By Pietro Cerone

This publication is the 1st in a set of study monographs which are dedicated to proposing contemporary examine, improvement and use of Mathematical Inequalities for exact capabilities. the entire papers integrated within the e-book have peen peer-reviewed and canopy a number of themes 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 quintessential illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev sensible to acquire bounds for a few classical certain services. The method is dependent upon deciding upon bounds on integrals of goods of capabilities. The innovations are used to procure novel and beneficial bounds for the Bessel functionality of the 1st sort, the Beta functionality, the Zeta functionality and Mathieu sequence.

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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. 10) (1 ≤) β [f (x1)] [f (x2)] f (x2) f (x1 ) ≤ exp αβ (x2 − x1) − f (αx1 + βx2) f (x2) f (x1 ) .

Math. ), 12 (2001), 15-21. [4] H. L. G. Ruehr, On Mathieu’s inequality, J. Math. Anal. , 218 (1998), 607–610. M. Apostol, Analytic Number Theory, Springer, New York, 1976. M. M. E. Crandall, Computational strategies for the Riemann zeta function, J. of Comput. , 121 (2000), 247–296. [7] I. Budimir, P. E. Peˇcari´c, Inequalities related to the Chebyshev functional involving integrals over different intervals, J. Ineq. Pure and Appl. , 2(2) Art. 22, (2001). au/v2n2/]. S. Bullen, A Dictionary of Inequalities, Addison Wesley Longman Limited, 1998.

18) ∞ x x 2 2 e− 2 x2µ−1 Jµ− 1 (rx) dx − 2K∗ . 24) 2 0 A result in Watson [56, p. 26) 1 2 2 e− 2 x2µ−1 Jµ− 1 (rx) dx = 1 2 Γ 2µ − π 2 π 2 r2µ−1 3 2 cos2µ−1 φ 1 2 4 0 + r2 cos2 φ dφ. 27) x µ− 12 0 1 2 2 Jµ− 1 (rx) − K∗ dx 2 Γ 2µ − = 3 π2 1 2 π 2 r2µ−1 2µ−1 cos 1 2 4 0 12 φ + r2 cos2 φ 2µ− 12 dφ − 2K∗2 . 11). 4. 12), B (x, y) is the Euler Beta function and hypergeometric function. 31) for 0 ≤ r ≤ 5. Proof. 29) Sµ (r) − π2 12µ r2 + 1 µ 4 π 2 1 2 Γ 2µ − κ 1 ≤ √ √ · 2µ−1 2 π 4 Γ (µ) 2µ cos2µ−1 φ dφ 1 2 4 0 2µ−1/2 + r2 cos2 φ 1 − r2 + 12 1 2µ 4 upon simplification.