By Ravi P. Agarwal, Donal O'Regan, Samir H. Saker

The publication is dedicated to dynamic inequalities of Hardy variety and extensions and generalizations through convexity on a time scale T. specifically, the publication includes the time scale models of classical Hardy variety inequalities, Hardy and Littlewood kind inequalities, Hardy-Knopp kind inequalities through convexity, Copson variety inequalities, Copson-Beesack sort inequalities, Liendeler kind inequalities, Levinson variety inequalities and Pachpatte sort inequalities, Bennett variety inequalities, Chan style inequalities, and Hardy variety inequalities with varied weight services. those dynamic inequalities comprise the classical non-stop and discrete inequalities as precise situations whilst T = R and T = N and will be prolonged to kinds of inequalities on various time scales corresponding to T = hN, h > zero, T = qN for q > 1, etc.In this booklet the authors the historical past and improvement of those inequalities. every one part in self-contained and you possibly can see the connection among the time scale models of the inequalities and the classical ones. To the easiest of the authors’ wisdom this is often the 1st e-book dedicated to Hardy-typeinequalities and their extensions on time scales.

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13) holds. t/t. 1. 2. 1). 1). 2). The results are adapted from [169]. 1. u/ be a nondecreasing positive convex function defined for u > 0. t//˛ p 1 . 4) Proof. t/ C ˛. t/˛ p . t/˛ p . t// 1 p 1 1 p ˛ p 1 . t/ C ˛. t/˛ p . t//˛ p 1 . t/˛ p . t/˛ p . t/˛ p 1 . t/˛ p 1 . 7). t/ D p˛ p 1 . t// : p ˛ p 1 . t/˛ p . t//˛ p 1 . t/˛ p . t/˛ p . t/˛ p . t//˛ p 1 . t/˛ . t//˛ p 1 . t/˛ p 1 . t/˛ . t/˛ . t/˛ p . t/˛ . t/ a ! s/s ! 8) 44 1 Hardy and Littlewood Type Inequalities By letting x ! 4), This completes the proof.

2. 1 and a D 1. 3) on time scales. 2. 19) Proof. 18). 19). The proof is complete. 2 then we obtain the following result. 1. Let T be a time scale with a 2 Œ0; 1/T and p > 1. 20) holds. t/t. 3. 4. 1) due to Hardy and Littlewood. 1). The constants are best possible. In Eq. c 1//p . In 1976, Copson [50, Theorems 1 and 3] proved the continuous counterparts of these inequalities. 4). 4). The results in this section are adapted from [162]. 1. t/ WD a c > 1. 7) Proof. t//c Z 1 . 6). 7).

T/˛ . t//˛ p 1 . t/˛ p 1 . t/˛ . t/˛ . t/˛ p . t/˛ . t/ a ! s/s ! 8) 44 1 Hardy and Littlewood Type Inequalities By letting x ! 4), This completes the proof. 1. 2. 2) of Pachpatte. 4). 1. t/ > 0. 3. 7]. 2. Let T be a time scale with a 2 Œ0; 1/T ; and ' be a positive nondecreasing convex function. t/ ! s/s R1 b ! 1 which is an essentially new even when T D R and when T D N. 2. u/ be a nonincreasing positive convex function defined for u > 0. 13) Proof. t/ C ƒ . t/ ƒ . 15) ƒ . t/ C p 1 p ƒ . t/ p 1 ƒ .

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