Man Kam Kwong

On the oscillation and nonoscillation of second order sublinear equations

An oscillation criterion and a nonoscillation criterion are given for the sublinear equation y + a(t) Iy YIsgn y = 0, 0 -y and nonoscillation for A 0, 4) 0 and X > -y or with JoTf(t) dt = 0 and X > 1 y, then (1) is oscillatory. When JoTf(t) dt = 0 and X -y in our motivating example. To supplement our first result, we prove a nonoscillation Received by the editors July 13, 1981. A MS (MOS) subject classifications (1970). Primary 34C 10, 34C 1 5. Kev words and phrases. Second order, nonlinear, differential equations, oscillation. The first author wishes to thank the Graduate School of Northern Illinois University for a grant that partially supports this research. i 1982 American Mathematical Society 0002-9939/82/0000-0667/

An improved Vietoris sine inequality

02.25 547 This content downloaded from 157.55.39.18 on Thu, 12 May 2016 06:34:22 UTC All use subject to http://about.jstor.org/terms 548 M. K. KWONG AND J. S. W. WONG theorem which allows us to settle the cases X 1. Integrating (5) twice, first over [s, t], then over [s, T], we obtain ep(T)z-8l(T) .(s)z(s) -(4(s)zt9 (s))(T s) (6) (A) + )fTf 4)(T)Z -3(T)Z2(T) dTdt = (3-( 1 )j t(T)a(T) ddt. Dividing by T and letting T oo, we see that, because the right-hand side tends to a limit and the integrands of the two integrals on the left-hand side as well as the first term are nonnegative, the following limits exist and are finite: (7) 0 lim .0 gA(t)z(t)/t. By (7) again (12) jT X > -y, we can choose +(t) = ty with any ,t such that 0 ,Ly + X > 0. Denote ,uy + X by 6. Then C is defined and (i(s) = se(cos s + o(l)). Since 402(S)S/42(S) = L2/S, (4) is satisfied and so (1) is oscillatory. The same argument works for a(t) = tAf(t) with JoTf(t) dt = 0 and X > -y. The following result extends the necessity part of Belohorecs Theorem, i.e. equation (1) has a nonoscillatory solution if a(t) satisfies 00 (13) a(t) ,> O, |tya(t) dt 0. If there exists a function F(t) E C[O, oo) such that I A(t) I 0 for all t > 1 and so y is nonoscillatory. For the sake of brevity, we omit the subscript m in the following discussion. This content downloaded from 157.55.39.18 on Thu, 12 May 2016 06:34:22 UTC All use subject to http://about.jstor.org/terms 550 M. K. KWONG AND J. S. W. WONG Suppose now that y(t) = 0 for some t > 1. Let T, be the smallest of such t. Let T2 be the smallest of all those t such that y(t) = 2m. (If no such t exists, let T2 = m*) Finally let T = min{ T,, T2). Then on [1, T), 0 0 on[ 1, T).) We now integrate the last integral in (18) above: (19) fIA(s)(Yy(s))ds (2yM)/( -r), we have in particular 0 < y(T) < 2m. This contradicts (16). REMARK. For X < -y and a(T) = tAsin t, we see that I f, a(s) ds I is less than a constant multiple of t. Then F(t) ctl satisfies the hypotheses of the theorem and so (1) is nonoscillatory. Another example is offered by a(t) = t-F(log t)Asin t, ,u < -2. We see that F(t) can be taken to be a multiple of t-F(log t)O. If F is any C nondecreasing function such that

New method for blowup of the Euler-Poisson system

We prove that if { a k } is a sequence of positive numbers, such that ( 2 j - 1 ) j + 1 2 j j a 2 j + 1 ? a 2 j ? 2 j - 1 2 j a 2 j - 1 for all j = 1 , 2 , ? , then for all n = 1 , 2 , ? , x ? 0 , π , ? k = 1 n a k sin ( k x ) ? 0 . An example is { a k } = { 1 , 1 2 , 1 2 , 3 4 2 , 1 3 , 5 6 3 , ? } = { 1 , 0.5 , 0.707 , 0.530 , 0.577 , 0.481 , ? } where a k = 1 / ( k + 1 ) / 2 for odd k , and ( k - 1 ) / ( k k / 2 ) , for even k . This improves the well-known Vietoris sine inequality, by relaxing the requirement that { a n } has to be a nonincreasing sequence.The proof is based on a Lukacs-type inequality and a result on positive trigonometric sums with convex coefficients (both established recently by the authors), the classical Sturm Theorem on the number of real roots of a polynomial, and a well-known comparison principle. The symbolic manipulation software MAPLE is used amply for various computations.

Sharp upper and lower bounds for a sine polynomial

In this paper, we provide a new method for establishing the blowup of C2 solutions for the pressureless Euler-Poisson system with attractive forces for RN (N ≥ 2) with ρ(0, x0) > 0 and Ω0ij(x0)=12∂iuj(0,x0)−∂jui(0,x0)=0 at some point x0 ∈ RN. By applying the generalized Hubble transformation u2009divu2009u(t,x0(t))=Na(t)a(t) to a reduced Riccati differential inequality derived from the system, we simplify the inequality into the Emden equation a(t)=−λa(t)N−1,a(0)=1,a(0)=u2009divu2009u(0,x0)N. Known results on its blowup set allow us to easily obtain the blowup conditions of the Euler-Poisson system.

Rogosinski-Szegö type inequalities for trigonometric sums

We prove that for all n ? 1 and x ? (0, π) we have α ? ? k = 1 n sin ( k x ) k ( n + 1 - k ) ? β with the best possible constant bounds α = 3 - 33 64 30 - 2 33 = - 0.18450 ? and β = 1 .

On a sine polynomial of Turán

We prove that the inequalities ? k = 1 n sin ( k x ) k + 1 ? 1 384 ( 9 - 137 ) 110 - 6 137 = - 0.044419686 . . . and ? k = 1 n sin ( k x ) + cos ( k x ) k + 1 ? - 1 2 are valid for all real numbers x ? 0 , π and all positive integers n . The constant lower bounds are sharp. Our theorems complement a classical result of Rogosinski and Szego, who proved in 1928 that the inequality ? k = 1 n cos ( k x ) k + 1 ? - 1 2 holds for all x ? 0 , π and n ? 1 .

On Jordan’s inequality

In 1935, P. Turan proved that

A refinement of Vietoris’ inequality for cosine polynomials

S_{n,a}(x)= sum_{j=1}^n{n+a-jchoose n-j} sin(jx)>0 quad{(n,ainmathbf{N}; 0<x<pi).}