certain inequalities involving the -fractional integral ...

Far East Journal of Mathematical Sciences (FJMS) © 2018 Pushpa Publishing House, Allahabad, India http://www.pphmj.com http://dx.doi.org/10.17654/MS103111879 Volume 103, Number 11, 2018, Pages 1879-1888

ISSN: 0972-0871

CERTAIN INEQUALITIES INVOLVING THE

(k, ρ) -FRACTIONAL INTEGRAL OPERATOR Gauhar Rahman, Kottakkaran Sooppy Nisar, Shahid Mubeen and Junesang Choi* Department of Mathematics International Islamic University Islamabad, Pakistan Department of Mathematics College of Arts and Science Prince Sattam Bin Abdulaziz University Riyadh 11991, Saudi Arabia Department of Mathematics University of Sargodha Sargodha, Pakistan Department of Mathematics Dongguk University Gyeongju 38066, Korea

Received: October 2, 2017; Accepted: January 7, 2018 2010 Mathematics Subject Classification: 26A33, 26D10, 26D15. Keywords and phrases: k -gamma function, k -fractional integral operator, (k, s ) -fractional integral operator, fractional integral inequalities, Grüss and Grüss type inequalities, Young’s inequality, weighted AM-GM inequality. *

Corresponding author

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G. Rahman, K. S. Nisar, S. Mubeen and J. Choi Abstract Since Grüss presented an interesting integral inequality in [9], its various generalizations and variants, which are called Grüss type inequalities, have been investigated. Very recently, certain Grüss type inequalities involving k, s  -fractional integral operator have been established. Motivated by the above mentioned works, we aim to present a Young’s type inequality and a weighted AM-GM type inequality involving the k,   -fractional integral operator. Some special cases of the main results here are also considered.

1. Introduction and Preliminaries During the last four decades, fractional calculus has attracted many researchers’ attention, due mainly to its potential of applications in a wide range of research fields from natural science to social science (see, e.g. [1, 2, 4, 13, 23, 25, 26, 28]). A number of fractional integral operators have been developed (see, e.g. [12]), which reduce to the classical Riemann-Liouville fractional integral operator

 I a f   x  

x 1  x  t  1 f t  dt   a



 x  a;    0 ,

(1.1)

where  denotes the familiar gamma function (see, e.g. [29, Section 1.1]). Katugampola [11] introduced and investigated the following fractional integral operator:

 a I x f   x  

1  x   x  t   1 t  1 f t  dt a  



    ,    0, a  x .

(1.2)

Obviously,  1a I x f   x    I a f   x . Here and in the following, let , ,   ,  be sets of complex numbers, real numbers, positive real numbers,

and positive integers, respectively, and let  0 :   0 and  0 :    0.

Certain Inequalities Involving the k,   -fractional …

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Díaz and Pariguan [5] introduced k -Pochhammer symbol defined (for x   and k    ) by

 x x  k   x  n  1k n    1 n  0

 x n, k : 

  x  nk   k k  x 

 n   0 .

(1.3)

Here k is the k -gamma function given by k  x  



k  x 1  t t e k dt 0

 x   0, k    

(1.4)

which satisfies the following identities: 1  x    x ,

k k  1, and k  x  k  xk  x .

(1.5)

For some properties, functions, and integral operators involving the k -gamma function, the reader may be referred, for example, to [14-16, 18-22, 26-28, 31]. Mubeen and Habibullah [20] introduced and investigated so-called k -fractional integral operator

 Ik f   x  

1



x

kk   0



 x  t  k 1 f t  dt

k, x    ;    0. (1.6)

Clearly,  I1 f   x    I 0 f   x  in (1.1). Sarikaya et al. [26] introduced and investigated so-called k, s  Riemann-Liouville fractional integral operator of order     

 ks I a

 s  11 k f  x  kk  

x

a



 x s 1  t s 1  k 1 t s f t  dt

a  x; k    , s  1    .

(1.7)

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G. Rahman, K. S. Nisar, S. Mubeen and J. Choi

By setting a  0 and s  1   in (1.7), we obtain 1



 x   k  k 1I 0 f   x    x  t   k 1 t  1 f t  dt :  k I  f   x  kk   0



 x, , , k    .

(1.8)

Grüss presented an interesting integral inequality (see [9], p. 236), which is asserted by Theorem A. Theorem A. Suppose that f , g : a, b   a  b  be two integrable functions such that   f  x    and   g  x    for all x  a, b, where , , ,  are constants. Then b b b 1 1 f  x  g  x  dx  f  x  dx g  x  dx ba a a b  a 2 a









1          . 4

(1.9)

Since then, various generalizations and variants of the Grüss’ inequality in Theorem A, which are called Grüss type inequalities, have been investigated (see, e.g. [6-8, 17, 24]). In particular, very recently, certain Grüss type inequalities involving k, s  -Riemann-Liouville fractional integral operator (1.7) have been established. Recall Young’s inequality a p bq   ab p q

a, b   0  ,

(1.10)

where p, q    with 1 p  1 q  1. We also recall the weighted AM-GM inequality pa  qb  a pb q

a, b   0 ; p, q    with p  q  1.

(1.11)

In this paper, we aim to establish a Young’s type inequality and a weighted AM-GM type inequality involving the k,   -fractional integral


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operator k I  in (1.8). Since the results presented here, being general, can be reduced to yield numerous inequalities, in particular, involving relatively simple integral operators. Only two special cases of the main results here are given. 2. Inequalities Involving the k,   -fractional Integral Operator k I  in (1.8) In this section, we present a Young’s type inequality and a weighted AM-GM type inequality involving the k,   -fractional integral operator   kI

in (1.8) which are asserted in Theorem 2.1 and Theorem 2.2,

respectively. Theorem 2.1. Let f , 1 ,  2 : 0,     be three integrable functions on 0,   such that 1t   f t   2 t 

t  0,  .

(2.1)

Also, let p, q    with 1 p  1 q  1. Then 

 x  k   1    I  2  f  p   x  pk   k    k 

 x  k   1    I  f  1 q   x   qk   k    k

  k I   2  f   x   k I   f  1   x   x, k, , ,     .

(2.2)

Proof. Substituting  2 t   f t  and f u   1 u  t , u     for a and b in (1.10), respectively, we have 1 1 2 t   f t  p   f u   1u q p q  2 t   f t   f u   1u 

t , u    .

(2.3)

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Multiplying both sides of the inequality (2.3) by 

2

  k  x





 t  k 1 x  u  k 1t  1u  1 k2k  k  

(2.4)

and integrating both sides of the resulting inequality with respect to the variables t and u, consecutively, from 0 to x, using (1.8) and the following easily derivable integral formula: x

0  x



t



  k 1 t  1dt



, , k, x     ,

k k  x 

(2.5) 

we obtain the desired inequality (2.2).

Theorem 2.2. Let f , 1, 2 : 0,     be three integrable functions on 0,   such that 1t   f t   2 t 

t  0,  .

(2.6)

Also, let p, q    with p  q  1. Then 

 x  k   p    I 2  f   x  k   k    k 

 x  k   q    I  f  1   x   k   k    k

  k I  2  f  p   x   k I   f  1 q   x 

 x, k, , ,     . (2.7)

Proof. Here, by using (1.11), a similar argument as in the proof of Theorem 2.1 will establish the result here. We omit the details.  3. Special Cases and Remarks The results presented in Section 2 can be reduced to yield those involving relatively simple fractional integral operators. Here, we


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demonstrate only two special cases of Theorems 2.1 and 2.2, which are asserted, respectively, in Corollaries 3.1 and 3.2. Using (1.5) and (2.5), we find 

 k I 1  x 

 x  k 1    k   k   

, , k, x    .

(3.1)

Corollary 3.1. Let f : 0,     be an integrable function on 0,   such that m  f t   M

t  0,  

(3.2)

for real constants m and M. Also, let x, k, , ,     . Then  



 x  k  x  k   2 M  m 1      I f  x  k   k k   k    k   k    k 2



 x  k   2 1    I f   x   2 k I  f   x   k I  f   x   k   k    k      x  k    x  k    1 1    k I f  x      k I f  x .  2 M  m     k   k     k   k      

(3.3) Proof. By setting p  q  2 in the result in Theorem 2.1 and using the linearity of the fractional integral operator with the aid of (3.1), we can obtain the result here. We omit the details.



Corollary 3.2. Let f : 0,     be an integrable function on 0,   such that m  f t   M

t  0,  

(3.4)

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for real constants m and M. Also, let x, k, , ,     . Then  



 x  k  x  k   M m 1      I f  x  k   k k   k    k   k    k 

 x  k   1    I f  x  k   k    k 1

1

 2 k I   M  f  2   x   k I   f  m  2   x .

(3.5)

Proof. By setting p  q  1 2 in the result in Theorem 2.2, similarly as in the proof of Corollary 3.1, we can get the result here. We omit the details.  References [1] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math. 10(3) (2009), Article 86, 5 pp. [2] P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mmes limites, Proc. Math. Soc. Charkov 2 (1882), 93-98. [3] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9(4) (2010), 493-497. [4] Z. Dahmani and L. Tabharit, On weighted Grüss type inequalities via fractional integration, J. Adv. Res. Pure Math. 2 (2010), 31-38. [5] R. Díaz and E. Pariguan, On hypergeometric functions and k -Pochhammer symbol, Divulg. Mat. 15(2) (2007), 179-192. [6] S. S. Dragomir, A generalization of Grüss inequality in inner product spaces and applications, J. Math. Anal. Appl. 237 (1999), 74-82. [7] S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math. 31(4) (2002), 397-415. [8] N. Elezović, Lj. Marangunić and J. Pečarić, Some improvements of Grüss type inequality, J. Math. Inequal. 1 (2007), 425-436.

Certain Inequalities Involving the k,   -fractional … [9] G.

1 ba

Grüss, b

Über

das

Maximum

1

b

des

absoluten

b

 a f t  g t  dt  b  a 2  a f t  dt  a g t  dt ,

Math.

1887

Betrages Z.

39

von (1935),

215-226. [10] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218(3) (2011), 860-865. [11] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1-15. [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. [13] V. S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. 301, Longman Scientific & Technical, Harlow, Co-published with John Wiley, New York, 1994. [14] C. G. Kokologiannaki, Properties and inequalities of generalized k -gamma, beta and zeta functions, Int. J. Contemp. Math. Sci. 5(13-16) (2010), 653-660. [15] C. G. Kokologiannaki and V. Krasniqi, Some properties of the k -gamma function, Matematiche (Catania) 68(1) (2013), 13-22. [16] V. Krasniqi, A limit for the k -gamma and k -beta function, Int. Math. Forum 5(33-36) (2010), 1613-1617. [17] X. Li, R. N. Mohapatra and R. S. Rodriguez, Grüss-type inequalities, J. Math. Anal. Appl. 267 (2002), 434-443. [18] M. Mansour, Determining the k -generalized gamma function k  x  , by functional equations, Int. J. Contemp. Math. Sci. 4(21) (2009), 1037-1042. [19] S. Mubeen and G. M. Habibullah, An integral representation of some k hypergeometric functions, Int. Math. Forum 7 (2012), 203-207. [20] S. Mubeen and G. M. Habibullah, k -fractional integrals and application, Int. J. Contemp. Math. Sci. 7 (2012), 89-94. [21] S. Mubeen and S. Iqbal, Grüss type integral inequalities for generalized RiemannLiouville k -fractional integrals, J. Inequal. Appl. 2016, Paper No. 109, 13 pp.

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[22] K. S. Nisar, S. R. Mondal and J. Choi, Certain inequalities involving the k -Struve function, J. Inequal. Appl. 2017, Paper No. 71, 8 pp. [23] K. S. Ntouyas, P. Agarwal and J. Tariboon, On Polya-Szego and Chebyshev types inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal. 10(2) (2016), 491-504. [24] B. G. Pachpatte, On multidimensional Grüss type inequalities, J. Inequal. Pure Appl. Math. 3(2) (2002), Art. ID 27, 6 pp. [25] I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, San Diego, CA, 1999. [26] M. Z. Sarikaya, Z. Dahmani, M. E. Kiris and F. Ahmad, k; s  -RiemannLiouville fractional integral and applications, Hacet. J. Math. Stat. 45(1) (2016), 77-89. [27] E. Set, J. Choi and A. Gözpinar, Hermite-Hadamard type inequalities for the generalized k -fractional integral operators, J. Inequal. Appl. 2017, Paper No. 206, 17 pp. [28] E. Set, M. Tomar and M. Z. Sarikaya, On generalized Grüss type inequalities for k -fractional integrals, Appl. Math. Comput. 269 (2015), 29-34. [29] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. [30] J. Tariboon, S. K. Ntouyas and W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci. 2014, Art. ID 869434, 6 pp. [31] J. Tariboona, S. K. Ntouyasb and M. Tomar, Some new integral inequalities for k -fractional integrals, Malaya J. Mat. 4(1) (2016), 100-110.

Gauhar Rahman: [email protected] Kottakkaran Sooppy Nisar: [email protected]; [email protected] Shahid Mubeen: [email protected] Junesang Choi: [email protected]