By B.G. Pachpatte

For greater than a century, the research of assorted kinds of inequalities has been the focal point of serious realization through many researchers, either within the concept and its purposes. particularly, there exists a really wealthy literature relating to the well-known Cebysev, Gruss, Trapezoid, Ostrowski, Hadamard and Jensen sort inequalities. the current monograph is an try to set up fresh growth relating to the above inequalities, which we are hoping will widen the scope in their purposes. the sector to be coated is intensely broad and it truly is very unlikely to regard all of those right here. the cloth incorporated within the monograph is fresh and difficult to discover in different books. it's available to any reader with a cheap heritage in actual research and an acquaintance with its similar components. All effects are awarded in an ordinary means and the ebook may also function a textbook for a complicated graduate path. The booklet merits a hot welcome to those that desire to study the topic and it'll even be most dear as a resource of reference within the box. will probably be helpful studying for mathematicians and engineers and in addition for graduate scholars, scientists and students wishing to maintain abreast of this significant sector of study.

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**Example text**

47 First we observe that 1 n 1 ∑ pi fi − Pn Pn i=1 = = 1 n ∑ pi Pn i=1 fi gi − fi 1 n ∑ pi fi gi − Pn i=1 1 Pn n ∑ pj fj j=1 n = n 1 Pn ∑ p jg j j=1 n 1 n n j=1 j=1 1 ∑ p j g j −gi Pn ∑ p j f j + Pn2 ∑ p j f j ∑ p j g j j=1 j=1 1 n ∑ pi fi Pn i=1 + gi − n 1 Pn 1 n ∑ pi Pn i=1 1 Pn 1 n ∑ pi fi gi − Pn i=1 ∑ p jg j − j=1 n ∑ pj fj j=1 1 Pn 1 n ∑ pi fi Pn i=1 1 n ∑ pi gi Pn i=1 1 Pn n ∑ pj fj j=1 n ∑ p jg j j=1 1 n ∑ pi gi Pn i=1 = Cn (p, f , g). 10) and the proof is complete. 2. By taking pi = 1 for i = 1, .

22) holds for n and let us prove it for n + 1. That is, we have to prove the equality (b − x)k+1 + (−1)k (x − a)k+1 (k) h (x) (k + 1)! n b h(t)dt = a ∑ k=0 b +(−1)n+1 a En+1 (x,t)h(n+1) (t)dt. 25) It is easy to observe that b En+1 (x,t)h(n+1) (t)dt = a x a (t − a)n+1 (n+1) h (t)dt + (n + 1)! x = 1 (t − a)n+1 (n) h (t) − (n + 1)! n! a x a b + = (t − b)n+1 (n) 1 h (t) − (n + 1)! n! x b x b x (t − b)n+1 (n+1) h (t)dt (n + 1)! (t − a)n h(n) (t)dt (t − b)n h(n) (t)dt (x − a)n+1 + (−1)n+2 (b − x)n+1 (n) h (x) − (n + 1)!

2. By taking pi = 1 for i = 1, . . 10), we get |Cn ( f , g)| 1 n ∑ n i=1 fi − 1 n n ∑ fj j=1 gi − 1 n n ∑ gj . 13) can be considered as the discrete version of the Gr¨usstype integral inequality given by Dragomir and McAndrew in [34]. ˇ The discrete Cebyˇ sev-type inequality established in [128] is given in the following theorem. 3. Let f = ( f1 , . . , fn ), g = (g1 , . . 2) and Δ fk = fk+1 − fk . 14) 48 Analytic Inequalities: Recent Advances Proof. 3) holds. 5). It is easy to observe that the following identity holds: fi − f j = i−1 i−1 k= j k= j ∑ ( fk+1 − fk ) = ∑ Δ fk .