By Aldo Ghizzetti (auth.), Aldo Ghizzetti (eds.)

A. Ghizzetti: a) Lezioni sui procedimenti di quasilinearizzazione e applicazioni. b) Nozioni fondamentali sulle equazioni alle differenze e sulle frazioni continue.- P. Wynn: 4 lectures at the numerical software of endured fractions.- W. Gautschi: energy and weak spot of three-term recurrence relation.- F.L. Bauer: Use of persevered fractions and algorithms on the topic of them.

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14), the sign :::; has to be replaced by ~ . 9. 17) holds. Further, for i < 0 and i + m > 1, the new inequality holds even if x(t) is complex-valued. 21), the sign :::; has to be changed to ~. 10. 24), the sign ~ is replaced by ~ . 25) is changed to ~ . 6 implies that m > 1. 6. 5. [6] Let i < 0, m > 1 and 0 < i + m < 1. 30) i'T hold. 31) where X (Jcxr {p{t))-1/(l+m-1) dt )l+m-1 < 00. 5) with d complex. Proof. Since J.. m < 0, we have lx(t)li/m > (lix'(s)i ds Y/m, a~ t ~ T. Thus, it follows that r Jcx lx{t)li/mlx'(t)l dt ~ i +m m ( r } 01 lx'{t)l dt )(l+m)/m .

A+ lt Then, u( a+) = 0 implies that liminf u 2 (t)p(t) v'((t)) t ..... a+ Proof. For a < c < t < /3, V t < 0. we have Thus, we can replace c by a in the integrals on the right, and on the left let c -+ a so that u(c) -+ 0. The resulting estimate for u(t) then holds for all t > a and yields the conclusion. 4. 2. Further, let w( a+) < oo, v' 2:: 0 near /3-, w' ~ 0, and (w') 2 pv + (pv')' < 0. 11) GENERALIZATIONS OF OPIAL'S INEQUALITY 32 Proof. 3, the result follows immediately. 3. 9}. Further, in this case, the equality condition p(u'v- v'u) = 0 gives rise to x = ct.

Redheffer's Generalization Let u(t), v(t) and p(t)v'(t) be absolutely continuous and v(t) Then, the following identity holds almost everywhere (pv')' (u-2pv')' (u'v- v'u) 2 -u 2 --+ , 2 = p p(u) v v v > 0 on (a, /3). 2) then we have provided two of the three integrals converge. 1. On (a,/3), let u, v and pv' be absolutely continuous with p 2:: 0, v > 0, and p( u') 2 integrable. a+ t-{3- = B > -oo. dt Ja 2:: -1a u -p-dt+B-A. 4), equality holds if and only if p(u1v- v1u) = 0 almost everywhere. 9) by a weighted sum of the factors.