By Pietro Cerone

This booklet is the 1st in a suite of study monographs which are dedicated to proposing contemporary examine, improvement and use of Mathematical Inequalities for precise services. the entire papers integrated within the ebook have peen peer-reviewed and canopy a number themes that come with either survey fabric of formerly released works in addition to new effects. In his presentation on particular features approximations and boundaries through essential illustration, Pietro Cerone utilises the classical Stevensen inequality and boundaries for the Ceby sev useful to procure bounds for a few classical unique features. The technique depends upon deciding upon bounds on integrals of goods of services. The strategies are used to procure novel and valuable bounds for the Bessel functionality of the 1st style, the Beta functionality, the Zeta functionality and Mathieu sequence.

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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. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974. [32] A. Elbert, Asymptotic expansion and continued fraction for Mathieu’s series, Period.

Cerone and S. S. 43) = = = a−b ln a−ln b is greater than the geometric (nh−mh ) k am (n−m) an √ m=1 ns+h · ms+h nm δk ≤ k k an an n=1 ns · n=1 ns+h k k am an 1 h h n=1 m=1 s+h+ 21 · s+h+ 21 (n − m) n − m 2 n m k k an an n=1 ns · n=1 ns+h k k k k an an an an h n=1 s+h+ 21 n · n n=1 s+h+ 21 − n=1 s+h+ 21 · n n=1 s+h+ 21 n n n n k k an an n=1 ns · n=1 ns+h k k k k an an an an n=1 s− 21 · n=1 s+h+ 21 − n=1 s+h− 21 · n=1 s+ 21 n n n n k k an an n=1 ns · n=1 ns+h k n=1 1 2 nh =: ∆k . 41). 9. 41): ψ (s) ψ (s + h) ψ s + h2 ψ s− 1 h 4 ≤ exp for s > ψ s+ and h > 0, which is of interest in itself.

Let f : I ⊆ R → (0, ∞) be a log-convex function. 3) f (x + h) ≤ f (x) f (x + 2h). Proof. 4) 1 x3 −x2 f (x2 ) ≥ f (x1 ) 1 x2 −x1 . 3). 1. 6) f (x + 1) ≤ f (x) f (x + 2), for any x ≥ a. 2. 7) exp h · f (x + h) f (x + h) f (x) ≥ ≥ exp h · . 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.