GRÜSS TYPE INEQUALITIES INVOLVING THE GENERALIZED GAUSS HYPERGEOMETRIC FUNCTIONS

In this paper, we establish certain generalized Grüss type inequality for generalized fractional integral inequalities involving the generalized Gauss hypergeometric function. Moreover, we also consider their relevances for other related known results. 1. Introduction and preliminares In 1935, D. Grüss, proved the following integral inequality which gives an estimation of a product in terms of the product of integrals ([]): (1.1) 1 (b a) b Ra f(x)g(x)dx 1 (b a) b R a f(x)dx 1 (b a) b R a g(x)dx 14 (L l) (M m) ; provided that f and g are two functions which are dened and integrable on [a; b] and satisfying the condition (1.2) l f(x) L; m g(x) M for all x 2 [a; b]. The costant 1 4 is the best possible. Denition 1. ([26; 27]) Let h (x) be an increasing and positive monotone function on [0;1), also derivative h (x) is continuous on [0;1) and h (0) = 0. The space X h (0;1) (1 p <1) of those real-valued Lebesque measurable functions f on [0;1) for which (1.3) jjf jjXp h = 1 R 0 jf(t)j h (x) dt 1 p <1; 1 p 1 and for the case p =1 (1.4) jjf jjX1 h = ess sup 1 t<1 h f(t)h 0 (x) i : In particular, when h (x) = x (1 p <1) the space X h (0;1) coincides with the Lp[0;1) space and also if we take h (x) = x k + 1 (1 p <1; k 0) the space X h (0;1) coincides with the Lp;k[0;1) space. Key words and phrases. Integral inequalities; Grüss inequality; fractional integrals and. 2010 Mathematics Subject Classication.26D10; 26A33. 1 2 ABDULLAH AKKURT, SEDA KILINÇ, AND HÜSEYIN YILDIRIM Denition 2. ([28]) Let (a; b) be a nite interval of the real line R and > 0. Also let h (x) be an increasing and positive monotone function on (a; b], having a continuous derivative h 0 (x) on (a; b). The leftand right-sided fractional integrals of a function f with respect to another function h on [a; b] for which (1.5) J a+;hf (x) := 1 ( ) x R a [h (x) h (t)] 1 h (t) f (t) dt; x a; < ( ) > 0 and (1.6) J b ;hf (x) := 1 ( ) b R x [h (t) h (x)] 1 h (t) f (t) dt; x b; < ( ) > 0: Denition 3. Let f 2 X h. For > 0; > 1; ; 2 R and h (x) be an increasing and positive monotone function on (0; x], having a continuous derivative h 0 (x) on (0; x): Then the generalized fractional integral I ; ; ; h(t) of order for real-valued continuous function f(t), is dened by (1.7) I ; ; ; h(t) ff (x)g = h (x) 2


Introduction and Preliminares
In 1935, D. Grüss, proved the following integral inequality which gives an estimation of a product in terms of the product of integrals [6]: The costant 1 4 is the best possible.
Definition 1. ( [26,27]) Let h (x) be an increasing and positive monotone function on [0, ∞), also derivative h ′ (x) is continuous on [0, ∞) and h (0) = 0.The space X p h (0, ∞) (1 ≤ p < ∞) of those real-valued Lebesque measurable functions f on [0, ∞) for which and for the case p In particular, when h (x) = x (1 ≤ p < ∞) the space X p h (0, ∞) coincides with the L p [0, ∞)−space and also if we take h (x) = x k+1 k + 1 (1 ≤ p < ∞, k ≥ 0) the space X p h (0, ∞) coincides with the L p,k [0, ∞)−space.Definition 2. ( [28]) Let (a, b) be a finite interval of the real line R and α > 0. Also let h (x) be an increasing and positive monotone function on (a, b], having a continuous derivative h ′ (x) on (a, b).The left-and right-sided fractional integrals of a function f with respect to another function h on [a, b] for which and and h (x) be an increasing and positive monotone function on (0, x], having a continuous derivative h ′ (x) on (0, x).Then the generalized fractional integral I α,β,η,δ h(t) of order α for real-valued continuous function f (t), is defined by where the function 2 F 1 (.) appearing as a kernel for the operator (1.7) is the Gaussian hypergeometric function defined by where, (1.9) Where N denotes the set of positive integers.The above integral (1.7) has the following commutative property: f (x) . (1.10) In the sequel, we use the following well-known result to establish our main results in the present paper: where and Ξ and /Z − 0 denotes the sets of complex numbers and nonpositive integers, respectively.
We define a fractional integral operator K α,β,η,δ h(t) associated with the Gauss hypergeometric function as follows.
f as follows: ) where I α,β,η,δ h(t) is the Gauss hypergeometric fractional integral of order α and is defined in the following.Definition 5. Two functions f and g are said to be synchronous functions on be an increasing and positive monotone function on (0, x], having a continuous derivative h ′ (x) on (0, x), h(0) = 0, we have where C is constant.
To prove (1.15), we again use the result (1.7), and (1.12) reduces to Using (1.8), (1.18) gets the following form: This completes the proof of the Lemma 6.
Lemma 7. Let g ∈ X 1 h and m, M ∈ R with m ≤ g(x) ≤ M .Then we have using u ∈ (0, x); x > 0, and integrating with respect to u from 0 to x, and then applying Definition 4 and Lemma 6, we obtain v ∈ (0, x) ; x > 0, then integrating with respect to v from 0 to x, we obtain the required result (1.20).This completes the proof of Lemma 7.
Proof.Let us define a function and then integrating twice with respect to u and v from 0 to x, we obtain the following result with the aid of (1.7), (1.12), and property (1.8): g(x). (2.3) Making use of the well-known Cauchy-Schwarz inequality for a linear operator, we find that 2 ).
(2.4) Since (2.5) Thus by using Lemma 7, we have (2.6) and (2.7) Using the inequalities (2.6) and (2.7), (2.4) reduces to the following form: Applying the well-known inequality 4ab ≤ (a + b) 2 ; and using a, b ∈ R in the right-hand side of the inequality (2.8), and simplifying it, we obtain the required result (2.1).This completes the proof of Theorem 8.
Proof.For the synchronous function f and g, the inequality (1.13) holds for all u, v ∈ [0, ∞).This implies that (2.10) Following the procedure of the Lemma 7 for applying the fractional integral K α,β,η,δ h(t) , after a little simplification, we arrive at the required result (2.9).This completes the proof of Theorem 9.

Concluding Remarks
We consider some consequences of the results derived in the previous section.
f and g are two functions which are defined and integrable on [a, b] and satisfying the condition l ≤ f (x) ≤ L, m ≤ g(x) ≤ M (1.2) for all x ∈ [a, b].