By George A. Anastassiou

ISBN-10: 1891891901

ISBN-13: 9781891891908

ISBN-10: 331921120X

ISBN-13: 9783319211206

ISBN-10: 3319211218

ISBN-13: 9783319211213

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 in depth record of references in step with chapter.

The book’s effects are anticipated to discover purposes in lots of parts of utilized and natural arithmetic, specially 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 topics, in addition to technological know-how and Engineering college libraries.

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**Extra info for Intelligent Comparisons: Analytic Inequalities**

**Sample text**

Hence | f (y)| dy = A R1 +R2 2 N 2π 2 N 2 |g (s)| s N −1 ds + R1 R2 R1 +R2 2 |g (s)| s N −1 ds ≤ N 2π 2 1 (α) ( p (α − 1) + 1) p N 2 R1 +R2 2 D αR1 + g Lq D αR2 − g R1 +R2 2 Lq R1 +R2 ,R2 2 ⎧ ⎨ 1 (α) ( p (α − 1) + 1) p ⎩ ⎛ ⎝ N −1 (−1) N +k−1 ⎛ ⎝ N −1 k=0 Lq N +α+ R1 +R2 ,R2 2 1 p ⎝ R1 +R2 2 N α+ 1 p 2 N −k+α− q1 (N − 1)! 107) 1 p ⎞ ⎠· ⎞ α+N − q1 + N −k = (N − 1)! −k 2 π 2 (N − 1)! N 2 R1 , (R1 + R2 ) (R2 − R1 ) k! s N −1 ds + s N −1 ds ⎛ D αR1 + g Lq ⎛ D αR2 − g α−1+ 1p α−1+ 1p k k! 108) 48 ⎧ ⎨ ⎩ 3 About Multivariate General Fractional Polya Integral Inequalities ⎛ D αR1 + g R1 , Lq ⎝ R1 +R2 2 N −1 ⎝ R1 +R2 ,R2 2 Lq k!

2 Main Results ⎡ ⎣ 35 k=0 1 α+ k! N 2 π R (α) (N − 1)! ⎫ ⎬ ⎤ N −1 N 2 +N −k 1 p ⎦ D αR− g R 2 ,R Lq ⎧ α ⎪ ⎨ D0+ g α+N − q1 1 ( p (α − 1) + 1) p 2 ⎡ N −1 1 ⎣ α+ p k! α+N − q1 −1 ⎪ ⎩ ⎭ Lq α+N − = 0, R2 1 q + ⎫ ⎬ ⎤ 1 α+ k=0 1 p + N −k ⎦ D αR− g Lq R 2 ,R ⎭ . 9. Then 1 p + 1 q = 1, α > q1 . All other terms and assumptions | f (y)| dy ≤ B(0,R) N 2 π R (α) N 2 1 p ( p (α − 1) + 1) 2 ⎡ (N − 1)! α+ ⎧ α ⎪ ⎨ D0+ g α+N − q1 1 ⎣ p α+N − q1 −1 α+N − 0, R2 1 q + ⎫ ⎪ ⎬ ⎤ N −1 k=0 ⎪ ⎩ Lq 1 α+ k! 1 p +N −k ⎦ D αR− g Lq R 2 ,R ⎪ ⎭ .

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. b]) of C m ([a, b]): a ≤ x ≤ b. We define the subspace Ca+ α Ca+ ([a, b]) = a+ (m) f ∈ C 1 ([a, b]) . 5) see [1], p.

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