By B.G. Pachpatte

For greater than a century, the learn of assorted forms of inequalities has been the focal point of significant cognizance by means of many researchers, either within the thought and its purposes. particularly, there exists a truly wealthy literature on the topic of the well-known Cebysev, Gruss, Trapezoid, Ostrowski, Hadamard and Jensen sort inequalities. the current monograph is an try to arrange fresh growth regarding the above inequalities, which we are hoping will widen the scope in their functions. the sphere to be lined is intensely large and it's very unlikely to regard all of those right here. the cloth integrated within the monograph is fresh and difficult to discover in different books. it truly is obtainable to any reader with a cheap heritage in actual research and an acquaintance with its comparable components. All effects are offered in an straight forward manner and the e-book may also function a textbook for a complicated graduate path. The publication merits a hot welcome to people who desire to research the topic and it'll even be most precious as a resource of reference within the box. it will likely be necessary interpreting for mathematicians and engineers and in addition for graduate scholars, scientists and students wishing to maintain abreast of this significant zone of analysis.

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30). 4 ˇ ¨ and Cebyˇ Inequalities of the Grusssev-type ˇ In this section we present some recent inequalities of the Gr¨uss-and Cebyˇ sev-type established by Pachpatte [106,111,117,127]. For suitable functions z, f , g : [a, b] → R and w : [a, b] → [0, ∞) an integrable function such that b a w(x)dx > 0, we use the following notation to simplify the details of presentation: D[z(x)] = z(x)(1 − λ ) + b A( f , g) = a z(a) + z(b) λ (b − a), 2 [g(x)D[ f (x)] + f (x)D[g(x)]] dx − 2 λ ∈ [0, 1], b b f (x)dx a g(x)dx , a 22 Analytic Inequalities: Recent Advances b B( f , g) = D[ f (x)]D[g(x)]dx a b − b f (x)dx b a a a b f (x)dx g(x)dx , a f (x)g(x)dx − a b 1 b a x f (x)dx f (x)g(x)dx − b 3 xg(x)dx , a a b w(x) f (x)g(x)dx− b w(x) f (x)dx w(x)g(x)dx , a w(x) f (x)g(x)dx− a ∞ b x f (x)dx b3 − a3 b and define z , a a S(w, f , g) = a b g(x)dx b T (w, f , g) = xg(x)dx a a H( f , g) = b f (x)dx b2 − a2 b + a b +(b − a) b D[ f (x)]dx g(x)dx a G( f , g) = b D[g(x)]dx + a b 1 b w(x) f (x)dx b a w(x)dx w(x)g(x)dx , a a = supt∈[a,b] |z(t)| < ∞.

In proving the inequalities in the next theorem, established in [106], we make use of the following variant of the well-known Lagrange’s mean value theorem given by Pompeiu in [145]. 1 (see [145]). [a, b] not containing 0 and for all pairs x1 = x2 in [a, b] there exists a point c in (x1 , x2 ) such that x1 f (x2 ) − x2 f (x1 ) = f (c) − c f (c). x1 − x2 For the proof of Pompeiu’s mean value theorem, we refer the interested readers to [57,147]. 2. 21) where l(t) = t, t ∈ [a, b] and M = (b − a) 1 − Proof.

30) ˇ Gr¨uss-and Cebyˇ sev-type inequalities × 27 b3 − a3 b3 − a3 b2 − a2 (b − a) − (b2 − a2 ) + (b − a) 3 2 3 . 31), we have H( f , g) = [ f (c) − c f (c)][g(d) − dg (d)]M. 21). The proof is complete. Let h : [a, b] → R be a differentiable function on [a, b] and h : [a, b] → R be integrable on [a, b]. Let w : [a, b] → [0, ∞) be some probability density function, that is, an integrable function satisfying b a w(t)dt = 1 with W (t) = t a w(x)dx for t ∈ [a, b], W (t) = 0 for t < a and W (t) = 1 for t > b.