GENERALIZED SIMPSON TYPE INTEGRAL INEQUALITIES MEHMET ZEKI SARIKAYA, SAKINE BARDAK, AND YONCA BAKI¸ S Abstract. In this paper, we have established some generalized Simpson type inequalities for convex functions. Furthermore, inequalities obtained in special case present a re…nement and improvement of previously known results.
1. INTRODUCTION The following inequality is well known in the literature as Simpson’s inequality. Theorem 1. Let f : [a; b] ! R be a four times continuously di¤ erentiable mapping on (a; b) and f (4) 1 = sup f (4) (x) < 1:Then, the following inequality holds: 1 f (a) + f (b) + 2f 3 2
a+b 2
1 b
a
Z
b
f (x) dx
a
1 f (4) 2880
1
(b
a)4 :
For recent re…nements, counterparts, generalizations and new Simpson’s type inequalities, see ([1][21]). In [2], Dragomir et. al. proved the following some recent developments on Simpson’s inequality for which the remainder is expressed in terms of lower derivatives than the fourth. Theorem 2. Suppose f : [a; b] ! R is a di¤ erent di¤ erentiable mapping whose derivative is continuous on (a; b) and f 0 2 L[a; b]: Then the following inequality Z b a+b b a 0 1 f (a) + f (b) 1 + 2f kf k1 (1.1) f (x) dx 3 2 2 b a a 3 holds, where kf 0 k1 =
Rb a
jf 0 (x)j dx:
The bound of (1.1) for L-Lipschitzian mapping was given in [2] by inequality was obtained in [2].
5 L(b 36
a): Also, the following
Theorem 3. Suppose f : [a; b] ! R is an absolutely continuous mapping on [a; b] whose derivative belongs to Lp [a; b]: Then the following inequality holds, Z b 1 f (a) + f (b) a+b 1 + 2f f (x) dx 3 2 2 b a a 1 2q+1 + 1 6 3(q + 1) where
1 p
+
1 q
1 q
(b
1
a) q kf 0 kp
=1
The aim of this paper is to establish new generalization of Simpson’s type inequalities for the class of functions whose derivatives in absolute value are convex. Furthermore, inequalities obtained in specialcase present a re…nement and improvement of previously known results. Key words and phrases. Simpson type inequalities, convex functions, integral inequalities. 2010 Mathematics Sub ject Classi…cation 26D07, 26D10, 26D15, 26A33. 1
2
M EHM ET ZEKI SARIKAYA, SAKINE BARDAK, AND YONCA BAKI¸ S
2. Main Results Let’s begain start the following Lemma which helps us to obtain the main results: Lemma 1. Let f : I R ! R be an absolutely continuous an I such that f 0 2 L1 [a; b] with a; b 2 I, a < b: Then the following equality holds: 2 Z 1 0 (b w) 1 t 1+t 1 t f w+ b dt 2 (b a) 0 3 2 2 2 2 Z 1 0 t (w a) 1 1+t 1 t + f w+ a dt 2 (b a) 0 2 3 2 2 1 1 f (w) + (b 6 3 (b a)
=
1 b where w = ha + (1
a
Z
w+b 2
w) f
+ (w
w+b 2
f (u) du a+w 2
h)b; h 2 [0; 1].
Proof. If we apply integration by parts, we have Z 1 1 t 1 t 1+t I1 = w+ b dt f0 3 2 2 2 0 =
=
a+w 2
a) f
2 b
w 2
6 (b
w)
1 3
t 2
f (w) +
f
1+t 1 t w+ b 2 2 2
3 (b
w)
w+b 2
f
1 0
1 b
w
2 (b
2
w)
Z
1
1+t 1 t w+ b dt 2 2
f
0
Z
w+b 2
f (u) du
w
and similary methots we can show that Z 1 t 1+t 1 1 t I2 = f0 w+ a dt 2 3 2 2 0 =
2 2 f (w) + f 6 (w a) 3 (w a)
a+w 2
2 2
(w
a)
Z
w
f (u) du: w+a 2
Therefore, we obtain the following result 2
2
(b w) (w a) I1 + I2 2 (b a) 2 (b a) b w b w a+w = f (w) + f 6 (b a) 3 (b a) 2 w a a+w w a f (w) + f + 6 (b a) 3 (b a) 2 =
1 1 f (w) + (b 6 3 (b a) Z w+b 2 1 f (u) du b a a+w 2
which completes the proof.
w) f
w+b 2
1 b
a 1 b
+ (w
Z
w+b 2
f (u) du
w
a
Z
w
f (u) du a+w 2
a) f
a+w 2
GENERALIZED SIM PSON TYPE INTEGRAL INEQUALITIES
3
Theorem 4. Let f : I R ! R be a di¤ erentiable mopping on I such that f 0 2 L1 [a; b] with 0 a; b 2 I, a < b: If jf j is a convex function on [a; b] then the following inequality holds: 1 1 f (w) + (b 6 3 (b a) Z
1
w) f
w+b 2
+ (w
a+w 2
w+b 2
f (u) du b a a+w 2 h i h 2 61 (b w) + (w a)2 jf 0 (w)j + 29 (b 64 (b
where w = ha + (1
a) f
2
w) jf 0 (b)j + (w
a)
i a)2 jf 0 (a)j
h)b; h 2 [0; 1].
Proof. From Lemma 1 and by using the convexity of jf 0 j ; we get 1 1 f (w) + (b 6 3 (b a) Z w+b 2 1 f (u) du a+w b a 2 2
(b w) 2 (b a)
Z
1
0
2
Z
t 2
w+b 2
+ (w
a) f
a+w 2
1+t 0 1 t 0 jf (w)j + jf (b)j dt 2 2
1
t 1 1+t 0 1 t 0 jf (w)j + jf (a)j dt 2 3 2 2 0 Z 1 Z 1 2 2 1 t (b w) 1 t (b w) 0 0 jf (w)j (1 + t)dt + jf (b)j (1 t)dt = 4 (b a) 3 2 4 (b a) 3 2 0 0 Z 1 Z 1 2 2 1 (w a) 1 (w a) t t 0 0 jf (w)j (1 + t)dt + jf (a)j (1 t)dt: + 4 (b a) 2 3 4 (b a) 2 3 0 0 +
(w a) 2 (b a)
1 3
w) f
By simple computation, we have Z
(2.1)
1
t 2
1 61 (1 + t)dt = 4 2 3 3 2
1
t 2
1 (1 3
0
and
Z
(2.2)
0
t)dt =
29 : 34 22
This completes the proof. Remark 1. If we choose h = 0 in Theorem 4, then we have w = b and Z b 1 1 a+b 1 (2.3) f (b) + f f (x) dx 6 3 2 b a a+b 2 61 jf 0 (b)j + 29 jf 0 (a)j 64 Similarly, if we choose h = 1 in Theorem 4, then we have w = a and Z a+b 2 1 1 a+b 1 (2.4) f (a) + f f (x) dx 6 3 2 b a a (b
a)
(b
a)
61 jf 0 (a)j + 29 jf 0 (b)j : 64
4
M EHM ET ZEKI SARIKAYA, SAKINE BARDAK, AND YONCA BAKI¸ S
Combining the inequalities (2.3) and (2.4), we have 1 f (a) + 4f 6 5 (b 72
a+b 2
1
+ f (b)
b
a
Z
b
f (x) dx
a
a) [jf 0 (a)j + jf 0 (b)j]
which is given by Sarikaya et al. in [13, (for s = 1)]. Theorem 5. Let f : I R ! R be a di¤ erentiable mopping on I such that f 0 2 L1 [a; b] where 0 q a; b 2 I, a < b: If jf j is a convex function on [a; b], q > 1; then the following inequality holds: (2.5)
f (w) 1 + (b 6 3 (b a) 1 b
a
Z
where w = ha + (1
w+b 2
+ (w
a+w 2
a) f
b+w 2
f (x) dx a+w 2
1 12 (b a) ( (b
w) f
1 p
1 + 2p+1 3 (p + 1)
q
2
w)
1 q
q
3 jf 0 (w)j + jf 0 (b)j 4
q
2
+ (w
a)
q
3 jf 0 (w)j + jf 0 (a)j 4
h)b; h 2 [0; 1].
Proof. From Lemma 1 and by Hölder’s inequality we get (2.6)
f (w) 1 + (b 6 3 (b a) 1 b
a
Z 2
(b w) 2 (b a)
+ (w
a+w 2
f (x) dx a+w 2
Z
1
1 3
0
(w a) 2 (b a)
Z
1
0
Z
2
(b w) 2 (b a)
1
0
2
(w a) + 2 (b a)
Z
t f0 2
1+t 1 t w+ b 2 2
t 2
1 f0 3
1 3
t 2
1
0
1+t 1 t w+ a 2 2 Z
1 p
p
1
f
0
t 2
1 3
p
1 3
t 2
p
1 p
Z
0
Z
0
1
dt =
dt
dt
1+t 1 t w+ b 2 2
0
1
f
0
1 t 1+t w+ a 2 2
By simple computation, we have
(2.7)
a) f
b+w 2
2
+
w+b 2
w) f
2 1 + 2p+1 : 6p+1 (p + 1)
q
1 q
q
1 q
:
1 q
)
GENERALIZED SIM PSON TYPE INTEGRAL INEQUALITIES
5
q
Since jf 0 j is a convex on [a; b], we know that for t 2 [0; 1] Z 1 q 1+t 1 t f0 (2.8) dt w+ b 2 2 0 Z 1 1+t 0 1 t 0 q q jf (w)j + jf (b)j dt 2 2 0 q
q
3 jf 0 (w)j + jf 0 (b)j 4
= and
Z
(2.9)
1
0
Z
1
q
1+t 1 t w+ a 2 2
f0
dt
1+t 0 1 t 0 q q jf (w)j + jf (a)j dt 2 2
0
q
q
3 jf 0 (w)j + jf 0 (a)j 4 Using the equations (2.7)-(2.9) in (2.6), then we obtain the (2.5). =
Remark 2. If we choose h = 0 in Theorem 5, then we have w = b and Z b f (b) 1 a+b b a 1 + 2p+1 1 (2.10) + f f (x) dx 6 3 2 b a a+b 12 3 (p + 1) 2
1 p
Similarly, if we choose h = 1 in Theorem 5, then we have w = a and Z a+b 2 f (a) 1 a+b 1 b a 1 + 2p+1 (2.11) + f f (x) dx 6 3 2 b a a 12 3 (p + 1)
q
q
jf 0 (a)j + 3 jf 0 (b)j 4
1 p
q
1 q
:
q
3 jf 0 (a)j + jf 0 (b)j 4
1 q
:
Combining the inequalities (2.10) and (2.11), we have 1 f (a) + 4f 6 b
1 + 2p+1 3 (p + 1)
a 12 "
a+b 2
q
+ f (b)
1 b
a
Z
b
f (x) dx
a
1 p
1 q
q
jf 0 (a)j + 3 jf 0 (b)j 4
q
q
3 jf 0 (a)j + jf 0 (b)j + 4
1 q
#
which is proved by Sarikaya et al. in [13, (for s = 1)]. Theorem 6. Let f : I R ! R be a di¤ erentiable mopping on I such that f 0 2 L1 [a; b] where 0 q a; b 2 I, a < b: If jf j is a convex function on [a; b], q 1; then the following inequality holds: f (w) 1 + (b 6 3 (b a) 1 b
a
Z
+ (w
w+b 2
+ (w
a) f
a+w 2
b+w 2
f (x) dx a+w 2
1 2 (b
w) f
a) 2
a)
1
1 q
"
q
q
61 jf 0 (w)j + 29 jf 0 (b)j (b w) 34 23 1# q q 61 jf 0 (w)j + 29 jf 0 (a)j q 34 23
5 36
2
1 q
6
M EHM ET ZEKI SARIKAYA, SAKINE BARDAK, AND YONCA BAKI¸ S
where w = ha + (1
h)b; h 2 [0; 1].
Proof. From Lemma 1 and by using power main inequality, we get f (w) 1 + (b 6 3 (b a) 1 b
a
Z
=
+ (w
a+w 2
a) f
b+w 2
f (x) dx a+w 2
Z
2
(b w) 2 (b a)
w+b 2
w) f
1
1 3
0
2
(w a) + 2 (b a)
Z
t 2
1
t 2
0
Z
1 q
1
1
1 3
0
1 3
1
Z
1 q
1
t 2
0
1+t 1 t w+ b 2 2
t f0 2
1 q
q
dt 1 q
q
1+t 1 t w+ a 2 2
1 f0 3
dt
:
q
Since jf 0 j is a convex function on [a; b] ; by using the equalities (2.1) and (2.2), we obtain f (w) 1 + (b 6 3 (b a) Z
1 b
a
f (x) dx 1
1 q
1
1 3
5 36 Z
q
jf 0 (w)j 2
0
1
2
(w a) + 2 (b a)
5 36 Z
q
jf 0 (w)j 2
1
0
Z
1
0
1 3
t (1 2
t 2
1 (1 3
t) dt
1 q
q
1 jf 0 (a)j (1 + t) dt + 3 2 1 q
"
Z
1
0
2
a)
2
(b
w)
which completes the proof. Remark 3. If we choose h = 0 in Theorem 6, then we have w = b and (2.12)
1 q
1 q
t) dt
61 29 q q jf 0 (w)j + 4 3 jf 0 (b)j 34 23 3 2 1) q 61 29 q q 0 0 jf (w)j + 4 3 jf (a)j 34 23 3 2
5 36
a)
q
t jf 0 (b)j (1 + t) dt + 2 2
t 2 1
1
+ (w
a+w 2
a) f
a+w 2
2
2 (b
+ (w
b+w 2
(b w) 2 (b a)
=
w+b 2
w) f
f (b) 1 + f 6 3
a+b 2 1
b
a 2
5 36
1 q
1 b
a q
Z
b
f (x) dx a+b 2
q
29 jf 0 (a)j + 61 jf 0 (b)j 34 23
1 q
:
1 q
GENERALIZED SIM PSON TYPE INTEGRAL INEQUALITIES
7
If we choose h = 1 in Theorem 6, then we have w = a and f (a) 1 + f 6 3
(2.13)
a+b 2 1 q
1
b
a 2
5 36
1 b
a
Z
a+b 2
f (x) dx
a
q
1 q
q
61 jf 0 (a)j + 29 jf 0 (b)j 34 23
:
Combining the inequalities (2.10) and (2.11), we have 1 f (a) + 4f 6 1
b
a+b 2 1 q
+ f (b)
"
q
1 b
a
Z
b
f (x) dx
a q
29 jf 0 (a)j + 61 jf 0 (b)j 2 34 23 1# q q 61 jf 0 (a)j + 29 jf 0 (b)j q 34 23 a
5 36
1 q
which is given by Sarikaya et al. in [13, (for s = 1)]. References [1] M. Alomari, M. Darus and S.S. Dragomir, New inequalities of Simpsonís type for sconvex functions with applications, RGMIA Res. Rep. Coll., 12 (4) (2009), Article 9. [2] S.S. Dragomir, R.P. Agarwal and P. Cerone, On Simpsonís inequality and applications, J. of Inequal. Appl., 5(2000), 533-579. [3] S.S. Dragomir. On Simpson’s quadrature formula for di¤ erentiable mappings whose derivatives belong to lp spaces and applications. J. KSIAM, 2 (1998), 57–65. [4] S.S. Dragomir, On Simpson’s quadrature formula for Lipschitzian mappings and applications Soochow J. Mathematics, 25 (1999), 175–180. [5] T. Du, Y. Li and Z. Yang, A generalization of Simpson’s inequality via di¤ erentiable mapping using extended (s; m)-convex functions, Applied Mathematics and Computation 293 (2017) 358–369 [6] S. Hussain andS. Qaisar, More results on Simpson’s type inequality through convexity for twice di¤ erentiable continuous mappings. Springer Plus (2016), 5:77. [7] B.Z. Liu, An inequality of Simpson type, Proc. R. Soc. A, 461 (2005), 2155-2158. [8] J. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991. [9] J. Pecaric., and S. Varosanec, A note on Simpson’s inequality for functions of bounded variation, Tamkang Journal of Mathematics, Volume 31, Number 3, Autumn (2000), 239–242. [10] S. Qaisar, C.J. He, S. Hussain, A generalizations of Simpson’s type inequality for di¤ erentiable functions using ( ; m)-convex functions and applications, J. Inequal. Appl. 2013 (2013) 13. Article 158. [11] H. Kavurmaci, A. O. Akdemir, E. Set and M. Z. Sarikaya, Simpson’s type inequalities for m and ( ; m)geometrically convex functions, Konuralp Journal of Mathematics, 2(1), pp:90-101, 2014. [12] M. E. Ozdemir, A. O. Akdemir and H. Kavurmac¬, On the Simpson’s inequality for convex functions on the coOrdinates, Turkish Journal of Analysis and Number Theory. 2014, 2(5), 165-169. [13] M. Z. Sarikaya, E. Set and M. E. Ozdemir, On new inequalities of Simpson’s type for s-convex functions, Computers and Mathematics with Applications 60 (2010) 2191–2199. [14] M.Z. Sarikaya, E. Set, M.E. Özdemir, On new inequalities of Simpson’s type for convex functions, RGMIA Res. Rep. Coll. 13 (2) (2010) Article2. [15] M. Z.Sarikaya, E. Set and M. E. Ozdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, Journal of Applied Mathematics, Statistics and Informatics , 9 (2013), No. 1. [16] M.Z. Sar¬kaya, T. Tunç and H. Budak, Simpson’s type inequality for F -convex function, Facta Universitatis Ser. Math. Inform., Vol. 32, No 5 (2017), 747–753. [17] E. Set, M. E. Ozdemir and M. Z. Sarikaya, On new inequalities of Simpson’s type for quasi-convex functions with applications, Tamkang Journal of Mathematics, 43 (2012), no. 3, 357–364. [18] E. Set, M. Z. Sarikaya and N. Uygun, On new inequalities of Simpson’s type for generalized quasi-convex functions, Advances in Inequalities and Applications, 2017, 2017:3, pp:1-11. [19] K. L. Tseng, G. S. Yang and S.S. Dragomir, On weighted Simpson type inequalities and applications Journal of mathematical inequalities, Vol. 1, number 1 (2007), 13–22.
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M EHM ET ZEKI SARIKAYA, SAKINE BARDAK, AND YONCA BAKI¸ S
[20] N. Ujevic, Double integral inequalities of Simpson type and applications, J. Appl. Math. Comput., 14 (2004), no:1-2, p. 213-223. [21] Z.Q. Yang, Y.J. Li andT. Du, A generalization of Simpson type inequality via di¤ erentiable functions using (s; m)convex functions, Ital. J. Pure Appl. Math. 35 (2015) 327–338. Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey E-mail address :
[email protected] E-mail address :
[email protected] E-mail address :
[email protected] 