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Midpoint and trapezoid type inequalities for multiplicatively convex functions

In this paper, we first prove two new identities for multiplicative differentiable functions. Based on this identity, we establish a midpoint and trapezoid type inequalities for multiplicatively convex functions. Applications to special means are also given.

The fundamental inequality for convex functions is undoubtedly the Hermite-Hadamard inequality, which can be stated as follows: For every convex function f on the interval [a, b] with a < b, we have (1.1) If the function f is concave, then (1.1) holds in the reverse direction see [17]. In [6], Dragomir and Agarwal established some trapezoid type inequalities for functions whose absolute value of the first derivatives are convex In [16], Pearce and Pečarić proved some midpoint type inequalities for functions whose absolute value of the first derivatives are convex Concerning some papers dealing with some quadrature see [9,10,11] and references therein. In 1967, Grossman and Katz, created the first non-Newtonian calculation system, called geometric calculation. Over the next few years they had created an infinite family of non-Newtonian calculi, thus modifying the classical calculus introduced by Newton and Leibniz in the 17th century each of which differed markedly from the classical calculus of Newton and Leibniz known today as the non-Newtonian calculus or the multiplicative calculus, where the ordinary product and ratio are used respectively as sum and exponential difference over the domain of positive real numbers see [8]. This calculation is useful for dealing with exponentially varying functions.
The complete mathematical description of multiplicative calculus was given by Bashirov et al. [4]. Also in the literature, there remains a trace of a similar calculation proposed by the mathematical biologists Volterra and Hostinsky [18] in 1938 called the Volterra calculation which is identified as a particular case of multiplicative calculation.
Recently, Ali et al. [1], gave the analogue of the Hermite-Hadamard inequality for multiplicatively convex functions Theorem 1.3. Let f be a positive and multiplicatively convex function on interval [a, b], then the following inequalities hold (1.4) Also, they proved the inequalities for the product and the quotient of two multiplicatively convex functions. In [2], Ali et al. studied the Hermite-Hadamard type inequalities for multiplicatively φ-convex and log-φ-convex functions. In [12] and [14],Özcan generalized the results obtained in [1] for s-convex and h-convex functions respectively. In [13],Özcan gave the analogue of Hermite-Hadamard type inequalities for multiplicatively peinvex functions. In [15],Özcan proposed the Hermite-Hadamard type inequalities for multiplicatively h-peinvex functions. In [3] Ali et al. investigate the Ostrowski and the Simpson type inequalities for multiplicatively convex functions.
Motivated by all the above papers, in this study we propose two new identities for multiplicatively differentiable functions, based on these identities we establish a midpoint and trapezoid type inequalities for multiplicatively convex functions. Applications to special means are also given.

Preliminaries
In this section we begin by recalling some definitions, properties and notions of derivation as well as multiplicative integration Definition 2.1 ([4]). Let f : R → R + be a positive function. The multiplicative derivative of the function f noted by f * is defined as follows

Remark 2.2.
If f has positive values and is differentiable at t, then f * exists and the relation between f * and ordinary derivative f ′ is as follows: The multiplicative derivative admits the following properties: ). Let f and g be multiplicatively differentiable functions, and c is arbitrary constant.
Then functions c f , f g, f + g, f /g and f g are * differentiable and . .
In [4], Bashirov et al. introduced the concept of the * integral called multiplicative integral Which the sum of the terms of the product is used in the definition of a classical Riemann integral of f over [a, b], while the product of the terms raised to the power is used in the definition of the multiplicative integral of f over [a, b].
The relationship between the Riemann integral and the multiplicative integral is as follows: Moreover, Bashirov et al. show that multiplicative integral has the following results and properties: Theorem 2.5 ([4]). Let f be a positive and Riemann integrable on [a, b], then f is multiplicative integrable on [a, b] and Theorem 2.6 (Multiplicative Integration by Parts [4]). Let f : [a, b] → R be multiplicative differentiable, let g : [a, b] → R be differentiable so the function f g is multiplicative integrable, and b a f

Main results
In order to prove our results, we need the following lemmas , then we have the following identity for multiplicative integrals

Proof.
Let and Using the integration by parts for multiplicative integrals, from I 1 we have Similarly, we have Multiplying above equalities we get the desired result. The proof is completed. on [a, b], then we have the following identity for multiplicative integrals where G is the geometric mean i.e. G (α, β) = αβ for α, β > 0.
Proof. Using integration by parts for multiplicative integrals, on the right side of (3.1) we have The proof is completed.
Proof. From Lemma (3.1), properties of multiplicative integral and the multiplicative convexity Then, we obtain The proof is completed.
Theorem 3.6. Let f : [a, b] → R + be a multiplicative differentiable mapping on [a, b] with a < b. If f * is multiplicative convex on [a, b], then we have where G is the geometric mean.
Proof. From Lemma (3.2), properties of multiplicative integral and the multiplicative convexity The proof is completed.

Applications to special means
We shall consider the means for arbitrary real numbers a, b The Arithmetic mean: A (a, b) = a+b 2 . The harmonic mean: H (a, b) = 2ab a+b , a, b > 0. The logarithmic means: L (a, b) = b−a ln b−ln a , a, b > 0 and a = b. The p-Logarithmic mean: L p (a, b) = b p+1 −a p+1   Proof. The assertion follows from Corollary (3.7) applied to the function f (t) = e