USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 297–301. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

On the Opial-Olech-Beesack inequalities ^{∗}

William C. Troy

Abstract

We investigate two integral inequalities. The first of these generalizes a result proved by Beesack [1] in 1962. We then use our inequality to generalize earlier results of Olech [3] and Opial [4] on a related problem.

1 Introduction.

In 1962 Beesack [1] proved the integral inequality

Theorem 1.1 Let b > 0. If y(x) is real, continuously differentiable on [0, b], and y(0) = 0,then

∫_{0}^{b} |y(x)y′(x)|dx ≤ \frac{b}{2} ∫_{0}^{b} |y′(x)|^{2} dx .  (1)

Equality holds only for y = mx where m is a constant.

Beesack used this result to obtain a simplification of proofs given earlier by Olech [3] and Opial[4] of the inequality

Theorem 1.2 Let c > 0, and let y(x) be real, continuously differentiable on [0, c], with y(0) = y(c) = 0. Then

∫_{0}^{c} |y(x)y′(x)|dx ≤ \frac{c}{4} ∫_{0}^{c} |y′(x)|^{2}dx.  (2)

Equality holds for the function satisfying y = x on [0, \frac{c}{2}], and y = c−x on [\frac{c}{2}, c].

In 1964 Levinson [2] gave a simpler proof of Theorem 1.1 His proof generalizes to the class of functions which are complex valued.

In this paper we have two goals. The first of these is to extend Levinson’s arguments and generalize Beesack’s inequality. This is done below in Theorem1.3. Our second goal is to use the results of Theorem1.3 and obtain a generalization of Theorem1.2. This is done in Theorem1.4.

∗Mathematics Subject Classifications: 41A44, 58E35.

Key words: Schwarz inequality, integral inequalities.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

Theorem 1.3 Let p > −1. (i) Let a and b be real with 0 ≤ a < b. If y(x) is continuously differentiable on [a, b], and y(a) = 0, then

∫_{a}^{b} t^{p}|y(t)y′(t)|dt ≤ \frac{1}{2\sqrt{p + 1}} ∫_{a}^{b}(b^{p+1} − at^{p})|y′(t)|^{2}dt.  (3)

(ii) Let b and c be real with 0 ≤ b < c. If y(x) is continuously differentiable on [b, c], and y(c) = 0, then

∫_{b}^{c} t^{p}|y(t)y′(t)|dt ≤ \frac{1}{2\sqrt{p + 1}} ∫_{b}^{c}(ct^{p} − b^{p+1})|y′(t)|^{2}dt.  (4)

Remarks:

(R1) If a = 0 then (3) reduces to

∫_{0}^{b} t^{p}|y(t)y′(t)|dt ≤ \frac{b^{p+1}}{2\sqrt{p + 1}} ∫_{0}^{b} |y′(t)|^{2}dt.  (5)

(R2) It remains an open problem to determine the sharpness of (3), (4) and (5).

We now state our second result which is a generalization of the inequality of Olech [3] and Opial [4] stated above in Theorem1.2.

Theorem 1.4 Let p > −1 and c > 0. If y(x) is continuously differentiable on [0, c], and y(0) = y(c) = 0, then

∫_{0}^{c} t^{p}|y(t)y′(t)|dt ≤ \frac{c^{p+1}}{4\sqrt{p + 1}} ∫_{0}^{c} |y′(t)|^{2}dt + \frac{c}{2\sqrt{p + 1}} ∫_{c_{p}}^{c}(t^{p} − c^{p})|y′(t)|^{2}dt, (6)

where c_{p} = \frac{c}{2^{1/(p+1)}} .

Remarks:

(R3) If p = 0 then (6) reduces to (2).

(R4) It remains an open problem to determine the sharpness of (6).

2 Proof of Theorem1.3

(i) We begin by defining the integral

I_{1} = ∫_{a}^{b} t^{p}|y(t)y′(t)|dt.

Then I_{1} can be written in the form

I_{1} = ∫_{a}^{b}(t^{\frac{p}{2}} (t − a)^{\frac{1}{2}} |y′(t)|)(t^{\frac{p}{2}} (t − a)^{\frac{−1}{2}} |y(t)|)dt,

and an application of the Schwarz inequality leads to

I_{1} ≤ I_{2}^{1/2}I_{3}^{1/2},  (7)

where

I_{2} = ∫_{a}^{b} t^{p}(t − a)|y′(t)|^{2}dt  (8)

and

I_{3} = ∫_{a}^{b} t^{p}(t − a)^{−1}|y(t)|^{2}dt .  (9)

Because y(a) = 0 and y(x) is continuously differentiable on [a, b], y(x) satisfies

|y(t)|^{2} = | ∫_{a}^{t} y′(η)dη|^{2},  a ≤ t ≤ b.  (10)

A further application of Schwarz’s inequality to (10) gives

|y(t)|^{2} ≤ (t − a) ∫_{a}^{t} |y′(η)|^{2}dη,  a ≤ t ≤ b.  (11)

Combining (9)and (11), we obtain

I_{3} ≤ ∫_{a}^{b} t^{p} ∫_{a}^{t} |y′(η)|^{2}dηdt.  (12)

Reversing the order of integration in (12) gives

I_{3} ≤ \frac{b^{p+1}}{p + 1} ∫_{a}^{b} |y′(t)|^{2}dt − \frac{1}{p + 1} ∫_{a}^{b} t^{p+1}|y′(t)|^{2}dt.  (13)

Next, we recall a well known result: if A ≥ 0, B ≥ 0 and λ > 0, then

(AB)^{1/2} ≤ \frac{λ}{2}A + \frac{1}{2λ}B.  (14)

We now combine (7), (8), (13) and (14), to obtain

I_{1} ≤ \frac{1}{2}(λ − \frac{1}{λ(p + 1)}) ∫_{a}^{b} t^{p+1}|y′(t)|^{2}dt + ∫_{a}^{b}(\frac{b^{p+1}}{2λ(p + 1)} − \frac{aλt^{p}}{2} )|y′(t)|^{2}dt,

(15)

where λ is any positive number. Setting λ = \frac{1}{\sqrt{p+1}} in (15), we obtain (3). This completes the proof of part (i). (ii) The proof of part (ii) follows the method used above. We give the details for the sake of completeness. Thus, we define

I_{4} = ∫_{b}^{c} t^{p}|y(t)y′(t)|dt.

Then I_{4} can be written in the form

I_{4} = ∫_{b}^{c}(t^{\frac{p}{2}} (c − t)^{\frac{1}{2}} |y′(t)|)(t^{\frac{p}{2}} (c − t)^{\frac{−1}{2}} |y(t)|)dt,

and once again an application of the Schwarz inequality leads to

I_{4} ≤ I_{5}^{1/2}I_{6}^{1/2},  (16)

where

I_{5} = ∫_{b}^{c} t^{p}(c − t)|y′(t)|^{2}dt  and  I_{6} = ∫_{b}^{c} t^{p}(c − t)^{−1}|y(t)|^{2}dt. (17)

Because y(c) = 0 and y(x) is continuously differentiable on [b, c], y(x) satisfies

|y(t)|^{2} = | ∫_{t}^{c} y′(η)dη|^{2},  b ≤ t ≤ c.  (18)

An application of Schwarz’s inequality to (18) gives

|y(t)|^{2} ≤ (c − t) ∫_{t}^{c} |y′(η)|^{2}dη,  b ≤ t ≤ c.  (19)

Combining (17) and (19), we obtain

I_{6} ≤ ∫_{b}^{c} t^{p} ∫_{t}^{c} |y′(η)|^{2}dηdt.  (20)

Reversing the order of integration in (20) leads to

I_{6} ≤ \frac{1}{p + 1} ∫_{b}^{c}(t^{p+1} − b^{p+1})|y′(t)|^{2}dt.  (21)

Next, we combine (16), (17) and (21), and apply (14) to arrive at

I_{4} ≤ \frac{1}{2}(\frac{1}{λ(p + 1)} − λ) ∫_{b}^{c} t^{p+1}|y′(t)|^{2}dt + ∫_{b}^{c}(\frac{cλt^{p}}{2} − \frac{b^{p+1}}{2λ(p + 1)})|y′(t)|^{2}dt,

(22)

where λ is any positive number. Setting λ = \frac{1}{\sqrt{p+1}} in (22), we obtain (4). This completes the proof of part (ii).

3 Proof of Theorem1.4

Let b be any positive number satisfying 0 < b < c. Then, from (3) and (4) we obtain

∫_{0}^{c} t^{p}|y(t)y′(t)|dt = ∫_{0}^{b} t^{p}|y(t)y′(t)|dt + ∫_{b}^{c} t^{p}|y(t)y′(t)|dt
≤ \frac{b^{p+1}}{2\sqrt{p + 1}} ∫_{0}^{b} |y′(t)|^{2}dt + \frac{1}{2\sqrt{p + 1}} ∫_{b}^{c}(ct^{p} − b^{p+1})|y′(t)|^{2}dt

This inequality can now be written in the form

∫_{0}^{c} t^{p}|y(t)y′(t)|dt = \frac{b^{p+1}}{2\sqrt{p + 1}} ∫_{0}^{c} |y′(t)|^{2}dt + \frac{1}{2\sqrt{p + 1}} ∫_{b}^{c}(ct^{p} − 2b^{p+1})|y′(t)|^{2}dt

(23)

Setting b = c_{p} = c/2^{1/(p+1)} in (23), we obtain (6) and Theorem1.4 is proved.

References

[1] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc., 104 (1962), pp. 470-475.

[2] N. Levinson, On an inequality of Opial and Beesack, Proc. Amer. Math. Soc.,15 (1964), pp. 565-566.

[3] C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math., 8 (1960), pp. 61-63.

[4] Z. Opial, Sur une inegalite, Ann. Polon. Math., 8 (1960), pp. 29-32.

William C. Troy Department of Mathematics University of Pittsburgh Pittsburgh, Pa. 15260 e-mail: troy@math.pitt.edu website: www.math.pitt.edu/∼troy