By Pietro Cerone

ISBN-10: 1600219195

ISBN-13: 9781600219191

ISBN-10: 1606926217

ISBN-13: 9781606926215

This ebook is the 1st in a set of study monographs which are dedicated to featuring fresh study, improvement and use of Mathematical Inequalities for unique services. the entire papers included within the ebook have peen peer-reviewed and canopy a number of issues that come with either survey fabric of formerly released works in addition to new effects. In his presentation on targeted services approximations and boundaries through crucial illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev practical to procure bounds for a few classical distinctive services. The technique depends upon identifying bounds on integrals of goods of capabilities. The ideas are used to procure novel and necessary bounds for the Bessel functionality of the 1st type, the Beta functionality, the Zeta functionality and Mathieu sequence.

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**Example text**

52) ψ (s) ψ (s + 2), s > 1. 11. Based on some numerical experiments conducted with a computer program, we conjecture that any Dirichlet series ψ with nonnegative coefficients has the property that the function 1/ψ is concave where it is defined. The following result gives an answer to the conjecture above in the case of the Zeta function. 12. The function 1/ζ is concave on the interval (1, ∞) Proof. 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.

10. 47) ψ (s1 ) ψ (s2 ) . 48) ψ (s) ψ (s) ≥ 2 ψ (s) 2 . 49) 2 ∞ an · ln n ns n=1 . Proof. 47). 49) holds true. 11. 50) ψ (s + 1) ≤ 2ψ (s) ψ (s + 2) , ψ (s) + ψ (s + 2) 48 P. Cerone and S. S. Dragomir for any s > 1. 52) ψ (s) ψ (s + 2), s > 1. 11. Based on some numerical experiments conducted with a computer program, we conjecture that any Dirichlet series ψ with nonnegative coefficients has the property that the function 1/ψ is concave where it is defined. The following result gives an answer to the conjecture above in the case of the Zeta function.

F (x + h) f (x) f (x) 40 P. Cerone and S. S. Dragomir Proof. 8) exp (x2 − x1) f (x2 ) f (x2) f (x1) ≥ ≥ exp (x2 − x1) . 7). 2. 9) exp f (x + 1) f (x) f (x + 1) ≥ ≥ exp , f (x + 1) f (x) f (x) for any x ∈ [a, ∞). Another result is as follows. 3. Let f : I ⊆ R → (0, ∞) be a log-convex function which is differentiable on ˚ I. 10) (1 ≤) β [f (x1)] [f (x2)] f (x2) f (x1 ) ≤ exp αβ (x2 − x1) − f (αx1 + βx2) f (x2) f (x1 ) . Proof. 12) f (x2 ) f (x2 ) f (x2) = −α (x2 − x1) . 10). 4. 13) (1 ≤) β [f (x)] [f (x + h)] f (x + h) f (x) ≤ exp αβh − f (x + βh) f (x + h) f (x) for any α, β > 0 with α + β = 1.

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