On a differential inequality of the type X" +/ (t)X ^ 0The aim of this paper is to prove the following

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

Ma r e k Ki e r l a n c z y k

(Warszawa)

On a differential inequality of the type X " + / ( t ) X ^ 0

The aim of this paper is to prove the following

Th e o r e m.

Let x ( t ) , f ( t ) be real functions determined on

such that

(1) x(t) is twice differentiable on [ a ,b ] , f (t) is bounded on [ a ,b ] ,

(2) x ”+ f ( t ) x ^ 0 on (a , b ),

(3) x(a) = x(b) = 0, x(f) is positive on (a ,b ), Let t0 e( a , b ) be a point such that

x (t0) = max x (t)

të[a,b]

and let M = sup / (t). Then M > 0 and the following inequalities hold:

te [a ,b]

(4) b + a - ^ ( b - a ) 2- S / M ^ ^ b + a + ^ / ( b - a ) 2-8 /M

U 2 " to ^ 2 ’

(5) t o ^ f o - v / 2 / У м .

P roof. At the beginning we shall show that M > 0. Namely, we have x'(a) ^ 0, x'(b) ^ 0, so for a certain <*e(a, b), x ”(Ç) < 0. From condition (2) it follows that

/( É )x ( É ) ^ > 0;

hence / (£) > 0 and therefore M = sup f (t) ^ / (£) > 0.

fe[a,b]

Let h(t) = x ( t ) +^ Mx ( t 0) t 2 for t e [ a , b ] . Then h"(t) ^ 0 on (a , b ) because h"(t) = x" (t) + Mx ( t 0) ^ x''(t)+f(t)x(t) ^ 0. Thus the function h(t) is convex on [a, b]. By the Jensen inequality,

h(t0) = x( t 0)-\—— x (t0)to M

= h b — a -10 M

t0 t0 — a a H—:--- b <

x(t 0)a2 +

b - t 0 b — a a M

h (a) +

b — a h(b) x(t 0)b2,

b — a 2 b — a 2

326 M. K i e r l a n c z y k

whence we obtain

M b — a ^ —— b2, or to - ( a + b ) t0 + a b + —- ^ 0;

b — a M

2

this completes the proof of (4).

In the proof of (5) we shall use the Taylor formula. We have

x{b) = x ( t 0) + ( b - t 0)x' (t0)+ ^ ?-° - - х"(в),

where 0e(to,b). But x'{t0) = 0 and x(b) = 0; hence x(t0)+ ( b - t 0)2

х"(в) = 0.

We know, however, that h" ^ 0. Then x"(0) > —Mx ( t 0), - — ° - x" (0)

^ — M ^ - x (t0). Hence

M 2 / 2

x{t0) - ^ - { b - t 0)2x( t 0) sÇ 0, t0 l i b - . Let us mention that (4) gives the estimation

(6) b — a ^ 2у/ 2/ у/ М.

From the paper of Z. OpiaU1) it is easy to obtain the similar estimation

(7) b — a ^ n/ y/ k,

where к = sup |/(f)|. In spite of the fact that 2 ^ / 2 < n. In general, how-

te[a,b]

ever, к > m, so both results (6) and (7) are incomparable.

(‘) Z. O p ia l, Sur une inégalité de C. de la Vallée Poussin, Ann. Polon. Math. 6 (1959), p. 87-91.