VNIV. BEOGRAD. PUDLo ELEKTROTEHN. FAK.
:Scr. Mat. Fiz. No 678 - No 71S (1980), 48-S3.
,687.
THE BEST CONSTANT IN SOME INTEGRAL
OF OPIAL TYPE*
INEQUALITIES
Gradimir V. Milovanovic and Igor Z. Milovanovic
Using some results from P. R. Beesack's paper [1) we will determine best
constants on Some concrete integral inequalities of Opial type (see (2)).
Let W;[a, b] be the space of all functions u which are locally absolutely
b
'continuous
Jru'2 dx < +
on (a, b) with
where x ~ r (x) is a given weight
00,
a
positive function.
Lemma 1. Let
-
00
-;;;'a<b-;;;,+ 00, and let p be positive and
continuous on
b
(a, b) with
r p(t)dt=P<
d
+
00.
Set r(x)=~.
Then, if 1 is an integral on
pW
b
.fa, b) with 1 (a) = 0 and
J rf'2 dx<
+ 00, we have
a
b
(1)
J (J
a
with
x
equality
"b
P (t) dt)
If' (x) 1 (x)
I dx
-;;;,
2:2
J
r (X)
If'
(x)12 dx,
a
a
if and only if 1 is given by
x
(2)
J
p(t) dt),
I(X)=Bsinh(;
a
where B is
,cothx=x.
arbitrary
constant
and
A
positive
. Presented May 15, 1980 by D W. BoYD.
48
solution
01 the
equation
The best constant
in some integral
49
inequalities
x
J
Proof. The hypotheses on I imply that I (x) = f' dt for a ~x<b.
If a
a
is finite this is equivalent to saying that I (a) = 0 and I is locally absolutely
x
continuous on [a, b). To prove (I), set u=yz,
where y(x)=sinh(VAoJp(t)dt)
a
for xE[a,
b) with Ao=A2/P2 (cothA=A,
A>O).
It is easy to verify that (rir = AoPY on (a, b).
Now, if a<~<~<b,
we have
~
ru'2 dx~
~
J
2
..
Jex
~
J r(y'
ryi zz' dx+
z)2dx
ex
x
Z2
= ryy'
I:
-
AO
~
x
( J pdt) (yz )21 : + 2 AOJ ( J pdt)
a
ex
a
If'
(x)
u' udx.
In the other hand
~
x
J ( aJ
p (t) dt)
~
If'
(x) I (x)
dx ~
J
ex
x
J (J
p (t) dt)
x
I( J If' (t) Idt) dx.
<X a
a
x
If we put u =
J II'
(1) Idt, then from above we obtain
a
~
x
J (J
(3)
ex
pet)
dt) If' (x) I (x) I dx
a
~
~--L
2)" 0
J
rf'2 (x) dx - --L r
2 ),,0 {
(L )-
x
AO
Y
ex
J
p dt I (X)2 ~ .
}
1
ex
a
It is easy to verify that I (~)2coth(VAo " P (t) dt
J
)- 0
as ~
-
a+ .
a
From (3), when ~_a+
the inequality (I).
and ~-b-,
and since cothA=A,
we obtain
The above proof shows that equality can hold in (I) only if z' = 0, or
By
for some constant B. Moreover, for any such f, we do have I (x) =
1=
x
=
Jf'
(t) dt, and IE W; [a, b] as one easily verifies, so that
I
is an admissible
a
function. By direct substitution
such f.
<4Publikacije
Elektrotehni~kol
fakulteta
one sees that
equality
does hold in (1) for
50
G. V. Milovanovic and I. Z. Milovanovic
The idea for the proof was obtained from the results in reference [1].
Also, this idea was used in paper [3].
The proof of Lemma 1 can be derived using a result from [4], changing
x
variables s = s(x) =
Jp (t) dt
in the integrals
in (1). Prof.
D. W. BOYD has
a
pointed to this fact.
REMARK.The approximative
1.1996786.
value of
the constant
A, with
seven
exact decimals, is
The following result can be proved similarly.
Lemma 2. Let -
00
~a<b~
+
00,
and let p be positive and continuous on
b
Jpdt
a
(a, b) with
P< +
1
Set r (x) = -.
p (x)
00.
If f is an integral on (a, b] with
b
Jrf'2 dx<
f (b) = 0, and
+ 00, we have
a
b
b
b
J(J
p (t) dt)
a
If'
(x) f (x)
x
Idx ~ 2:' J r (X) If'
a
(x)
dx,
12
b
with equality
if and only if f is given by f (x) = B sinh
(; Jp (t) dt ), where B
x
is arbitrary constant, and A positive solution of the equation coth x = x.
We now state:
.
b
Theorem 1. Let p be positive and continuous on (a, b) with
J pdt=P<
+
00.
a
Set r(x)=~,p (x) and let a<~<b.
Then for all FEW; [a, b] the inequality
b
(4)
J
~
s(x)IF'
(x)(F(x)-F(~))
I dX~2~'
max
a
holds, where
~
Jp (t) dt
s (x) =
x
x
JP (t) dt
(~~x~b),
~
and the number A is the same as in Lemmas
b
b
{(J pdtf, (J Pdty}J rF'(X)2
a
~
a
1 and 2.
dx,
51
The best constant in some integral inequalities...
Equality holds in (4) if and only if
!;
J
p (t) dt
x
B! h (q) sinh A
!;
(
f P (t) dt
J
a
x
P (t) dr
f
A!;
B'! h ( - q) sinh
b
f!; P (t)dt
(
J
where B!, B~, Bz are arbitrary constants, h (q) is Heaviside's function
!;
fp(t)dt=
b
fp(t)dt.
a
and q =
!;
Proof. Let a< ~ <b.
hand side of the equality
b
f
Applying
Lemma
!;
sex) IF' (x)(F(x)~F(~»
Idx= f (f
a
x
a
b
+.f
!;
2 and Lemma
1 on the right
!;
Idx
pet) dt)1 F'(x)(F(x)-F(~»
x
(f!; pet)
dt)1 F' (x)(F(x)-F(~)
I dx
we obtain (4). Notice that x 1-+f (x) = F(x) - F(~) has the required
at x = ~.
Corollary
1. Let functions
p and r satisfy
conditions
behaviour
as in Theorem
1 and let
be Juch that
!;
b
J
(5)
pdt =
a
f pdt.
!;
Then
b
J
(6)
S (x)
IF' (x) (F(x)
I
dx ~
- F(~»
p2
8 A2
J
r (x) F' (X)2 dx,
a
a
with equality if and only if
B! sinh (2/
!;
p (t) dt)
J
x
x
B~sinh
(2/ fp(t)dt)
!;
where B!, B~., Bz are arbitrary constants.
4.
b
'
.
~
52
G. V. .M:iJoyanovicand I. 2. Milovanovic
Proof. Since
with regard to (5), we have Q=~
2
P. Then, Corollary I follows from Theorem 1.
REMARK 1. Notice that (6) holds only for the
all ~.
From Theorem
~
single
such
(5) holds,
and
not
for
I it follows:
Corollary 2. For every FEW1
2
[- I, I] the inequality
1
1
JI
x (F(x)
-
F (0)) F' (x)
I
dx ~
2 ~2
-I
holds, with equality
that
-I
J
F' (X)2 dx,
if and only if
F(x)=H 1 +
(-I ~x~O),
(0 ~ x ~ I),
-Hsinh(Ax)
.
{
H' sinh (Ax)
where H, H', HI are arbitrary constants.
Theorem 2. Let $: R + -+ R + be a concave and nondecreasing function, and let
functions p, r, f satisfy the conditions as in Lemma 1.
Then the inequality
a
(7)
x
a
J(J
p(t) dt) $ (If' (x)f (x) I) dx~Mp
o
$
0
(2:;
p
A2
J
r(x)f'
(X)2 dx)
0
a
holds, where P and A are as in Lemma
I and Mp =
J (a - t) p (t) dt.
o
.~
Proof.
Let (,) (x) =
J pdt.
o
Using the JENSENintegral inequality for concave
function $, we have
a
a
!
a
I
(,)(x) $ ( f' (x) f (x) I) dx ~ Ci)(x) dx $
I
)'
(
J
(,) (x)f'
(x)f(x)
dx
0
J
o
a
(,)(x) dx
'
]
The best constant
in some integral
53
inequalities
i.e.
a
J
a
<u(x) tI> ( I j' (x) f (x) I ) dx ~ Mp tI>
(~p
o
Knowing that tI> is a nondecreasing
tain
J
<u(x) j' (x) f (x) dx
I
c
).
function applying Lemma
1 we ob-
(7).
Corollary 3. If we take
inequality (7) becomes
a
(8)
I
in Theorem 2 x ~ tI>(x) = xq (0 < q < 1), then the
x
J(J
a
p(t)dt)
If'
(x) f (x) Iqdx~M/-q
000
(2:2
nI
r(x)j'
We obtain
the inequality (l) when q -+ 1 in the
q-+O, the inequality (8) becomes an equality.
(X)2 dxt
inequality
(8). When
REFERENCES
1. P. R. BEESACK:Integral inequalities involving a function and its derivative. Amer.
Math. Monthly 78 (1971), 705-741.
2. D. S. MITRINOVIC(with P. M. VASIC): Analytic inequalities. Berlin-Heidelberg-New
York, 1970.
3. G. V. MILOVANOVIC
and I. Z. MILOVANOVIC:
On generalization of certain results by
A. Ostrowski and A. Lupa$. These Publications N2634-N2 677 (1979), 62---:69r
4. D. W. BoYD and J. S. W. WONG: An Extension of Opial's Inequality. J. Math.
Anal. AppI. 19 (1967), 100-102.
Faculty of Electronic Engineering
Department
of M.1thema1ics
18000 Nis, Yugoslavia
NAJBOLJA
KONSTANTA
U NEKIM INTEGRALNIM
OPIALOVOG
TIPA
NEJEDNAKOSTIMA
G. V. Milovanovic i I. Z. Milovanovic
Koriseenjem jednog rezultata P. R. BEEsAcKa([1]) u radu se odreduju najbolje konstante u nekim konkretnim nejednakostima OPIALovOgtipa (videti [2]).