CC BY
83Probl. Anal. Issues Anal. Vol.8(26), No 2, 2019, pp. 3-27 DOI: 10.15393/j3.art.2019.5750
UDC 517.51
George A. Anastassiqu
MULTIVARIATE IYENGAR TYPE INEQUALITIES FOR RADIAL FUNCTIONS
Abstract. Here we present a variety of multivariate Iyengar type inequalities for radial functions defined on the shell and ball. Our approach is based on the polar coordinates in RN, N ^ 2, and the related multivariate polar integration formula. Via this method we transfer well-known univariate Iyengar type inequalities and univariate author's related results into multivariate Iyengar inequalities.
Keywords: Iyengar inequality, Polar coordinates, radial function, Shell, Ball.
2010 Mathematical Subject Classification: 26D10, 26D15.
1. Background. In the year 1938, Iyengar [5] proved the following interesting inequality.
Theorem 1. Let f be a differentiable function on [a,b\ and | f/ (x)| ^ M1. Then
b
j f (x) dx
а
2 (b - a) (f (a) + f (b))
Mi (b — a)2 4
(f (b) — f (a))2 4M1 '
(1)
In 2001, X.-L. Cheng [4] proved that Theorem 2. Let f G C2 ([a,b\) and If" (x)| ^ M2. Then
b
/i i
f (x) dx — 2(b — a) (f (a) + f (b)) + ^(b — a)2 (f/ (b) — f/ (a))
а
z M2 ^ (b — a) д 2
^ 24-(b — a) — l6MrAi-
(g) Petrozavodsk State University, 2019
where
Д, = f' (a) - f - f(a)) + f' (6)
(6 — a)
In 1996, Agarwal and Dragomir [1] obtained a generalization of (1):
Theorem 3. Let f : [a, 6] ^ R be a differentiable function such that for all x E [a, b] with M > m we have m ^ f/ (x) ^ M. Then
f (x) dx — ^ (b — a) (f (a) + f (b))
(f (b) — f (a) — m (b — a)) (M (b — a) — f (b) + f (a))
3)
2 (M — m)
In [7], Qi proved
Theorem 4. Let f : [a, b] ^ R be a twice differentiable function such that for all x E [a, b] with M > 0 we have |f" (x)| ^ M. Then
f (x) dx — (f (a) + f (b)) (b — a)+ (1 +gQ ) (f/(b) — f/ (a)) (b — a)2
where
Q2
M(b — a)3 , 2
« (24 ' f1 — 3Q-
f/ (a) + f/ (b) — 2 (!Ц=Ж ))
5)
M2 (b — a)2 — (f/(b) — f/(a))2 ■
In 2005, Zheng Liu, [6], proved the following:
Theorem 5. Let f : [a,b] ^ R be a differentiable function such that f is integrable on [a, b] and for all x E [a, b] with M > m we have
f/ (x) — f/ (a) f/ (b) — f/ (x)
m ---L2^2 < M and m < € M. (6)
xa
bx
b
b
2
Then
b
j f (x) dx
а
(f (a) + f (b)) 2
(b — a) +
1 + P2 8
(f/ (b) — f/ (a))(b
a)
2
1 + 3P 2
(m + M) (b - a)3
48
^ (M - m)(b - a)3 ^ — зр2^
where
P2
f' (a) + f' (b) - 2 (ref))
[M-m)2 (b - a)2 - (f' (b) - f' (a) - (*+*) (b - a))
2
2
(7)
(8)
We need
Remark. We define the ball B (0, R) = {x E RN : |x| < R} C RN N ^ 2, R > 0, and the sphere
SN-1 := {x E RN : |x| = 1}
where |-| is the Euclidean norm. Let dx be the element of surface measure on SN-1 and
xN = dx
sn-i
2n ^
f(f
be the area of SN 1.
For x E RN - {0} we can write uniquely x = rx, where r = | x| > 0 and x = X E SN-1, |x| = 1. Note that f dy = is the Lebesgue
B(0,R)
measure on the ball, that is the volume of B (0,R), which exactly is
П Nr RN
Vol (B (0,R)) = r( „ .
Following [8, pp. 149-150, exercise 6], and [9, pp. 87-88, Theorem 5.2.2] we can write for F : B (0, R) ^ R a Lebesgue integrable function that
R
F (x) dx
B(0,R)
F (rx) rN 1 dr\dx
SN-i 0
(9)
and we use this formula a lot.
Typically here the function f : B (0, R) ^ R is radial; that is, there exists a function g such that f (x) = g (r), where r = |x|, r E [0,R], Vx E B (0, R).
Remark. Let the spherical shell A := B (0, R2) - B (0, R1), 0 < R1 < R2, A C Rn, N ^ 2, x E A. Consider that f : A ^ R is radial; that is, there
exists g such that f (x) = g (r), r = |x|, r E [R1,R2\, V x E A. Here x can be written uniquely as x = ru, where r = |x| > 0 and ш = X E SN-1, |u| = 1, see ( [8, p. 149-150], and [2, p.421]), furthermore for F : A ^ R a Lebesgue integrable function we have that
J F (x) dx = J
A SN-1
#2
F (ru) rN 1dr)du
Ri
(10)
Here
Vol (A)
Un R - RN) N
n^ (r2n - RN)
г (N + 1)
(11)
In this article we derive multivariate Iyengar type inequalities on the shell and ball of RN, N ^ 2, for radial functions. Our results are based on Theorem 1 - Theorem 5 and several other results by the author.
2. Main Results. We present the following multivariate Iyengar type inequalities on the shell and the ball:
We start with
Theorem 6. Let the spherical shell A := B (0, R2) — B(0, R\), A C RN, N ^ 2, 0 < R1 < R2. Consider f : A ^ R that is radial, that is, there exists g such that f (x) = g (r), r = |x|, r E [R1 ,R2], Vx E A; x = ru, ш E SN-1. We assume that g E C1 ([Ru R2]).
Then
f (y) dy — (R2 — R1) [g (R1) RiN-1 + g (R2) RN-1]
,N-11 n 2
2 J г f N
A
N П 2
2Г (N
(g (s) sN-1)' 1„,[R.,R2|(R2 — R1)2— (g (R2) rN-1 — g (R1) R
N-1)2n
(g (s) sn-1)
I те, [Ri ,#2]
:i2)
N
Proof. Here g E C 1([R1, R2]) and clearly h(s) := g(s)sN 1 E C 1([R1, R2]),
N > 2. We set llh'l^,[R1 ,r2| = M. By (1) we get
#2
fh (x)
R1
2 (R2 — R1) (h (R1) + h (R2))
M (R - R1)2 (h (R2) - h (Ri))2
4
4M
(g (s)
N-i)
&>, [Ri,R2] 1
(R2 - Ri)2 (g (R2) RN-1 - g (Ri) RN-1)2
4|l (g (s) SN-1)' Щ [ЯьЯ2]
=: A. (13)
Equivalently, we have
R2
1
- A ^ g (s) sN-1ds - 2 (R2 - Ri) (g (Ri) RN-1 + g (R2) RN-1) < A,
Ri
R2
(14)
- A ^ If (su) sN 1ds -
Ri
Hence it holds
R2 - R1
1 (g (Ri) Rf-1 + g (R2) RN-1) < A.
(15)
R2 - R1
-Ajdu ^ J ^ J f (su)sN 1 d^jdu-
SN-1 SN-1 Ri
^ (g (Ri) RN-1 + g (R2) RN-1) J du ^ A J du, (16)
SN-1 SN-1
that is (by (10))
- aЛЬ) < ff (y) dy- (R - Ri) [g (Ri) Rf ■+
Г (f
A
n-n 2 , , 2n 2
+ g (R2) Rf-1] < A^. (17)
Therefore we get
A
f (y) dy - (R2 - Ri) [g (Ri) Rf-1 + g (R2) Rf-1] r^N
^ A
2n N
г (f
N 2 ,
(18)
N
The theorem is proved. □
We give
Corollary. (to Theorem 6) Let f : B(0, R) ^ R be radial, that is, there exists a function д such that f (x) = д (r), where r = |x|, r E [0,R], R > 0, V x E B (0, R); N ^ 2. We assume that д E C1 ([0, R]). Then
B(0,R)
f (y)dy - Rng(R)
N П 2
N П 2
2Г (N
s)s
N-П/
,R2 -
g2 (R) R2(n-1)
t°’R] || (g (s) sN-1)'
Proof. Similar to Theorem 6, use of (9). □
We also give
те, [0, R] _
(19)
Theorem 7. Let f : A ^ R be radial; that is, there exists g such that f (x) = g (r), r = |x|, r E [R1, R2], V x E A; x = ru, ш E SN-1, N ^ 2. We assume that g E C2 ([R1, R2]).
Then
f (y) dy -
A
(R2 - R1) (g (R1) RN-1 + g (R2) RN-1) +
+
(R2 - R1)2
4
s) sN 1)/(r2) - (g(s) sN 1) (R1))
n- 1)//
N П 2
s) s
I те, [Ri,R2]
12
(R2 - R1 )3 -
,8 ||(g(s) sN 1)// L, [r1,r2])
(R2 - R1)
A?
г (N
N П 2
N),
г (N
(20)
where
a (()N-1)/. 2 (g (R2) RN- - g (R) rn-1) ,(()N-1)/ .
A,:=(g (s) s ) (Ri)--sR R^------------------ + (g(s)s ) №).
(21)
Proof. Here g E C2([R1, R2]) and clearly h(s):= g(s)sN-1 E C2([R1, R2]),
N ^ 2. We set ||h//|Ц [R1,R2] = M. By (2) we get
R2
h (s) ds — ^ (R2 — Ri) (h (Ri) + h (R2)) + 8(R2-
Ri
where
- R1)2 (h'(R2) - h'(Ri))
/ M (R R )3 (R2 - R1) Д 2 (22)
< 24№ - Rl) - 16M Д1' (00)
Д1 = h (Ri) -2(h R h\Rl,) + h (R2)
(R2 — R1)
(03)
That is
R2
g (s) SN 1ds - (R2 - Rl) (g (R1) Rf 1 + g (R2) rN 1) +
Ri
(R2 - R1)2
s) sN 1)/ (R2) - (g (s) sN 1)/ (R1))
^ 24 (R2 R1) 16M
3 (R2 - R^2 =: ^
(04)
where
Д1 := (g (s) sN _^/ (R1) -
N _ n/,D, 2 (g (R2) RN -1 - g (R1) rN -4
(R2 - R1)
+ (g (s) sN ^ (R2) . (05)
Equivalently, we have R2
-ф < f f (sw) sN_ 1ds - <Rr-R1^^> »N_1 . -1'
(g (R1) RN_1 + g (R2) RN -1) +
Ri
+
(R2 - R1)2
s) sN 1)/ (R2) - (g (s) sN 1)/ (R^) ^ ф (26)
Hence it holds
- ф / dw ^ I I J f (sw) sN 1ds I dw-
SN-1 SN-1
(R2 - Ri)
2
, (R -
+ 8
[g (Ri) RN-i + g (R2) RN-i R^( (g(s)sN-i) (r2) -
+
- (g(s)s
N-i
J du ^ ф
du, (27)
SN-1 SN-1
that is (by (10))
2nN f
W'TTTNT < f (y) dy -
Г If
A
(R - Ri)
(g (Ri) RN-i + g (R2) RN-i) +
+
(R2 - Ri)2
s)sN i)/(r2) - (g(s) sN i)/(Ri))
2n ^ ^ ! 2n ^
r(N) ^ ф r(N) •
Therefore we get
(28)
f (y)d?y-
A
(R2 - R)( g(Ri)RN-i + g(R2)RN-i)+
+ №-Ry( (g(s)sN-i)/(r2)_
4
- (g(s)sN ^/ (Ri)
N П 2
Г (т
< Ф|n_L• (29)
Г( N )■
The theorem is proved. □
We give
Corollary. (to Theorem 7) Let f : B (0, R) ^ R be radial, that is, there exists a function g such that f (x) = g (r), where r = |x|, r G [0,R], R > 0, V x G B (0, R); N ^ 2. We assume that g G C2 ([0, R]). Then
f (У) dV -
B(0, R)
R2 ( ( )
Rn g (R) + T ((g (s) sN-i)/(R) -
n-n/,nл n 2
- (g (s) sN-i)/ (0))
JF( N)
II (g(s)
n- n//
12
[°,R] r3_____
N
__________
8|| (g(s)sN-1)"
-A*1
*2
I <ж, [0, R]
N П 2
Г (ft
(30)
where
д; := (g (s) sN-1)' (0) - 2g (R) RN-2 + (g (s) sN-1)' (R). (31)
If N > 2, then (g (s) sN-1)' (0) = 0.
Proof. Similar to Theorem 7, use of (9). □
We present
Theorem 8. Consider f : A ^ R that is radial; that is, there exists g such that f (x) = g (r), r = |x|, r G [R;, R2], V x G A; ж = ru, ш G SN-1, N ^ 2. We assume that g G C1 ([R;, R2]).
Then
A
f (y) dy - (R2 - R;) [g (R;) RN-1 + g (R2) RN-1]
N П 2
<
< (g (R2) R2N-1 - g (R1) RN-1 - m (R2 - R1)) X
w (m (R2 - R1) - g (R2) rN-1 + g (R1) rN-1) / nN
X (M - m) VT(f)
where M > m with
(32)
m < (g (s) sN ^' < M, V s G [R1, R2]. (33)
Proof. Here g G C1 ([R1, R2]) and clearly h (s) := g (s) sN-1 G C1 ([R1, R2]), N ^ 2. We assume here m < h' (s) < M, V s G [R1, R2] with M > m. By (3) we get
/h (s) ds
2(R2 -
R1) (h (R1) + h (R2))
<
(34)
Ri
< (h (R2) - h (R1) - m (R2 - R1)) (M (R2 - R1) - h (R2) + h (R1))
< 2 (M - m)
That is
R2
g (s) sN-1ds
Ri
2 (R2 - R1) (g (R1) rN-1 + g (R2) rN-1)
<
< (g (R2) RN-1 - g (Ri) RN-1 - m (R2 - R1)) x
(M (R2 - R1) - g (R2) RN-1 + g (R1) RN-1) = X 2 (M - m) '
Equivalently, we have
(35)
R2
- P < / g (s) sN 1ds
Ri
R2
- P < f (sw) sN 1ds
Ri
Hence it holds
R2 - R1
2
(g (R1) RN-1 + g (R2) RN-1) < P,
(36)
RdrR1) (g (R1) RN-1 + g (R) RN-1) < p.
(37)
-P
dw <
R2
f(sw)sN 1ds
dw—
S»-1 S^-1 Ri
(R 2 RM (g (R1) RN-1 + g (R2) RN-1) / dw < p J dw, (38)
SN-1 sN-1
that is (by (10))
- p< /f (y) dy - (R2 2 RM (g(R1)RN 1+
+ g (R2) r
\ 2n N 2n N
,N-Л 2n 2 < p 2n 2 (39)
Г(N) < PГ(N). ( )
Therefore we get
N N
П 2 2n 2
f (y) d,y - (r2 - R1) (g (R1) rN-1 + g (r2) rN-1) ^-7^ < p.
The theorem is proved. □
We give
A
Corollary. (to Theorem 8) Let f : B (0, R) ^ R be radial, that is, there exists a function g such that f (x) = g (r), where r = |x|, r E [0,R], R > 0, V x E B (0, R); N ^ 2. We assume that g E C1 ([0, R]). Then
f (y) dy - RN g (R)
N П 2
A
?N-
Г (N
(g (R) RN-1 - mR) (MR - g (R) R
n-1 w
П 2
г (N
N
(41)
(42)
(M - m)
where M > m with
m ^ (g (s) sN-1 ) ^ M, V s E [0, R].
Proof. Similar to Theorem 8, use of (9). □
We continue with
Theorem 9. Let f : A ^ R be radial; that is, there exists g such that f (x) = g (r), r = |x|, r E [R1,R2], V x E A; x = ru, ш E SN-1, N ^2. We assume that g E C2 ([R1,R2]). We call
Ml := У (g (s) sN"
Then
I те, [R,, R2]'
f (y)dy - [ (g№)<-1 + g№)R2N-1) № - R1) + (1 +Q1) x
A
x ((g(s) sN ^' (r2) - (g(s) s
N- n'
1))(R2 - R1)2]
N
2 1 П 2
г (N
M1 (R2 - R1)3 12
(1 3Q1) , N), (43)
г (N
where
Q1
(g (s) sN-1)' (R1)+(g (s) sN-1)' (R2)- 2( J(K])KVR(,K1)<-1 'l2
Mf (R2 - R1)2- ((g (s) sN-1)' (R2) - (g (s) sN-1)' (R1))
Proof. Similar to the proof of Theorem 7 by the use of Theorem 4. □ We give
N
2
Corollary. (to Theorem 9) Let f : B (0, R) ^ R be radial, that is, there exists a function g such that f (x) = g (r), where r = |x|, r E [0,R], R > 0, V x E B(0, R); N ^ 2. We assume that g E C2 ([0, R]). We call
M := II (g (s) sN-1)" L, [0,r]■ Then
f (y)dy -
A
g(R)RN +
- (g(s) sN ^' (0)] R
(i + Q2) 4
[(g (s) sN-1)' (R) -
4
N П 2
Г If
< MR3 (1 - 3Q2)ANt. (45)
Г (N
where
Q2 :
(g (s) sN-1)' (0) + (g (s) sN-1)' (R) - 2g (R) RN-2 M2R2 - ((g (s) sN-1)' (R) - (g (s) sN-1)' (0))2
(46)
Proof. Similar to Corollary to Theorem 7. □
We present
Theorem 10. Here all as in Theorem 6 and M1 > m1. Assume that
m1 <
(g(s)
N1
(s) - (g(s) sN 1
s - R1
(R1)
< m1
and (
(g (s) sN-1
m1 i
for all s E [R1, R2]. Then
(r2) - (g (s) sN ^'(s) R2 - s
< M1,
(47)
(48)
f (y) dy - [(g (R,) RN-1 + g (R2) RN-1) (R2 - R1) +
A
442s
s) sN-1) (R2) - (g (s) sN-1) (R1)) (R2 - R1 )2 -
^—24 ^ (m1 + M1) (R2 - R1)"
N П 2
Г (f
i
< (M1 - m1) (R2 - R1)3 (1 - 3p2)
N П 2
Г (N
N
(49)
where
pi
(g (s) sN-g (Щ + (g (s) sN(Д2) - 2 (
N-1 \12
()2 (R2-R1)2- [(g(s)sN :1)'(R2)-(g(s)sN :1)'(R,)-( KR2-R1)] 2
(50)
Proof. Similar to Theorem 6 by the use of Theorem 5. □
We give
Corollary. (to Theorem 10) Here all as in Corollary to Theorem 6 and M2 > m2. Assume that
(g (s) sN-1)' (s) - (g (s) sN-1)' (0)
m 2 C
C M2
(51)
and
(g (s) sN 1)7 (R) - (g (s) sN 1)7 (s) Л/Г m2 C ---------------------------------------------- C M2,
Rs
for all s E [0, R]. Then
(52)
f (y) dy - [g (R) Rn+ ( (g (s) sN-1)7 (R) -
A
- (g(s) sN 1)7 (0))r2 -
, 1 1 3 p2,
2 1 +3 2 (m2 + M2)R3
C
24
(M2 - m2)R3
24
N_
П 2
Г (N
C
N_
о П 2
(1 - 3P22bvc^v, (53)
>2)______
2) Г (N),
where
P2
(g (s) sN:1)7 (0) + (g (s) sN:1)7 (R) - 2g (R) R
n- n7
N2
12
««)2 R2- [(g (s) sN:1)7 (R)-(g (s) sN:1)7 (0)- () R]
2
(54)
Proof. Similar to Corollary to Theorem 6, based on Theorem 5. □
We continue with some author’s results to be used later in this article:
Theorem 11. [3] Let n E N, f E ACn ([a, b]) (i. e. f(n 1) E AC ([a, b]), absolutely continuous functions). We assume that f(n) E L^ ([a, b]). Then
i) V t E [a, b]
n— 1
f (x) dx -Y^
1
k=0
(k + 1)! L
f(k) (a) (t-a)k+1 + (-1)kf(k) (b) (b-t)k+1
Ь<х,([а,Ь])
(n + 1)!
[(t - a)n+1 + (b - t)n+1] ,
(55)
ii) at t = Ap, the right hand side of (55) is minimized, and we get:
/n—1
f (x) dx - ^
k=0
1 (b - a)
(k + 1)! 2k+r
k+1
a) + (-1)kf(k) (6)]
LTO([a, b]) (b a)
n+1
iii
(n +1)! 2n
a) = f(k) (b) = 0, for all k = 0,1,..., n - 1, we obtain
b
(56)
f ( x) dx
Ьж([а, b]) (b a)
n+1
(n +1)! 2n
(57)
which is a sharp inequality,
iv) more generally, for j = 0,1, 2,..., N E N, it holds
/n—1
f (x) dx - ^
k=0
i /1 \ k+1
1 ( b — a\
(k + 1)! N
+ (-1)k (N - j)k+1 f(k) (b)]
[j k+1f(k) (a) +
/ 7 \ n+1
Ьж([а,Ь]) I b - a \
(V )'
X
v
(n + 1)!
x [jn+1 + (N - j)n+1] , (58) a) = f(k) (b) = 0, k = 1,..., n - 1, from (58) we get:
b
b - a N
b
LTO([a, b])
(n + 1)!
b — a N
n+1
[jn+1 + (N — j)"+1] ,
for j = 0,1,2,..., N e N,
vi) when N = 2 and j = 1, (59) turns to
b
J f (x) dx
a
b — a 2
(f (a) + f (b))
(59)
11^)^LTO([a,b]) (b — a)
(n + 1)! 2n
(60)
vii) when n =1 (without any boundary conditions), we get from (60) that
b
f ( x) dx
a
b — a 2
(f (a) + f (b))
(b — a)2
ro, [a, b] 4
(61)
a similar to Iyengar inequality (1).
Theorem 12. [3] Let f e ACn ([a, b]), n e N. Then i) V t e [a, b]
г n-1 1
J f (x)dx — (k +1); [f(k)(a) (1— a)‘+1 +
+ (—1)kf (k)(b) (b — t)k+1 ]
Li([a,b])
n!
[(t — a)n + (b — t)n] , (62)
ii) at t = a+b, the right hand side of (62) is minimized, and we get:
/n-1
f (x) dx — ^ k=0
1 (b — a)
(k + 1)! 2k+r
k+1
a) + (—1)kf(k) (b)]
Li([a,b]) (b — a)T
n!
n- 1
(63)
in
f(k) (b) = 0, for all k = 0,1,... ,n — 1, we obtain
f (x) dx
<
Li(M) (b — а)Г
П!
In— 1
(64)
which is a sharp inequality,
iv) more generally, for j = 0,1, 2,..., N E N, it holds
jf(x)dx — E (k^TT): (V) jk+1f'"'la
k=0
+ (—1)k (N—j )k+1 f (k)(b)
Li([a, b]) f b a
n:
N
[jn + (N—j)n] ,
(65)
v
a) = f(k) (b) = 0, k = 1, . . . ,n - 1, from (65) we get:
b
f(x)dx-
b — a N
[jf (a) + (N — j) f (b)]
Li([a,b])
n:
b — a N
[jn + (N — j )n]
for j = 0,1,2,..., N E N,
vi) when N = 2 and j = 1, (66) turns to
(66)
f (x)dx — (b 0 a) (f (a) + f (b))
Li([a,b]) (b — a)
n! 2n-- (67)
vii) when n =1 (without any boundary conditions), we get from (67) that
f (x)dx — ( b—- ) (f (a) + f (b))
7 llLi([a,b]) (b — a). (68)
b
b
n
n
b
n
b
Theorem 13. [3] Let f E ACn ([a, b]), n E N; p, q > 1 : p + 1 = 1, and f(n) E Lq ([a, b]). Then
i) V t E [a, b]
r n-1 1
J f (x) dx-g (j+jjy [f<k)(a)(t - a)k+1 + (-l)kf<k)(b)(6 - t)k+1 ]
Lq <[a,b])
(n - l)! (n + M (p (n - l) + l)
(t - a)n+p + (b - t)n+p
(69)
ii) at t = a++b, the right hand side of (69) is minimized, and we get:
/n-1
f (x) dx - g
k=0
l (b - a) (k + l)! 2k+r
k+1
a) + (-l)kf<k) (b)]
Lq <[a,b])
(b - a)n+p
,n-1 ’
p 2 q
iii
(n - l)! (n + (p (n - l) + l)p 2
= f <k) (b) = 0, for all k = 0, l,..., n - l, we obtain
(70)
f (x) dx
Lq <[a,b])
(b - a)n+ p
(n - l)! (n + M (p (n - l) + l)p 2
n- - ’
p 2 q
(71)
which is a sharp inequality,
iv) more generally, for j = 0, l, 2,..., N E N, it holds
/ n-1
f (x) dx - g
k=0
1 / 7 \ k+1
l b — a\
(k + l)! V N
+ (-l)k (N - j)k+1 f<k)(b)
Lq <[a b])
(n - l)! (n + p) (p(n - l) + l)
X
b - a\n+ p
X 1 - )
jn+p + (N - j)
n+1
v
a) = f <k) (b) = 0, k = l,..., n - l, from (72) we get:
b
b-a
N
ь
<
Lq ([a,b])
(n - 1)! (n + i) (p (n - 1) + 1)
7 \ П+1
b — a\ p
N
1 1 jn+ P + (N - j)n+ P
for j = 0,1,2,..., N e N,
vi) when N = 2 and j = 1, (73) turns to
(73)
f (x) dx - (b 0 a) (f (a) + f (b))
2
Lq([a,b])
(b — a)n+p
(n - 1)! (n + pj (p (n - 1) + 1)p 2
(74)
vii) when n =1 (without any boundary conditions), we get from (74)
that
f (x) dx -
ba
(f (a) + f (b))
'ILq(M]) (b - a)1+ p
1 + 1
p
i
2 P
(75)
1
P
b
q
b
Next, we extend Theorems 11-13 to the multivariate case over shells and balls for radial functions. The proving method is the same as in our earlier results of this article, as such we omit these next proofs.
We present (use of Theorem 11)
Theorem 14. Consider f : A ^ R which is radial; that is, there exists g such that f (x) = g (r), r = |x|, r e [R1,R2], V x e A; x = ru, ix e SN-1, N ^ 2. We assume that g (s) sN-1 e ACn ([R1,R2]) and (g (s) sN-1)(n) e L^ ([R1, R2]), n e N. Then
i) V t e [R1, R2]
(n-1 1
f (y) dy -i £
A
\ k=0 \k
(k + 1)!
N- 1)(k)
(g (s) sN-1)(k) (R1) (t - R1)k+1 +
2n N
+ (-1)k (g (s) sN-1)(k) (R2)(R2 - t)
N- 1)(n) M
k+1
2n N
г (f
<
s) s
lLTO([Ri,R2])
г (N
(n + 1)!
[(t - R1)n+1 + (R2 - t)n+1] , (76)
ii) at t = Rl+R2, the right hand side of (76) is minimized, and we get:
f (y) dy -\ ^
A
. k=0
1 (R2 - Ri))
(k + 1)! 2k
(g (s) $N ^ ^ (Ri) +
+ (-1)k (g (s) sN-i)(k) (R2)
N-i)(n)
N П 2
г (f
N П 2
s) s
L^([Ri,R2]) (R2 - Ri)
n+i
Г (N
(n + 1)!
~)n— 1
(77)
iii) if (g(s)sN i)(k)(Ri) = (g(s)sN l)(k)(R2) = 0, for all k = 0,1,... ,n- 1, we obtain
/ (У) dy
A
N II (g (S) SN-1)(П) Hlm([Ri,R2|) (R2 - Ri)
П
n+i
Г <N
(n + 1)!
ni
(78)
which is a sharp inequality,
iv) more generally, for j = 0,1,2,..., N E N, it holds
//(y)dy - {Eyip. (‘ [jk+1 (g(s)sN-l)(k) (Ri) +
A k=0
0 N
+ (-1)k (N - j)k+1 (g (s) sN-i)(k) (Д*)] } ^
2n N
s> sN~1)(П) IL([RiWR'2 - Ri
Г (f
(n + 1)!
N
n+i
(jn+i + (N - j)n+i)
(79)
v) if (g (s) sN-i)(k) (Ri) = (g (s) SN-i)(k) (R2) = °, k = 1, . . . , n - 1,
from (79) we get:
A
f (y) dy - j (^R2 - R^ [jg (Ri) rN i +
2n N
Г (f
2n N
Г (f
x
" (g (S) ^1i^^') (“)”+1 (j“+1 + (N - j)”+I) ,
(80)
for j = 0,1,2,..., N E N,
vi) when N = 2 and j = 1, (80) turns to
f (y) dy - (R - Ri) (g (Ri) RN-1 + g Ш R-1)
?n-1) n 2
A
Г (N
nN "(g (s) 1 ^)"Ьто([Д1>Д2]) (Ri - Ri)n+1
Г (N) (n +1)! 9n-1
(81)
vii) when n =1 (without any boundary conditions), we get from (81)
that
N
f (y) dy - (R2 - R1) (g (R1) RN-1 + g (Ri) RN-1)
,N-1) П 2
A
N П 2
W
s) s
N- 1)'
(Ri - R1)
2 1 Г (f
i
LTO([R 1,Д2 ])
(82)
which is related to (12).
We present (use of Theorem 12)
Theorem 15. Consider f : A ^ R which is radial; that is, there exists g such that f (x) = g (r), r = |x|, r E [R1, R2], V x E A; x = ru, ш E SN-1, N ^ 2. We assume that g (s) sN-1 E ACn ([R1, R2]), n E N. Then i)
i) V t E [R1, R2]
( n-1
f (y)dy - \ Y
A
\ k=0
(k + 1)!
(g (s) sN-1)(k) (R1) (t - R1)fc+1 +
9 N
+ (-1)k (g (s) sN-1)(k) (Ri) (Ri - t)k+1] } ^
2nN "(g(s) sN-1)(n)
lLi([Ri,R2])
г in
n!
[(t - R1)n + (Ri - t)n] , (83)
N
1
ii) at t = R, the right hand side of (83) is minimized, and we get:
' П— 1
/ (y) dy -
A
. k=0
1 (R - Ri)k+1
(k + 1)! 2k
(g (s) $N ^ ^ (Ri) +
N
+ (-1)fc (g (s) sN ^ ) (R^ } r”^NJ
N П 2
rTf
s) s ) IIli([Ri,R2]) (R2 - R1)n
n
~>n-2
(84)
iii) if (g (s) sN 1)(k) (R1) = (g (s) sN 1)(k) (R2) = 0, for all k = 0,1,...
n 1, we obtain
/ (y) dy
A
N П 2
rTf
s) s ) 11 Li([Ri,R2]) (R2 - R1)n
n!
~)n— 2
(85)
which is a sharp inequality,
iv) more generally, for j = 0,1,2,..., N E N, it holds
J / (y) dy - {E (k:+^ (‘ [jk+1 (g (s) sN-1)(k) (R1) +
+ (-1)k (N - j)k'+1 (g (s) sN~1)<k> №)] }
2nN II (g W SN-1)(n) ||li([Ri,R21) ( R'2 - R1
N
Г (N
n!
N
(jn + (N - j)n)
(86)
v) if (g (s) sN-1)(k) (R1) = (g (s) sN-1)(k) (R2) = 0, k =1,...,n - 1, from (86) we get:
A
f (y) dy - j (^Ra N R^j jg (R1) rN 1+
2n N
Г
2n N
Г
x
|| (g (s) s
N -i
(n) ||
IIli([Ri,R2])
n!
R2 — Ri
N
n
(jn + (N — j)n),
for j = 0,1,2,..., N e N,
vi) when N = 2 and j = 1, (87) turns to
(87)
A
f (y) dy — (R2 — Ri) (g (Ri) RN-i + g (R2) RN-i)
N П 2
ПN || (g W 8N 1)(П) |Li([r„r2]) (R2 - Ri)’
Г (f
n!
n2
(88)
vii) when n =1 (without any boundary conditions), we get from (88)
that
f (y) dy — (R2 — Ri) (g (Ri) RN-i + g (R2) RN-i)
,n-1) n 2
A
Г If
N
2n N
WT
1 (g w
N-i
We present (use of Theorem 13)
Li([Ri,R2]) (R'2 Ri)
(89)
Theorem 16. Consider f : A ^ R which is radial; that is, there exists g such that f (x) = g (r), r = |x|, r e [Ri,R2], V ж e A; x = гш, ш e SN-i, N ^ 2. We assume that g (s) sN-i e ACn ([Ri,R2]) and (g (s) sN-i)(n) e Lq ([Ri,R2]), where p, q > 1 : i + i = 1, and n e N. Then
i) V t e [Ri, R2]
ni
f (y) dy -
(g (s) sN-i)(k) (Ri)(t — Ri)k+i +
A
. k=0
(k + 1)! L
9 N
+ (— 1)k (g (s) SN-i)(k) (Л2) (R2 — t)k+i] } ^
2n N
1 (g (s)
,N-i)(n)
I Lq ([Ri,R2 ])
Г (N
2) (n — 1)! n +
n — 1) + 1)
(t — Ri)n+1 + (R2 — t)
n+1
1
i
p
ii) at t = Rl+R, the right hand side of (90) is minimized, and we get:
A
f (y) dy
N П 2
(n-1 1 (R2 — R1)k+1 (g (s) sN-1)(k)
(k+1)! 2k
+ (—1)k (g (s) sN-1)(k) №)]} N П 2
Г (N)
II (g (s) sN-1)(n) 11 Lq([Rl, R2]) (R2 — - R1)n+
n — 1)! (n + (p (n — 1) + 1)p 2n 1 q
1
P
(Ri) +
5
(91)
iii) if (g(s)sN 1)(fc) (R1) = (g(s)sN 1)(fc) (R2) = 0, for all k = 0,1,...,
n — 1, we obtain
f (y) dy
A
ПN II (g (s) sN 1^) IIl,([Ri,R2]) (R2 — Rl)n+
Г(2) (n — 1)! ^n + (p (n — 1) + 1)p 2
1 0n-1-1 ’
p 2 q
(92)
which is a sharp inequality,
iv) more generally, for j = 0,1,2,..., N E N, it holds
f (y) dy £ 1
A
к k=0
(k + 1)^ N
R2 — R1
k+1
jk+1 (g (s) sN- 1)(k) (R1) +
0 N
+ (—1)k (N — j)k+1 (g (s) sN- 1)(k) (R2)] } ^
2n N
s) s
n- 1)(n)
I Lq ([Ri,R2])
Г (Ю (n — 1)! (n + 1 j (p (n — 1) + 1)
R2 — R1
N
n+1
X
X (jn+p + (N — j)
n+1
(93)
v) if (g(s)sN 1)(k) (R1) = (g(s) sN 1)(k) (r2) = 0, k = 1,...,n — 1,
from (93) we get:
Jf (y) dy — j (Ra—R1) jg (R1) rN 1+
+ (N - j) g (R) Rf-1]}
2n N
Г (f
2n N
^ (f)X
Г ( NN
X ■
(g (s) sf-1)'”’
I Lq([Ri,R2])
(n - 1)! (n + M (p (n - 1) + 1)
R-2 - Ri
N
n+
P jn+ P + (N - j)n+p
(94)
for j = 0,1,2,..., N e N,
vi) when N = 2 and j = 1, (94) turns to
f (y) dy - (R - Ri) (g (Ri) Rf-1 + g (R2) R
n-i) n 2
A
Г (f
N П 2
(g (s) Sf-1)(n)
Lq ([Ri,R2 ]) (R2 - R1 )
n+1
Г(2) (n - 1)! (n + pj (p (n - 1) + 1) 1 2
~)П— 1— -
q
(95)
vii) when n =1 (without any boundary conditions), we get from (95)
that
-
N
f (y) dy - (R2 - R1) (g (R1) Rf—1 + g (R2) Rf—1
A
N П 2
o- N 2 q n 2
(g (s)
W-1
iLq ([Ri,R2])
г (f
(1 + P)
(R2 - ВД
1 +1
We continue with
(96)
Remark. Theorems 14-16 can easily be converted to results for the ball B (0,R), R > 0. Their corresponding same assumptions will be for f : B (0, R) ^ R which is radial. All we need to do then is set R1 = 0 and R2 = R, and we get a plethora of interesting similar results for the ball that are simpler. Due to lack of space we omit this tedious task.
References
[1] Agarwal R. P., Dragomir S.S. An application of Hayashi’s inequality for differentiable functions. Computers Math. Applic., 1996, no. 6, pp. 95-99.
[2] Anastassiou G. A. Fractional Differentiation Inequalities.Research Monograph. Springer, New York, 2009.
[3] Anastassiou G. A. General Iyengar type inequalities. J. of Computational Analysis and Applications, 2020, vol. 28, no. 5, pp. 786-797.
[4] Xiao-Liang Cheng. The Iyengar-type inequality. Applied Math. Letters, 2001, no. 14, pp. 975-978.
[5] Iyengar K. S. K. Note on an inequality. Math. Student, 1938, no. 6, pp. 75 -76.
[6] Zheng Liu. Note on Iyengar’s inequality. Univ. Beograd Publ. Elektrotechn. Fak., Ser. Mat., 2005, no. 16, pp. 29-35.
[7] Qi F. Further generalizations of inequalities for an integral. Univ. Beograd Publ. Elektrotechn. Fak., Ser. Mat. 1997, no. 8, pp. 79-83.
[8] Rudin W. Real and Complex Analysis. International Student edition, Mc Graw Hill, London, New York, 1970.
[9] Stroock D. A Concise Introduction to the Theory of Integration. Third Edition, Birkhauser, Boston, Basel, Berlin, 1999.
Received November 20, 2018.
In revised form, March 15, 2019.
Accepted March 18, 2019.
Published online March 27, 2019.
Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A.
E-mail: [email protected]
