Advances in Inequalities for Special Functions by Pietro Cerone

By Pietro Cerone

This publication is the 1st in a set of study monographs which are dedicated to providing fresh learn, improvement and use of Mathematical Inequalities for designated features. the entire papers included within the publication have peen peer-reviewed and canopy more than a few themes that come with either survey fabric of formerly released works in addition to new effects. In his presentation on exact features approximations and boundaries through vital illustration, Pietro Cerone utilises the classical Stevensen inequality and limits for the Ceby sev practical to acquire bounds for a few classical unique capabilities. The method depends on deciding upon bounds on integrals of goods of services. The suggestions are used to acquire 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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3(2) Art. html]. trapezoid inequal(2002). B. Conrey, The Riemann hypothesis, Notices of the AMS (2003), 341–353. [27] D. Cvijovi´c and J. Klinowski, Integral representations of the Riemann zeta function for odd-integer arguments, J. 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.

2. 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

2) for the Mathieu series. 31). 2482358. 2) seem to be superior for the remainder of the r values. 28). 2. 28). The two figures are provided to cater for the different vertical scale. 6 Concluding Remarks In the paper the usefulness of some recent results in the analysis of inequalities, has been demonstrated through application to some special functions. Although these techniques have been applied in a variety of areas of applied mathematics, their application to special functions does not seem to have received much attention to date.

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