By George A. Anastassiou
This monograph provides contemporary and unique paintings of the writer on inequalities in actual, sensible and fractional research. The chapters are self-contained and will be learn independently, they contain an intensive record of references in line with chapter.
The book’s effects are anticipated to discover purposes in lots of parts of utilized and natural arithmetic, particularly in traditional and partial differential equations and fractional differential equations. As such this monograph is appropriate for researchers, graduate scholars, and seminars of the above matters, in addition to technology and Engineering collage libraries.
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Example text
1 + N − k) k N −1 k=0 (−1) N +k−1 + 1 = (R1 + R2 )k (R2 − R1 ) N +α−k . k! S. S. 89). 21 Let f : A → R be radial; that is, there exists a function g such that f (x) = g (s), s = |x|, s ∈ [R1 , R2 ], ∀ x ∈ A, α > 0, m = [α]. , m − 1. When 0 < α < 1 the last boundary conditions are void. Then N A D αR1 + g R +R R1 , 1 2 2 D αR2 − g N 2 A N −1 ∞, π 2 (N − 1)! | f (y)| dy ≤ f (y) dy ≤ k=0 · (−1) N +k−1 (R1 + R2 )k (R2 − R1 ) N −k+α k! 97) (R1 + R2 )k (R2 − R1 )α+N −k k! (α + 1 + N − k) . 98) R2 .
2 Let f (x) be differentiable and not identically constant on [a, b] with f (a) = f (b) = 0 and M = sup f (x) . 2) is the best constant. A. 1007/978-3-319-21121-3_3 23 24 3 About Multivariate General Fractional Polya Integral Inequalities The above motivate the current chapter. 2). For the last we need the following fractional calculus background. Let α > 0, m = [α] ([·] is the integral part), β = α − m, 0 < β < 1, f ∈ C ([a, b]), [a, b] ⊂ R, x ∈ [a, b]. The gamma function is given by (α) = ∞ −t α−1 dt.
We treat both the radial and non-radial cases in all possibilities. We give also estimates for the related averages. It follows [4]. 1 Introduction We mention the following famous Polya’s integral inequality, see [10], [11, p. 62], [12] and [13, p. 83]. 1 Let f (x) be differentiable and not identically a constant on [a, b] with f (a) = f (b) = 0. Then the exists at least one point ξ ∈ [a, b] such that f (ξ) > b 4 (b − a)2 f (x) d x. 1). 2 Let f (x) be differentiable and not identically constant on [a, b] with f (a) = f (b) = 0 and M = sup f (x) .
