ON HERMITE-HADAMARD TYPE INEQUALITIES FOR INTERVAL-VALUED MULTIPLICATIVE INTEGRALS

. In this work, we de(cid:133)ne multiplicative integrals for interval-valued functions. We establish some new Hermite-Hadamard type inequalities in the setting of interval-valued multiplicative calculus and give some examples to illustrate our main results. We also discuss special cases of our main results which are the extension of already established results.


Introduction
The Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard, (see [14], [32, pp. 137]) is one of the most well established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that, if F : I ! R is a convex function on the interval I of real numbers and a; b 2 I with a < b, then (1.1) Both inequalities in (1.1) hold in the reversed direction if F is concave. We note that Hermite-Hadamard inequality may be regarded as a re…nement of the concept of convexity and it follows easily from Jensen's inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable variety of re…nements and generalizations have been studied, see [1, 2, 7, 8, 11, 15-18, 26, 31, 36-40] and reference therein.
The main purpose of this paper is to de…ne *integral/multiplicative integral for interval-valued functions and to obtain Hermite-Hadamard inequality via these integrals.
The overall structure of the study takes the form of six sections including introduction. The remainder of this work is organized as follows: we …rst recall the interval calculus by giving the several de…nitions and properties in Section 2. In section 3, we de…ne multiplicative integral for interval-valued functions and give some basic properties of this newly de…ne integral. In Section 4, we de…ne logarithmically interval-valued h-convex functions and discuss special cases and properties of this class of functions. In section 5, we obtain Hermite-Hadamard inequalities and related inequalities for our new class of convex functions by utilizing our newly de…ne integral. At the end, in section 6, we give concluding remarks about our work.

Interval Calculus
A real valued interval X is bounded, closed subset of R de…ned by X = X; X = t 2 R : X t X where X, X 2 R and X X: The numbers X and X are called the left and the right endpoints of interval X; respectively. When X = X = a, the interval X is said to be degenerate and we use the form X = a = [a; a]. Also, we call X positive if X > 0 or negative if X < 0: The set of all closed intervals of R, the sets of all closed positive intervals of R and closed negative intervals of R is denoted by R I , R + I and R I respectively. The Hausdor¤-Pompeiu distance between the intervals X and Y is de…ned by It is known that (R I ; d) is a complete metric space [4]. Now, we give the de…nitions of basic interval arithmetic operations for the intervals X and Y as follows: Scalar multiplication of the interval X is de…ned by The opposite of the interval X is The subtraction is given by Use of monotonic functions For example, F (x) = e x ; x 2 R and F (x) = ln x; x > 0 then we have exp (X) = exp (X) ; exp X ln (X) = ln (X) ; ln X : In general, X is not additive inverse for X i.e X X 6 = 0: ON HERM ITE-HADAM ARD TYPE INEQUALITIES   1431 The de…nitions of operations lead to a number of algebraic properties which allows R I to be quasilinear space (see, [24]). They can be listed as follows (see, [4,[22][23][24]27]): (1) (Associativity of addition) (X + Y ) + Z = X + (Y + Z) for all X; Y; Z 2 R I ; (2) (Additive element) X + 0 = 0 + X = X for all X 2 R I ; What's more, one of the set property is the inclusion " " that is given by Considering together with arithmetic operations and inclusion, one has the following property which is called inclusion isotony of interval operations: Let be the addition, multiplication, subtraction or division. If X; Y; Z and T are intervals such that X Y and Z T; then the following relation is valid X Z Y T:

*Integral of Interval-Valued Functions
In this section, we de…ne *integral or multiplicative integral for the interval-valued functions and give properties of this new integral. Throughout in this section, we shall use F (t) = F (t); F (t) is positive interval-valued function, IR is the notation for the interval-valued integrals and I means the multiplicative integral. First, we recall that the concept of *integral is denoted by R b a (F (x)) dx which introduced by Bashirov et al. in [5]. In multiplicative integrals we replace the sum by product and the product by raising to power of a function F on [a; b] : We give the following relation between Riemann integral and *integral: a ln(F (x))dx : For further details of *integral reader can read [5].
Now we recall the concept of Interval-valued integral given by R. E. Moore in [28].
The interval-valued Riemann integral of function F is de…ned by Let's de…ne interval-valued *integral or multiplicative integral (I R): A function F is said to be an interval-valued function of t on [a; b] if it assigns a nonempty interval to each t 2 [a; b] A partition of [a; b] is any …nite ordered subset P having the form P : a = t 0 < t 1 < ::: < t n = b: The mesh of a partition P is de…ned by where F : [a; b] ! R I is a positive function: We call P (F; P 1 ; ) a Riemann product of F corresponding to P 1 2 P ( ; [a; b]) : for every Riemann product P of F corresponding to each and is denoted by The collection of all functions that are I R integrable on [a; b] will be denote by The following theorem gives relation between I R-integral and multiplicative integral (I -integral): and It is seen easily that if F (t) It is very easy to notice that if positive function F is interval-valued integrable (IR integrable), then F is I R integrable and As we know that ln F is (IR) integrable on [a; b] and continuity of the exponential we have imply the above statement and conversely, we have Now we give some properties of *integral for interval valued functions. We consider F and G are positive interval-valued functions then the following equalities hold: Proof. Now we give the proofs of above properties.

Logarithmically Interval-valued Convex Functions
In show the set of all interval-valued h-convex functions. The usual notion of convex interval-valued function corresponds to relation (4.1) with h(t) = t [35]. Also, if h(t) = t s in (4.1); then De…nition 2 gives the other interval-valued convex function de…ned by Breckner [6].
for all     ; then following double inequality holds: By setting x = ta + (1 t) b and y = tb + (1 t) a, we get Integrating inequality (5.2) with respect to t over [0; 1], we have and by changing the variable of integration, we have which is the …rst inequality in (5.1). Now we have to prove second inequality in (5.1), for this …rst we note that which is the second inequality in (5.1). The proof of the theorem is completed.
where G (F (a) ; F (b)) referes to the geometric mean of F (a) and F (b) : Remark 10. If F = F in Corollary 3, then Corollary 3 reduces to [30,Corollary 3.4]. ; then following double inequality holds: Proof. Since F; G 2 SXL h; [a; b]; R + I , so F and G can be written as By setting x = ta + (1 t) b and y = tb + (1 t) a in (5.6) and (5.7), we have From (5.8) and (5.9), we have and integrating (5.10) with respect to t over [0; 1], we have By changing the variable in last inequality, we have which is the …rst inequality in (5.5).
To prove the second inequality in (5.5), …rst we note that F; G 2 SXL h; [a; b]; R + I , so we have h(1 t) (5.12) and From (5.12) and (5.13), we have and integrating inequality (5.14) with respect to t over [0; 1], we get By changing the variable of integration, we have which is the second inequality in (5.5).
The proof of theorem is completed.
Corollary 6. If we use h (t) = t in Theorem 3, then following new inequalities for logarithmically interval-valued convex functions hold: where G (:; :) referes to the geometric mean.

Corollary 7.
Under the assumptions of Theorem 3, if we let h (t) = t s , then following inequalities for the s-logarithmically interval-valued convex functions hold: Corollary 9. If we put h (t) = t 1 in …rst inequality of Theorem 3, then we have following inequalities for the logarithmically interval-valued Q-convex functions Corollary 10. Under the assumptions of Theorem 3, if we set h (t) = t s in …rst inequality, then following inequalities for s-logarithmically interval-valued Q-convex functions hold Integrating inequality (5.15) with respect to t over [0; 1] ; we have and by changing the variable of integration, we have Now from Theorem 2, one has which completes the proof.
Corollary 11. Under the assumptions of Theorem (4), if we set h (t) = t in Theorem 4, then following inequalities for the logarithmically interval-valued convex functions hold where G (:; :) referes to the geometric mean.

Conclusion
In this paper, authors de…ne multiplicative integral for the interval-valued functions and derived some new Hermite-Hadamard and related inequalities for logarithmically interval-valued h-convex functions by utilizing our newly de…ne integral. Authors also gave some more new results in the special cases of our main results. Interested readers can obtain more results by using the notions used in this paper. The results in this paper can be a new contribution in the …eld of Hermite-Hadamard integral inequalities.