schwarz inequality k pm α2k pm m=1 2 k k ≥ pm αm m=1 m=1 with pm ≥ 0, αm ∈ R, m ∈ {1, . 58) ζ (s) ζ (s) − ζ (s) 2 ≥ ζ (s) ln ζ (s) (> 0) , for any s ∈ (1, ∞) . 59) ζ (s) ζ (s) ≥ 2 ζ (s) 2 which is equivalent with the fact that 1/ζ is concave on the interval ζ −1 (e) , ∞ . 58) is trivially satisfied on this interval as well. Figure 1: The plot for ζ (x) ζ (x) − 2 ζ (x) 2 on (1, ζ −1 (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. Applications for the Zeta function are also provided. 1991 Mathematics Subject Classification: Primary 26D15, 11M38, 11M41. Key words and phrases: Dirichlet series, Zeta function, Lambda function, Logarithmic convexity.