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Hardy's distinguished inequalities involving series and integrals [1, 2], which are called discrete and continuous inequalities, respectively, led to many papers dealing with their alternative proofs, various generalizations, improvements and applications. Among numerous mathematicians interested in such inequalities, we choose to mention (related to our work) Copson, Bennett and Leindler. Besides Hardy's discrete and continuous inequalities (together with their generalizations, we may call all of them as Hardy-Copson inequalities), their reverse versions, which are called Bennett-Leindler inequalities, are found to be attractive to mathematicians who obtain extended and refined inequalities in reverse directions. In order to avoid proving the results twice, once for continuous functions and once for functions defined on discrete sets, delta and nabla time scale unifications of these inequalities have appeared in the literature as well. Since these unifications are not adequate in the theoretical research of some differential and difference equations and in certain computational applications such as adaptive computing and multiscale methods [3], the diamond alpha, $\diamond_{\alpha},$ calculus, which utilizes convex linear combinations of delta and nabla derivatives and integrals, has been introduced by Sheng et al. [3]. In this talk we present some important steps of the development of Hardy-Copson and Bennett-Leindler inequalities with their various modifications and extensions. In addition to discrete and continuous generalizations, we show delta, nabla and diamond alpha unifications of Hardy-Copson and Bennett-Leindler inequalities as well as complements of these unifications. 

This is a joint work with Billur Kaymakçalan

References
[1] G. H. Hardy, Notes on a theorem of Hilbert. Math. Z. 6 (1920), no. 3-4, $314-317$
[2] G. H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals. Messenger Math. $54(1925), 150-156$.
[3] Q. Sheng, M. Fadag, J. Henderson, J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications. Nonlinear Anal. Real World Appl. $7(2006)$, no. $3,395-413 .$

Link to the seminar is:

https://teams.microsoft.com/l/meetup-join/19%3a0ae91f7b86a24e8fa1d818a6f74b22a1%40thread.tacv2/1617974725596?context=%7b%22Tid%22%3a%22066690f2-a8a6-4889-852e-124371dcbd6f%22%2c%22Oid%22%3a%22a343f6fd-86f8-4abe-95cf-3c7f4ad5f0ca%22%7d