tailieunhanh - Dynamic Shum inequalities
Recently, various forms and improvements of Opial dynamic inequalities have been given in the literature. In this paper, we give refinements of Opial inequalities on time scales that reduce in the continuous case to classical inequalities named after Beesack and Shum. These refinements are new in the important discrete case. | Turk J Math (2017) 41: 55 – 66 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Dynamic Shum inequalities Ravi AGARWAL1 , Martin BOHNER2,∗, Donal O’REGAN3 , Samir SAKER4 1 Department of Mathematics, Texas A&M University–Kingsville, Kingsville, TX, USA 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA 3 School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway, Ireland 4 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt Received: • • Accepted/Published Online: Final Version: Abstract: Recently, various forms and improvements of Opial dynamic inequalities have been given in the literature. In this paper, we give refinements of Opial inequalities on time scales that reduce in the continuous case to classical inequalities named after Beesack and Shum. These refinements are new in the important discrete case. Key words: Opial’s inequality, Beesack’s inequality, Shum’s inequality, time scales 1. Introduction In 1960, Olech [13] extended an inequality of Opial [14] and proved the following result. Theorem (Opial Inequality) If f ∈ C1 ([0, h], R) with h > 0 satisfies f (0) = 0 , then ∫ h |f (t)f ′ (t)| dt ≤ 0 h 2 ∫ h (f ′ (t)) dt. 2 () 0 This inequality has generated a lot of research, both in the continuous and the discrete cases (see the monograph [3] and the references therein). In 1962, Beesack [4] refined Theorem as follows (see also [15, Theorem 3]). Theorem (Beesack Inequality) If f ∈ C1 ([0, h], R) with h > 0 satisfies f (0) = 0 , then ∫ 0 h h |f (t)f (t)| dt + 2 ′ where ∫ g(t) := 2 t ∫ h 0 g(t) h dt ≤ t2 2 ∫ h (f ′ (t)) dt, 2 () 0 |f (τ )f ′ (τ )| dτ − (f (t)) ≥ 0. 2 0 We note that Theorem implies Theorem . In 2001, Bohner and Kaymak¸calan [6] extended Theorem to
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