k -Fractional Integrals and Application
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1 Int. J. Contem. Math. Sciences, Vol. 7,, no., Fractional Integrals and Alication S. Mubeen National College of Business Administration and Economics Gulberg-III, Lahore, Paistan G. M. Habibullah National College of Business Administration and Economics Gulberg-III, Lahore, Paistan Abstract The urose of this aer is to introduce a variant of fractional integrals to be called -fractional integral which is based on -gamma function. When, it turns out to be the usual Riemann-Liouville fractional integral. Use of -fractional integrals is also illustrated. Mathematics Subect Classification: 33B5, 33D5, 6A33 Keywords: -Gamma Function, -Beta Function, -Fractional Integration Recently, in a series of research ublications, Diaz et al. [,, 3] have introduced -gamma and -beta functions and roved a number of their roerties. They have also studied -zeta function and -hyergeometric functions based on Pochhammer -symbols for factorial functions. It has been followed by wors of Mansour [7], Koologiannai [5], Kransniqi [6] and Merovci [8] elaborating and strengthening study of -gamma and -beta functions. The -gamma functions is defined by
2 9 S. Mubeen and G. M. Habibullah n n ( n) n, n! = lim, () n = +, > n, is the Pochhammer -symbols for factorial function. It has been shown that the Mellin transform of the eonential function the -gamma function, elicitly given by t t e = t e dt. () Clearly, = lim, = and ( ) + =. This gives rise to -beta function defined by y B ( y, ) = t ( t) dt (3) y y so that B (, y) = B, B, y = + y. (4) and Most of the functions involving or based on gamma function can be refined by using -gamma function. For eamle, -Zeta and -Mittag-Leffler functions could be defined resectively by the formulae ζ (, s) =,, >, s> (5) s +, and E =,, >. (6) ( + ) Following Diaz [], we define -hyergeometric function F by the series ( ), F ((, ) ;( γ, ) ; ) =. ( γ )!, Its integral reresentations can be determined as follows F, ; γ, ; ( ) γ ( γ ) ( γ ) ( γ ) = B ( +, γ )! γ + = t ( t) dt! is
3 - fractional integrals and alication 9 ( γ ) γ γ t t t e dt =. (7) The -gamma also leads to another interesting direction, -fractional integral defined by I ( f ) = ( t) f () t dt. ( ) (8) Note that when, it then reduces to the classical Riemann-Liouville fractional integral [4,.69] I ( f ) = ( t) f ( t) dt. ( ) (9) {Thining simle, I f eists in C if f C, where C be the class of all functions which are continuous and integrable on the interval (, )}. Define formally I ( f ), <, to be the solution, if eists, of the equation f = I g. Clearly, I f = I g imlies f = g. Also, a use of (4) leads to + = =. () Moreover, and ( ) ( ) I I f I f I I f ( ) ( ) ( ) I = I u = u + We can etend the -fractional integral by λ, λ λ ( ) = ( ) () ( ) (). () I f t t f t dt. (3) A change of order of integration and use of (4) roves the relation ( ) + λ,, λ +, λ, I f : L L I I f I f Furthermore, Schur s inequality [9,.96]. =. (4), > is a simle consequences of, γ, As an alication, we now show that the integral oerator H ( f )
4 9 S. Mubeen and G. M. Habibullah (, ;, ; ) () + H, γ f = t F γ t f t dt, (5) is bounded in L. To do this, we first rove the following lemma. Lemma: Let t T ( f ) = t e f () t dt, ( > ), (6) * t and T ( g) = t e g() t dt, ( > ) (i) If >, >, then T ( f) * (ii) () () = () (). (7) and T * ( g) are bounded in L ; f t T g t dt g t T f t dt; (8) * (iii) () ( γ ) ((, );(, ); ) γ T t t F t γ ϕ = γ where ϕ () γ t = t t, < t < < ; =, t. (9) Proof: Write V( f) f in L. Now note that Since =. Then V f = f and V( f) is an isometry t ( ) = ( )(), ( > ) T V f t e V f t dt + u y = y e f y dy. u e du is finite when > and >, ( ):, so is T ( f ) T V ( f ) T V f L L To rove (iii), let >. Then =.,
5 - fractional integrals and alication 93 * ( ϕ )() = ( ) γ ty T t y y e dy ( γ ) ( γ ) γ = t F, ;, ; t. (( ) ( γ ) ) Theorem: If,, γ > > >, then H ( f ) Proof: Emloy -fractional integral to get γ + γ γ, { } = ( γ ) γ is bounded in L., I T f t t T f t dt. It, thus, imlies that, + γ * = () ( ) ( γ ) + γ ( γ ) γ f tt t t dt * = f () tt ( ϕ() t) dt. (, ;, ; ) () + H γ f = t F γ t f t dt γ + ( γ ) * f () tt ( ϕ)() ( γ ) ( γ ) γ I, { T ( f )}( ) ( ) = =. Hence, if >, >, γ >, then ( γ ) γ, γ, ( ) tdt H f = I T ( f) C T ( f) C f, where C is a constant deending uon,, and γ. Consequently, it follows that H, γ ( f ) is bounded in L.
6 94 S. Mubeen and G. M. Habibullah References [] R. Diaz and C. Teruel, q-generalized, gamma and beta functions, J. Nonlinear Math. Phys., (5), [] R. Diaz and E. Pariguan, On hyergeometric functions and Pochhammer -symbol, Divulg. Mat., 5 (7), [3] R. Diaz, C. Ortiz and E. Pariguan, On the -gamma q -distribution, Cent. Eur. J. Math., 8 (), [4] A.A. Kilbas, H.M. Srivastava, and J.J. Truillo, Theory and Alications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 4, Elsevier, Amsterdam, 6. [5] C. G. Koologiannai, Proerties and inequalities of generalized - gamma, beta and zeta functions, Int. J. Contem. Math. Sciences, 5 (), [6] V. Krasniqi, A limit for the -gamma and -beta function, Int. Math. Forum, 5 (), [7] M. Mansour, Determining the -generalized gamma function by functional equations, Int. J. Contem. Math. Sciences, 4 (9), [8] F. Merovci, Power roduct inequalities for the function, Int. J. of Math. Analysis, 4 (), 7-. [9] G. O. Oiiolu, Asects of the Theory of Bounded Linear Oerators, Academic Press INC., London, 97. Received: June,
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